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diff --git a/0tFIT4oBgHgl3EQf3Cv8/content/tmp_files/2301.11380v1.pdf.txt b/0tFIT4oBgHgl3EQf3Cv8/content/tmp_files/2301.11380v1.pdf.txt
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+Electron Impact Ionization in the Icy Galilean Satellites’ Atmospheres
+Shane R. Carberry Mogan1,∗, Robert E. Johnson2,3, Audrey Vorburger4, Lorenz Roth5
+1Space Sciences Laboratory, University of California, Berkeley, Berkeley, USA; 2University
+of Virginia, Charlottesville, USA; 3New York University, New York, USA; 4University of
+Bern, Bern, Switzerland; 5KTH Royal Institute of Technology, Stockholm, Sweden;
+∗Corresponding author: Shane R. Carberry Mogan (CarberryMogan@Berkeley.edu)
+Abstract
+Electron impact ionization is critical in producing the ionospheres on many plan-
+etary bodies and, as discussed here, is critical for interpreting spacecraft and tele-
+scopic observations of the tenuous atmospheres of the icy Galilean satellites of Jupiter
+(Europa, Ganymede, and Callisto), which form an interesting planetary system. For-
+tunately, laboratory measurements, extrapolated by theoretical models, were devel-
+oped and published over a number of years by K. Becker and colleagues (see Deutsch
+et al. 2009) to provide accurate electron impact ionization cross sections for atoms
+and molecules, which are crucial to correctly interpret these measurements. Because
+of their relevance for the Jovian icy satellites we provide useful fits to the complex,
+semi-empirical Deutsch–M¨ark formula for energy-dependent electron impact ioniza-
+tion cross-sections of gas-phase water products (i.e., H2O, H2, O2, H, O). These are
+then used with measurements of the thermal plasma in the Jovian magnetosphere to
+produce ionization rates for comparison with solar photo-ionization rates at the icy
+Galilean satellites.
+1
+Introduction
+Since Galileo Galilei’s revolutionary discovery that Jupiter, the largest planet in the solar
+system, has four large planetary bodies revolving around it – the “Galilean” satellites: Io,
+Europa, Ganymede, and Callisto – our fascination with this planetary system has only
+grown with the advancement of observational technologies. Several spacecraft have been
+sent directly to or have at least passed by the Jovian system. In the 1970s, Pioneer 10 & 11
+and Voyager 1 & 2 utilized Jupiter’s gravity to enhance their trajectories and observed the
+giant planet and its moons up-close. From 1995–2003 Galileo orbited Jupiter, and made
+several close encounters with the namesake moons. At the time of this writing, the Juno
+spacecraft is currently orbiting Jupiter and has recently made close flybys of Ganymede and
+Europa with 2 flybys of Io forthcoming. In addition to these in-situ observations, the Hubble
+Space Telescope (HST), which has been situated in Earth’s orbit since 1990, has been used
+to observe and contribute new information to our understanding of this system. To better
+understand the Jovian system, the Galilean satellites, and their interconnected dynamics, as
+well as address certain prevailing mysteries, three forthcoming missions, ESA’s JUpiter ICy
+moons Explorer, NASA’s Europa Clipper, and CNSA’s Gan De, will send spacecraft back
+to the Jovian system.
+Our focus here is on the icy Galilean satellites (see Table 1) – Europa, Ganymede, and
+Callisto – and their tenuous atmospheres for which the dominant constituents are water
+1
+arXiv:2301.11380v1  [astro-ph.EP]  26 Jan 2023
+
+products: H2O, O2, H2, H, and O. Because these objects orbit Jupiter within its giant mag-
+netic field, they are exposed to an ambient plasma. Interactions between this plasma and the
+icy Galilean satellites’ atmospheres to a large extent determine the nature of the latter. One
+key aspect of this interaction is the role of the plasma electrons in dissociating, ionizing, and
+exciting the gas-phase water products via impacts, which can produce observable emission
+features (e.g., Hall et al. 1998, Feldman et al. 2000, Roth et al. 2017, Roth 2021, Roth et al.
+2021). In order to calculate the ionization rates to be used in future simulations of these at-
+mospheres, we use the extensive laboratory data of electron impact ionization cross-sections
+accumulated and summarized by K. Becker and his colleagues over a number of years as
+reviewed in Deutsch et al. (2009) and references herein. Since the various laboratory mea-
+surements can exhibit differences and typically cover only a limited range of energies, the
+group developed a fitting procedure also presented in those papers. In this method, the
+Deutsch–M¨ark (DM) formula, atomic orbital basis sets are used in expressions to fit and
+extrapolate the measurements, as well as to generate cross-sections when measurements are
+unavailable. In this way, they created a large number of energy-dependent electron impact
+ionization cross-section distributions. Although the DM formula provides a broad range of
+useful data, as well as allows the calculation of results for molecular targets to be determined
+from the constituent atomic orbitals, it is not easily implemented. For example, in detailed
+molecular kinetic simulations much simpler calculations are more often made, such as those
+recently implemented in Carberry Mogan et al. (2022) for describing electron impacts in Cal-
+listo’s atmosphere. More readily useful expressions are needed. Therefore, here we present
+much more simplified fits to the results obtained via the DM formula, which we then use with
+electron density and temperature data at the icy Galilean satellites to calculate ionization
+rates in their atmospheres. These rates are in turn needed to help interpret past, present,
+and future spacecraft and telescopic observations of these topical planetary bodies soon to
+be visited by several new spacecraft.
+Before we discuss the derivation of the electron impact ionization cross-section fits (Sec-
+tion 3) and present the corresponding ionization rates for each species considered (Section
+4), we first review the local space environment of the icy Galilean satellites, particularly the
+Jovian magnetospheric plasma in which they are embedded, as well as the observations of
+water products in their atmospheres.
+2
+Background
+The strong dynamo generated within the interior of Jupiter produces its magnetic field, which
+has a rotation period of ∼10 h. Jupiter’s magnetosphere, i.e., the environment controlled by
+the planet’s magnetic field, is filled with charged particles. While the volcanic moon Io is the
+main source, materials from the icy Galilean satellites’ surfaces are also a source of charged
+particles (either directly ionized or lost as neutrals and ionized later in the magnetosphere)
+(Johnson et al., 2004, Kivelson et al., 2004), which are picked-up, accelerated, and then
+become “trapped” by this rapidly rotating field.
+The combination of this fast rotating
+magnetic field and the large number of charged particles trapped within it gives rise to the
+plasma detected in Jupiter’s immense magnetosphere, which can extend as far as 45–100 RJ
+from the planet, where RJ = 71, 492 km is the radius of Jupiter. The Jovian magnetosphere
+2
+
+is typically divided up into three regions: the inner (<10 RJ), the middle (10–40 RJ), and the
+outer (>40 RJ) regions (Khurana et al., 2004); Europa resides within the inner region, while
+Ganymede and Callisto reside in the middle region. The Jovian magnetospheric plasma is
+typically described as being comprised of two populations: a “cold” thermal plasma with
+energies < 1 keV and a “hot” energetic plasma with energies ≥ 1 keV (Krupp et al., 2004,
+Bagenal and Delamere, 2011). Both populations are composed of electrons, as well as H+,
+On+, and Sn+ ions (Bagenal and Sullivan, 1981, Broadfoot et al., 1981, Nerney et al., 2017).
+The sulfur and oxygen ions, both of which are of various charge states, primarily originate
+from the volcanic moon Io (Bagenal and Dols, 2020). On the other hand, the hydrogen
+ions originate from two different sources depending on the magnetospheric region: in the
+inner region, they mainly originate in Jupiter’s atmosphere; and in the middle and outer
+magnetosphere, the solar wind can gain access to the magnetosphere, becoming the main
+provider of hydrogen ions thereafter.
+An additional plasma contribution (electrons, H+,
+On+) is associated with sputtering of the icy satellite’s surfaces (Cooper et al., 2001, Johnson
+et al., 2004, Szalay et al., 2022). Here we focus on the thermal electrons as these particles
+are the most relevant to ionization. Physical parameters for thermal electrons at the orbits
+of Europa, Ganymede, and Callisto based on Voyager and Galileo data are listed in Table
+1.
+Table 1: Physical Parameters of Europa, Ganymede, and Callisto, as well as of the Jovian
+magnetosphere at their orbits
+Parameters [units]
+Europa
+Ganymede
+Callisto
+Radius [km]
+1,561
+2,631
+2,410
+Mass [(×1023) kg]
+0.48
+1.5
+1.1
+Distance from Jupiter [RJ]
+9.38
+15.0
+26.3
+Orbital Velocity [km s−1]
+13.7
+10.9
+8.20
+Thermal Electron Density [cm−3]
+18–290 a,b
+1–10 a,c
+0.01–1.1 a,c
+Thermal Electron Temperature [eV]
+10–30 b
+100 c
+100 c
+Thermal Electron Flux d [cm−2 s−1]
+(0.4–1.1)×1011 e
+(0.7–6.7)×109
+(0.08–7.4)×108
+Plasma Azimuthal Velocity [km s−1]
+90 a
+150 a
+200 a
+Relative Plasma Velocity f [km s−1]
+76 a
+139 a
+192 a
+a Values taken from Kivelson et al. (2004) and references therein.
+b Values taken from Bagenal et al. (2015).
+c Values taken from Neubauer (1998) and references therein.
+d Thermal electron flux, φe = neve, where ne is the thermal electron density, ve =
+�
+8kBTe/π/me is the
+mean Maxwellian speed of the electrons, with kBTe the thermal electron temperature, kB the Boltzmann
+constant, and me the mass of an electron.
+e φe calculated according to the “low/hot” (kBTe = 30 eV, ne = 63 cm−3) and “high/cold”
+(kBTe = 10 eV, ne = 290 cm−3) plasma components from Table 5 in Bagenal et al. (2015).
+g Relative speed between plasma azimuthal velocity and the satellites’ orbital speeds.
+3
+
+As a result of the large relative velocities between the azimuthal velocity of the Jovian
+magnetosphere and the icy Galilean satellites’ orbital velocities (Table 1), the satellites
+are continuously overtaken and bombarded by the magnetospheric plasma.
+The spatial
+distribution of this bombardment is determined by the flow rate of the plasma particles
+past the satellite as well as their thermal velocities relative to the local magnetic field lines
+(Johnson et al., 2004). However, intrinsic or induced electric and magnetic fields as well
+as the interactions with the tenuous atmospheres and ionospheres at these satellites can
+radically modify the local fluxes of impinging particles at Ganymede (e.g., from Paranicas
+et al. 2022) and at Callisto (e.g., from Strobel et al. 2002). The cyclotron radii or gyro-
+radii of these charged particles depend on their mass, speed, and charge, as well as the
+local magnetic field strength. Due to their small mass, electrons primarily have gyro-radii
+much smaller than the satellite radius, and thus preferentially impact the satellites’ trailing
+hemispheres and poles as they move up and down the rotating field lines (Johnson et al.,
+2004).
+Following the Pioneer discovery of intense plasma trapped in the Jovian magnetic field
+(Smith et al., 1974, Wolfe et al., 1974, Trainor et al., 1974, Frank et al., 1976) a series of ex-
+periments were carried out to measure the effect this could have on the icy surfaces of Europa,
+Ganymede, and Callisto (Brown et al., 1978, Lanzerotti et al., 1978). These experiments
+showed that the ejection of water molecules from low-temperature ices by incident energetic
+particles, a process referred to as “sputtering”, is dominated by electronic excitations and
+ionizations produced in the ice (“electronic” sputtering), rather than by “knock-on” colli-
+sions of the ions with water molecules (“nuclear” sputtering), the hitherto typically studied
+sputtering process. Subsequent experiments led to the discovery that additional molecular
+species can form in and be released from the ice, namely H2 and O2, in a process referred to
+as “radiolysis” (Brown et al., 1982, Boring et al., 1983, Reimann et al., 1984, Brown et al.,
+1984), which occurs as bonds in H2O molecules are broken by the electronic energy deposited
+by the impinging charged particles and the fragmented molecules recombine. Moreover, the
+number of radiolytic products ejected from the icy surface per each incident charged particle
+(i.e., the “sputter yield”) was shown to display a strong temperature dependence.
+These discoveries had immense implications for the icy Galilean satellites: magneto-
+spheric plasma-induced sputtering could erode their surfaces, and the ejected atoms and
+molecules could migrate significant distances as well as escape the local gravitational en-
+vironment of its host satellite or form gravitationally bound atmospheres (Johnson, 1990).
+Indeed Europa was predicted to have a tenuous, predominantly O2 atmosphere due to the
+radiolytic decomposition of its icy surface by the incident Jovian plasma particles (Johnson
+et al., 1982, 1983, Johnson, 1990), which has been borne out by extensive HST observations
+(e.g., Roth et al. 2016, and references therein). Tenuous O2 atmospheres produced via similar
+processes have also been detected at Ganymede and Callisto: following the HST detection of
+O emissions indicative of an O2 atmosphere at Europa (Hall et al., 1995), airglow emissions
+were detected by HST in Ganymede’s O2 atmosphere (Hall et al., 1998) as well as in Europa’s
+atmosphere thereby confirming the earlier observation; O emissions were detected by HST
+in Callisto’s atmosphere (Cunningham et al., 2015), which were suggested to be induced by
+photoelectron impacts in a near-surface, O2-dominated atmosphere; and recently atomic O
+emissions have been detected at Ganymede (Roth et al., 2021) indicative of being produced
+via dissociative excitations of O2 (and H2O).
+4
+
+Although H2 can more readily escape from the atmospheres of these bodies than can the
+concomitant radiolytically produced O2, a steady-state H2 atmospheric component can also
+form (e.g., Carberry Mogan et al. 2022). Atomic H, the dissociated product of H2, has also
+been detected at Callisto (e.g., Roth et al. 2017, Carberry Mogan et al. 2022). Moreover,
+Carberry Mogan et al. (2022) and Roth et al. (2023) recently suggested that the H detected
+in the extended atmospheres of Europa (Roth et al., 2017, 2023) and Ganymede (Barth et al.,
+1997, Feldman et al., 2000, Alday et al., 2017, Roth et al., 2023) are also indicative of an H2-
+source. Although H2 is able to escape from these satellites’ atmospheres, it does not escape
+from the Jovian system; and since its lifetime is longer than the satellites’ orbital periods
+(e.g., Smyth and Marconi (2006), Leblanc et al. (2017), Carberry Mogan et al. (2022)), a
+detectable toroidal cloud of neutral H2 co-rotating with the bodies can form (e.g., Szalay
+et al. 2022).
+The sputtering and radiolytic sources of the icy Galilean satellites’ atmospheres compete
+with other sources, such as sublimation of water ice and the subsequent photochemistry of
+the newly formed water vapor (e.g., Yung and McElroy 1977, Kumar and Hunten 1982).
+However, with increasing distance from Jupiter, the plasma density and, as a result, the
+corresponding atmospheric source decreases (Johnson et al., 2004). For example, although
+gas-phase H2O has not been directly observed at Callisto, the outermost Galilean satellite,
+observed geomorphological features have been interpreted to be caused by sublimation of
+the surface ice rather than by sputtering (Spencer and Maloney, 1984, Spencer, 1987, Moore
+et al., 1999). Further, whereas sublimation has been suggested to be the primary source of
+Ganymede’s H2O atmosphere (Roth et al., 2021, Vorburger et al., 2022), sputtering has been
+suggested to be a primary source of Europa’s H2O atmosphere (Addison et al., 2021).
+Below we focus on deriving thermal electron impact ionization rates in these icy satellites’
+atmospheres, which are needed to help understand their evolution.
+3
+Method
+The Deutsch-M¨ark (DM) formula was developed to employ and extrapolate laboratory data
+to allow users to estimate reasonably accurate electron impact ionization cross-sections,
+σ(E), over a large range of energies, E. The “modified” DM formula (Appendix A) from
+Deutsch et al. (2004), hereafter referred to as “DM2004,” is a revised version of the original
+formula developed by Deutsch and M¨ark (1987) and used to calculate cross-sections up to
+energies ≲ keV. As discussed by Deutsch et al. (2000), hereafter referred to as “DM2000,”
+and references therein, σ(E) for atoms (e.g., H and O) can be converted to σ(E) for molecules
+composed of those atoms (e.g., H2O, O2, and H2).
+Here we present more easily usable fits to these complex models for species of interest to
+the planetary science community, and of particular interest at icy satellites. An exponentially
+modified Gaussian distribution is used to compute a non-linear least-squares fit to σ(E)
+derived for H, O, H2, O2, and H2O using DM2000 and DM2004. The resulting equation is
+as follows (in units of ×10−16 cm2):
+σ(E) = α
+2δ exp
+� γ2
+2δ2 + β − log10(E)
+δ
+� �
+erf
+�log10(E) − β
+√
+2γ
+−
+γ
+√
+2δ
+�
++ δ
+|δ|
+�
++ ϵ,
+(1)
+5
+
+where the coefficients α, β, γ, δ, and ϵ in Eq. 1 for H, O, H2, O2, and H2O are listed in
+Table 2.
+Table 2: Coefficients in Eq. 1 for H, O, H2, O2, H2O
+Species
+α
+β
+γ
+δ
+ϵ
+H
+0.951653
+1.40862
+0.271538
+0.804953
+-0.0397646
+O
+1.91899
+1.72847
+0.333420
+0.864596
+-0.121378
+H2
+2.27256
+1.39600
+0.277991
+1.08255
+-0.278929
+O2
+3.20403
+1.63262
+0.241759
+0.799914
+-0.0876336
+H2O
+4.41745
+1.48743
+0.291951
+1.01222
+-0.527864
+4
+Results
+Electron impact ionization cross-sections, σ(E), for H, O, H2, O2, and H2O as derived by
+DM2000 and DM2004, as well as the corresponding fits calculated via Eq.
+1 with the
+coefficients from Table 2, are illustrated in Figure 1. These fits, of course, account for the
+ionization threshold energies occurring around 10 and 20 eV for the species considered (e.g.,
+see Tables A.1–A.2 in Appendix A). By about ∼20 eV, the difference in σ(E) derived by
+DM2000/DM2004 and by the corresponding fits for all of the species considered fall below
+10%, except for that of O, which drops below 10% between 20 and 30 eV. From these lower
+bounds up to 1 keV, the difference remains below 10% for all species considered except for
+H2, for which it exceeds 10% by ∼800 eV but is only ∼14% by 1 keV. Thus, between ∼20 eV–
+1 keV (i.e., between the ionization energy of the species considered, (e.g., Tables A.1–A.2 in
+Appendix A), and the maximum energy of the thermal electrons in the Jovian magnetosphere
+at the icy Galilean satellites, Table 1), the fits provided here can determine electron impact
+ionization cross-sections within ∼10% accuracy of those determined via the more complex
+DM models, which have been extensively tested, modified, and improved over the years,
+demonstrating agreement with experimental data better than ∼20–35% (Deutsch et al. 2009
+and references therein).
+With the thermal electron fluxes and temperatures listed in Table 1, we use the fits for
+electron impact ionization cross-sections presented in Fig. 1 to determine the corresponding
+ionization rates in the water product atmospheres of Europa, Ganymede, and Callisto, which
+are presented in Table 3. The difference in σ(E) derived by DM2004 and by the corresponding
+fit for O electron impact ionization cross-section at 20 eV is ∼32%, making the latter (Eq. 1)
+in the region of Europa’s orbit not as accurate as those of the other species, for which
+the difference is always < 10% (Table 3). However, by 30 eV the difference for all species
+drops to < 5%. The differences for electron impact ionization rates at incident electron
+energies of 100 eV at Ganymede and Callisto are < 3% for all species. We compare these
+electron impact ionization rates to the analogous photoionization rates derived according
+6
+
+Figure 1: Top panel: Electron impact ionization cross-section, σ(E), for H (red lines), O
+(blue lines), H2 (green lines), O2 (cyan lines), and H2O (magenta lines) as a function of
+incident electron energy, E (x-axis). The dashed colored lines are from Deutsch et al. (2000)
+(“DM2000”) for H2, O2, and H2O and from Deutsch et al. (2004) (“DM2004”) for H and O;
+and the solid colored lines are the corresponding fits (“Fit”) calculated via Eq. 1 with the
+coefficients from Table 2. Note the values for σ(E) begin at the ionization energies of the
+species considered (e.g., Tables A.1–A.2 in Appendix A), hence the blank spaces below these
+energies. Bottom panel: The dash-dotted colored lines represent the difference between σ(E)
+calculated via DM2000/DM2004 and via Eq. 1, |fit−DM2000/2004|
+DM2000/2004
+×100 (“Error”).
+to solar activity (Huebner and Mukherjee, 2015) in Table B.1 in Appendix B. Since the
+electron temperature for the “high/cold” plasma component at Europa (Table 1) is less than
+the ionization energies of the atoms and molecules considered (e.g., Tables A.1 and A.2 in
+Appendix A), we show ionization rates over a temperature range of 20–30 eV (“medium” to
+“low/hot” plasma components from Bagenal et al. 2015). At Europa and Ganymede, electron
+impact ionization rates are greater than the photoionization rates for all species. On the
+other hand, at Callisto, where there is the most uncertainty in electron densities (see e.g.,
+Table 1), the upper bound electron impact ionization rates are greater than the upper bound
+photoionization rates for all species, but the lower bound electron impact ionization rates
+are less than the lower bound photoionization rates. We note, however, that the electron
+impact ionization rates relate to the upstream plasma properties, and the effective electron
+impact ionization strongly depends on the details of the interaction of the plasma flow with
+the moons’ atmospheres and ionospheres, which cool as well as divert the plasma around the
+moons (e.g., Saur et al. 1998, 2004, Rubin et al. 2015, Dols et al. 2016).
+7
+
+Energy, E[eV]
+101
+102
+103
+7
+10-15
+cm
+H
+10-17
+0
+H2
+02
+H20
+10-19
+100
+Fit
+90
+DM2000
+80
+0054320
+/DM2004
+Error
+Error[
+102
+103
+101
+Energy, E[eV]Table 3: Electron impact ionization cross-sections and rates in the water product atmo-
+spheres of Europa (E), Ganymede (G), and Callisto (C)
+Species
+Satellite
+Cross-Sectiona [cm−2]
+Rateb [s−1]
+Errorc [%]
+H
+E
+(3.07–5.11)×10−17
+(2.57–5.57)×10−6
+2.96–4.93
+G
+5.48×10−17
+(0.384–3.67)×10−7
+1.34
+C
+4.38×10−10 – 4.06×10−8
+O
+E
+(0.495–2.72)×10−17
+(0.414–2.97)×10−6
+4.92–32.6
+G
+1.03×10−16
+(0.719–6.88)×10−7
+1.36
+C
+8.22×10−10 – 7.60×10−8
+H2
+E
+(3.99–7.52)×10−17
+(3.34–8.20)×10−6
+4.64–6.86
+G
+9.48×10−17
+(0.664–6.35)×10−7
+2.04
+C
+7.58×10−10 – 7.01×10−8
+O2
+E
+(1.98–7.80)×10−17
+(1.66–8.50)×10−6
+1.22–6.91
+G
+2.22×10−16
+(0.155–1.49)×10−6
+2.12
+C
+1.77×10−9 – 1.64×10−7
+H2O
+E
+(0.401–1.24)×10−16
+(0.336–1.36)×10−5
+4.63–9.82
+G
+2.00×10−16
+(0.140–1.34)×10−6
+1.16
+C
+1.60×10−9 – 1.47×10−7
+a Electron impact ionization cross-sections are calculated via Eq. 1 at 20–30 eV for Europa and 100 eV
+for Ganymede and Callisto. Note we only consider temperatures kBTe ≥ 20 eV at Europa because the
+minimum temperature, kBTe = 10 eV (Table 1), is lower than the ionization energies of the species
+considered (e.g., Tables A.1–A.2 in Appendix A).
+b Electron impact ionization rates are calculated as the product of the electron impact ionization cross-
+sections and the range in thermal electron fluxes given in Table 1. Note, since the minimum temperature
+at Europa, kBTe = 10 eV, is lower than the ionization energies of the species considered, the lower
+bound thermal electron flux is calculated according to the “medium” plasma component (kBTe = 20 eV,
+ne = 158 cm−3) from Bagenal et al. (2015), Table 5 therein.
+c The differences in the cross-sections derived by DM2000/DM2004 and by the corresponding fits, the
+“errors,” are interpolated from that illustrated in Fig. 1 at 20–30 eV for Europa (with the lower value
+calculated at 30 eV) and 100 eV for Ganymede and Callisto.
+5
+Conclusion
+The importance to the space physics community of data on atomic and molecular processes
+driven by an ambient plasma cannot be overstated. There are so many difficult but important
+observations whose interpretation is limited by the uncertainties in the atomic and molecular
+database or by the limited range of energies and species studied in the laboratory. Therefore,
+8
+
+the combination of laboratory measurements with detailed, physically-based extrapolation
+procedures, as carried out by K. Becker and colleagues, will continue to be incredibly useful.
+Because the accurate DM formula used to develop useful electron impact cross sections
+over a large range of energies requires a considerable understanding of atomic physics, here
+we present more readily useful fits to their detailed analyses for use by the space physics
+community in order to prepare for the expected data from the forthcoming observations of
+plasma-atmosphere interactions at the icy Jovian satellites. These are used to show the
+relative importance of electron impact ionization in the icy Galilean satellites’ atmospheres
+as compared to photo-ionization.
+Finally, this study can be summarized as followed:
+• Fits to the Deutsch–M¨ark formula for energy-dependent electron impact ionization
+cross-sections have been derived for H2O, H2, O2, H, O from the species’ minimum
+ionization energies up to 1 keV.
+• These cross-sections are used in tandem with electron data at the orbits of Europa,
+Ganymede, and Callisto to determine the corresponding electron impact ionization
+rates in these bodies’ water-product atmospheres.
+• At Europa and Ganymede the electron impact ionization rates are shown to exceed
+the photoionization rates, whereas at Callisto, where the electron densities vary the
+most, likely a result of the moon being inside or outside of the Jovian plasma sheet,
+the electron impact ionization rates can be more or less than the photoionization rates.
+9
+
+Appendix
+A
+DM Formula
+The “modified” DM formula (Deutsch et al., 2004) derives the total energy-dependent
+electron-impact ionization cross section, σ(E), of an atom as:
+σ(E) =
+�
+n,l
+πgn,lr2
+n,lξn,lb(q)
+n,l(u) [ln(cn,lu)/u] .
+(2)
+Here rn,l is the radius of maximum radial density of and ξn,l is the number of electrons in
+the atomic subshell characterized by quantum numbers n and l; gn,l is a weighting factor
+originally determined from a fitting procedure; u = E/En,l, where E is the incident energy
+of the electrons and En,l is the ionization energy in the (n, l) subshell; and cn,l is a constant
+determined from measured cross-sections for various values of n and l. Tables A.1 and A.2 list
+values for the various terms in Eq. 2 for H and O atoms, respectively. The energy-dependent
+function b(q)
+n,l(u) [ln(cn,lu)/u] allows the DM formula to be applied up to keV-energy regimes,
+with b(q)
+n,l(u) written as the following:
+b(q)
+n,l(u) =
+A1 − A2
+1 + (u/A3)p + A2,
+(3)
+where A1−3 and p are constants determined from measured cross-sections for various values
+of n and l, and the superscript (q) refers to the number of electrons in the (n, l) sub-
+shell. Tables A.3 and A.4 list values for the various terms in the energy-dependent function
+b(q)
+n,l(u) [ln(cn,lu)/u] for H and O atoms, respectively. We refer the reader to the review by
+Deutsch et al. (2009) for how σ(E) of atoms are used to calculate σ(E) of molecules com-
+posed of those atoms; i.e., how to estimate σ(E) of H2, O2, and H2O from σ(E) of H and
+O.
+10
+
+Table A.1: Various terms in Eq. 2 for electron impact ionization cross-section of H atoms
+n
+l
+ξn,l
+En,l a [eV]
+rn,l a [(×10−11) m]
+gn,l b
+1
+0
+1
+13.6
+5.29
+2.81
+a En,l and rn,l are taken from Tables 2 and 4 in Desclaux (1973), respectively.
+b gn,l is determined by dividing En,l from the “reduced weighting factor” gn,lEn,l = 38.20 for 1s1 in
+Deutsch et al. (2000).
+Table A.2: Various terms in Eq. 2 for electron impact ionization cross-section of O atoms
+n
+l
+ξn,l
+En,l a [eV]
+rn,l a [(×10−11) m]
+gn,l b
+1
+0
+2
+563
+0.684
+0.124
+2
+0
+2
+34.1
+4.63
+0.587
+2
+1
+4
+16.7
+4.41
+1.79
+a En,l and rn,l are taken from Tables 2 and 4 in Desclaux (1973), respectively.
+b gn,l is determined by dividing En,l from the “reduced weighting factors” gn,lEn,l = 70.00, 20.00, and
+30.00 for 1s2, 2s2, and 2p4, respectively, in Deutsch et al. (2000).
+Table A.3: Various terms in the energy-dependent function b(q)
+n,l(u) [ln(cn,lu)/u] (Eqs. 2–3)
+for electron impact ionization cross-section of H atoms
+n
+l
+q
+cn,l
+A1
+A2
+A3
+p
+1
+0
+1
+1.00
+0.31
+0.87
+2.32
+1.95
+Table A.4: Various terms in the energy-dependent function b(q)
+n,l(u) [ln(cn,lu)/u] (Eqs. 2–3)
+for electron impact ionization cross-section of O atoms
+n
+l
+q
+cn,l
+A1
+A2
+A3
+p
+1
+0
+2
+1.01
+0.23
+0.86
+3.67
+2.08
+2
+0
+2
+1.01
+0.23
+0.86
+3.67
+2.08
+2
+1
+4
+1.02
+-0.15
+1.17
+4.05
+1.31
+11
+
+B
+Photoionization Rates
+Table B.1 lists the range of photoionization rates determined by Huebner and Mukherjee
+(2015) for a “quiet” Sun (i.e., solar minimum) – “active” Sun (i.e., solar maximum), which
+are then scaled to the average Jovian system’s distance from the Sun, 5.2 AU, ignoring any
+possible absorption with depth into the atmosphere.
+Table B.1: Photoionization rates at 5.2 AU
+Species
+Rate [s−1]
+H
+(2.68–6.36)×10−9
+O
+(0.880–2.44)×10−8
+H2
+(2.00–4.25)×10−9
+O2
+(1.73–4.36)×10−8
+H2O
+(1.22–3.06)×10−8
+Acknowledgments
+S.R.C.M. acknowledges the support provided by NASA through the Solar System Workings
+grant 80NSSC21K0152, A.V. acknowledges the support provided by the Swiss National Sci-
+ence Foundation, and L.R. was supported by the Swedish National Space Agency through
+grant 2021-00153 and by the Swedish Research Council through grant 2017-04897.
+Author Contribution Statement
+All authors contributed equally to this work.
+Data Availability Statement
+All data generated or analyzed during this study are included in this published article.
+12
+
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+
diff --git a/0tFIT4oBgHgl3EQf3Cv8/content/tmp_files/load_file.txt b/0tFIT4oBgHgl3EQf3Cv8/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..d4ef22999fd8d2e76e449a08fe1d073d90e86e25
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@@ -0,0 +1,1187 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf,len=1186
+page_content='Electron Impact Ionization in the Icy Galilean Satellites’ Atmospheres  Shane R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Carberry Mogan1,∗, Robert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Johnson2,3, Audrey Vorburger4, Lorenz Roth5  1Space Sciences Laboratory, University of California, Berkeley, Berkeley, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2University  of Virginia, Charlottesville, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 3New York University, New York, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 4University of  Bern, Bern, Switzerland;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 5KTH Royal Institute of Technology, Stockholm, Sweden;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  ∗Corresponding author: Shane R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Carberry Mogan (CarberryMogan@Berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='edu)  Abstract  Electron impact ionization is critical in producing the ionospheres on many plan-  etary bodies and, as discussed here, is critical for interpreting spacecraft and tele-  scopic observations of the tenuous atmospheres of the icy Galilean satellites of Jupiter  (Europa, Ganymede, and Callisto), which form an interesting planetary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' For-  tunately, laboratory measurements, extrapolated by theoretical models, were devel-  oped and published over a number of years by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Becker and colleagues (see Deutsch  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2009) to provide accurate electron impact ionization cross sections for atoms  and molecules, which are crucial to correctly interpret these measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Because  of their relevance for the Jovian icy satellites we provide useful fits to the complex,  semi-empirical Deutsch–M¨ark formula for energy-dependent electron impact ioniza-  tion cross-sections of gas-phase water products (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', H2O, H2, O2, H, O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' These are  then used with measurements of the thermal plasma in the Jovian magnetosphere to  produce ionization rates for comparison with solar photo-ionization rates at the icy  Galilean satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  1  Introduction  Since Galileo Galilei’s revolutionary discovery that Jupiter, the largest planet in the solar  system, has four large planetary bodies revolving around it – the “Galilean” satellites: Io,  Europa, Ganymede, and Callisto – our fascination with this planetary system has only  grown with the advancement of observational technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Several spacecraft have been  sent directly to or have at least passed by the Jovian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' In the 1970s, Pioneer 10 & 11  and Voyager 1 & 2 utilized Jupiter’s gravity to enhance their trajectories and observed the  giant planet and its moons up-close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' From 1995–2003 Galileo orbited Jupiter, and made  several close encounters with the namesake moons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' At the time of this writing, the Juno  spacecraft is currently orbiting Jupiter and has recently made close flybys of Ganymede and  Europa with 2 flybys of Io forthcoming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' In addition to these in-situ observations, the Hubble  Space Telescope (HST), which has been situated in Earth’s orbit since 1990, has been used  to observe and contribute new information to our understanding of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' To better  understand the Jovian system, the Galilean satellites, and their interconnected dynamics, as  well as address certain prevailing mysteries, three forthcoming missions, ESA’s JUpiter ICy  moons Explorer, NASA’s Europa Clipper, and CNSA’s Gan De, will send spacecraft back  to the Jovian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Our focus here is on the icy Galilean satellites (see Table 1) – Europa, Ganymede, and  Callisto – and their tenuous atmospheres for which the dominant constituents are water  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='11380v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='EP]  26 Jan 2023  products: H2O, O2, H2, H, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Because these objects orbit Jupiter within its giant mag-  netic field, they are exposed to an ambient plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Interactions between this plasma and the  icy Galilean satellites’ atmospheres to a large extent determine the nature of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' One  key aspect of this interaction is the role of the plasma electrons in dissociating, ionizing, and  exciting the gas-phase water products via impacts, which can produce observable emission  features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1998, Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2000, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2017, Roth 2021, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' In order to calculate the ionization rates to be used in future simulations of these at-  mospheres, we use the extensive laboratory data of electron impact ionization cross-sections  accumulated and summarized by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Becker and his colleagues over a number of years as  reviewed in Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2009) and references herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Since the various laboratory mea-  surements can exhibit differences and typically cover only a limited range of energies, the  group developed a fitting procedure also presented in those papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' In this method, the  Deutsch–M¨ark (DM) formula, atomic orbital basis sets are used in expressions to fit and  extrapolate the measurements, as well as to generate cross-sections when measurements are  unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' In this way, they created a large number of energy-dependent electron impact  ionization cross-section distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Although the DM formula provides a broad range of  useful data, as well as allows the calculation of results for molecular targets to be determined  from the constituent atomic orbitals, it is not easily implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' For example, in detailed  molecular kinetic simulations much simpler calculations are more often made, such as those  recently implemented in Carberry Mogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2022) for describing electron impacts in Cal-  listo’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' More readily useful expressions are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Therefore, here we present  much more simplified fits to the results obtained via the DM formula, which we then use with  electron density and temperature data at the icy Galilean satellites to calculate ionization  rates in their atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' These rates are in turn needed to help interpret past, present,  and future spacecraft and telescopic observations of these topical planetary bodies soon to  be visited by several new spacecraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Before we discuss the derivation of the electron impact ionization cross-section fits (Sec-  tion 3) and present the corresponding ionization rates for each species considered (Section  4), we first review the local space environment of the icy Galilean satellites, particularly the  Jovian magnetospheric plasma in which they are embedded, as well as the observations of  water products in their atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  2  Background  The strong dynamo generated within the interior of Jupiter produces its magnetic field, which  has a rotation period of ∼10 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Jupiter’s magnetosphere, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', the environment controlled by  the planet’s magnetic field, is filled with charged particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' While the volcanic moon Io is the  main source, materials from the icy Galilean satellites’ surfaces are also a source of charged  particles (either directly ionized or lost as neutrals and ionized later in the magnetosphere)  (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004, Kivelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004), which are picked-up, accelerated, and then  become “trapped” by this rapidly rotating field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  The combination of this fast rotating  magnetic field and the large number of charged particles trapped within it gives rise to the  plasma detected in Jupiter’s immense magnetosphere, which can extend as far as 45–100 RJ  from the planet, where RJ = 71, 492 km is the radius of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The Jovian magnetosphere  2  is typically divided up into three regions: the inner (<10 RJ), the middle (10–40 RJ), and the  outer (>40 RJ) regions (Khurana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Europa resides within the inner region, while  Ganymede and Callisto reside in the middle region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The Jovian magnetospheric plasma is  typically described as being comprised of two populations: a “cold” thermal plasma with  energies < 1 keV and a “hot” energetic plasma with energies ≥ 1 keV (Krupp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004,  Bagenal and Delamere, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Both populations are composed of electrons, as well as H+,  On+, and Sn+ ions (Bagenal and Sullivan, 1981, Broadfoot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1981, Nerney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  The sulfur and oxygen ions, both of which are of various charge states, primarily originate  from the volcanic moon Io (Bagenal and Dols, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' On the other hand, the hydrogen  ions originate from two different sources depending on the magnetospheric region: in the  inner region, they mainly originate in Jupiter’s atmosphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and in the middle and outer  magnetosphere, the solar wind can gain access to the magnetosphere, becoming the main  provider of hydrogen ions thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  An additional plasma contribution (electrons, H+,  On+) is associated with sputtering of the icy satellite’s surfaces (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2001, Johnson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004, Szalay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Here we focus on the thermal electrons as these particles  are the most relevant to ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Physical parameters for thermal electrons at the orbits  of Europa, Ganymede, and Callisto based on Voyager and Galileo data are listed in Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Table 1: Physical Parameters of Europa, Ganymede, and Callisto, as well as of the Jovian  magnetosphere at their orbits  Parameters [units]  Europa  Ganymede  Callisto  Radius [km]  1,561  2,631  2,410  Mass [(×1023) kg]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='48  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1  Distance from Jupiter [RJ]  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='38  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='0  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='3  Orbital Velocity [km s−1]  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='7  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='9  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='20  Thermal Electron Density [cm−3]  18–290 a,b  1–10 a,c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='01–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1 a,c  Thermal Electron Temperature [eV]  10–30 b  100 c  100 c  Thermal Electron Flux d [cm−2 s−1]  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='4–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1)×1011 e  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='7–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='7)×109  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='08–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='4)×108  Plasma Azimuthal Velocity [km s−1]  90 a  150 a  200 a  Relative Plasma Velocity f [km s−1]  76 a  139 a  192 a  a Values taken from Kivelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2004) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  b Values taken from Bagenal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  c Values taken from Neubauer (1998) and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  d Thermal electron flux, φe = neve, where ne is the thermal electron density, ve =  �  8kBTe/π/me is the  mean Maxwellian speed of the electrons, with kBTe the thermal electron temperature, kB the Boltzmann  constant, and me the mass of an electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  e φe calculated according to the “low/hot” (kBTe = 30 eV, ne = 63 cm−3) and “high/cold”  (kBTe = 10 eV, ne = 290 cm−3) plasma components from Table 5 in Bagenal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  g Relative speed between plasma azimuthal velocity and the satellites’ orbital speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  3  As a result of the large relative velocities between the azimuthal velocity of the Jovian  magnetosphere and the icy Galilean satellites’ orbital velocities (Table 1), the satellites  are continuously overtaken and bombarded by the magnetospheric plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  The spatial  distribution of this bombardment is determined by the flow rate of the plasma particles  past the satellite as well as their thermal velocities relative to the local magnetic field lines  (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' However, intrinsic or induced electric and magnetic fields as well  as the interactions with the tenuous atmospheres and ionospheres at these satellites can  radically modify the local fluxes of impinging particles at Ganymede (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', from Paranicas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2022) and at Callisto (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', from Strobel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The cyclotron radii or gyro-  radii of these charged particles depend on their mass, speed, and charge, as well as the  local magnetic field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Due to their small mass, electrons primarily have gyro-radii  much smaller than the satellite radius, and thus preferentially impact the satellites’ trailing  hemispheres and poles as they move up and down the rotating field lines (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=',  2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Following the Pioneer discovery of intense plasma trapped in the Jovian magnetic field  (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1974, Wolfe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1974, Trainor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1974, Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1976) a series of ex-  periments were carried out to measure the effect this could have on the icy surfaces of Europa,  Ganymede, and Callisto (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1978, Lanzerotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' These experiments  showed that the ejection of water molecules from low-temperature ices by incident energetic  particles, a process referred to as “sputtering”, is dominated by electronic excitations and  ionizations produced in the ice (“electronic” sputtering), rather than by “knock-on” colli-  sions of the ions with water molecules (“nuclear” sputtering), the hitherto typically studied  sputtering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Subsequent experiments led to the discovery that additional molecular  species can form in and be released from the ice, namely H2 and O2, in a process referred to  as “radiolysis” (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1982, Boring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1983, Reimann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1984, Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=',  1984), which occurs as bonds in H2O molecules are broken by the electronic energy deposited  by the impinging charged particles and the fragmented molecules recombine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Moreover, the  number of radiolytic products ejected from the icy surface per each incident charged particle  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', the “sputter yield”) was shown to display a strong temperature dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  These discoveries had immense implications for the icy Galilean satellites: magneto-  spheric plasma-induced sputtering could erode their surfaces, and the ejected atoms and  molecules could migrate significant distances as well as escape the local gravitational en-  vironment of its host satellite or form gravitationally bound atmospheres (Johnson, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Indeed Europa was predicted to have a tenuous, predominantly O2 atmosphere due to the  radiolytic decomposition of its icy surface by the incident Jovian plasma particles (Johnson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1982, 1983, Johnson, 1990), which has been borne out by extensive HST observations  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2016, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Tenuous O2 atmospheres produced via similar  processes have also been detected at Ganymede and Callisto: following the HST detection of  O emissions indicative of an O2 atmosphere at Europa (Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1995), airglow emissions  were detected by HST in Ganymede’s O2 atmosphere (Hall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1998) as well as in Europa’s  atmosphere thereby confirming the earlier observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' O emissions were detected by HST  in Callisto’s atmosphere (Cunningham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2015), which were suggested to be induced by  photoelectron impacts in a near-surface, O2-dominated atmosphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and recently atomic O  emissions have been detected at Ganymede (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2021) indicative of being produced  via dissociative excitations of O2 (and H2O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  4  Although H2 can more readily escape from the atmospheres of these bodies than can the  concomitant radiolytically produced O2, a steady-state H2 atmospheric component can also  form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Carberry Mogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Atomic H, the dissociated product of H2, has also  been detected at Callisto (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2017, Carberry Mogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Moreover,  Carberry Mogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2022) and Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2023) recently suggested that the H detected  in the extended atmospheres of Europa (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2017, 2023) and Ganymede (Barth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=',  1997, Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2000, Alday et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2017, Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2023) are also indicative of an H2-  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Although H2 is able to escape from these satellites’ atmospheres, it does not escape  from the Jovian system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and since its lifetime is longer than the satellites’ orbital periods  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Smyth and Marconi (2006), Leblanc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2017), Carberry Mogan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2022)), a  detectable toroidal cloud of neutral H2 co-rotating with the bodies can form (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Szalay  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  The sputtering and radiolytic sources of the icy Galilean satellites’ atmospheres compete  with other sources, such as sublimation of water ice and the subsequent photochemistry of  the newly formed water vapor (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Yung and McElroy 1977, Kumar and Hunten 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  However, with increasing distance from Jupiter, the plasma density and, as a result, the  corresponding atmospheric source decreases (Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' For example, although  gas-phase H2O has not been directly observed at Callisto, the outermost Galilean satellite,  observed geomorphological features have been interpreted to be caused by sublimation of  the surface ice rather than by sputtering (Spencer and Maloney, 1984, Spencer, 1987, Moore  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Further, whereas sublimation has been suggested to be the primary source of  Ganymede’s H2O atmosphere (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2021, Vorburger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2022), sputtering has been  suggested to be a primary source of Europa’s H2O atmosphere (Addison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Below we focus on deriving thermal electron impact ionization rates in these icy satellites’  atmospheres, which are needed to help understand their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  3  Method  The Deutsch-M¨ark (DM) formula was developed to employ and extrapolate laboratory data  to allow users to estimate reasonably accurate electron impact ionization cross-sections,  σ(E), over a large range of energies, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The “modified” DM formula (Appendix A) from  Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2004), hereafter referred to as “DM2004,” is a revised version of the original  formula developed by Deutsch and M¨ark (1987) and used to calculate cross-sections up to  energies ≲ keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' As discussed by Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2000), hereafter referred to as “DM2000,”  and references therein, σ(E) for atoms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', H and O) can be converted to σ(E) for molecules  composed of those atoms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', H2O, O2, and H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Here we present more easily usable fits to these complex models for species of interest to  the planetary science community, and of particular interest at icy satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' An exponentially  modified Gaussian distribution is used to compute a non-linear least-squares fit to σ(E)  derived for H, O, H2, O2, and H2O using DM2000 and DM2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The resulting equation is  as follows (in units of ×10−16 cm2):  σ(E) = α  2δ exp  � γ2  2δ2 + β − log10(E)  δ  � �  erf  �log10(E) − β  √  2γ  −  γ  √  2δ  �  + δ  |δ|  �  + ϵ,  (1)  5  where the coefficients α, β, γ, δ, and ϵ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1 for H, O, H2, O2, and H2O are listed in  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Table 2: Coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1 for H, O, H2, O2, H2O  Species  α  β  γ  δ  ϵ  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content='121378  H2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content='278929  O2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content='0876336  H2O  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content='527864  4  Results  Electron impact ionization cross-sections, σ(E), for H, O, H2, O2, and H2O as derived by  DM2000 and DM2004, as well as the corresponding fits calculated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  1 with the  coefficients from Table 2, are illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' These fits, of course, account for the  ionization threshold energies occurring around 10 and 20 eV for the species considered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=',  see Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' By about ∼20 eV, the difference in σ(E) derived by  DM2000/DM2004 and by the corresponding fits for all of the species considered fall below  10%, except for that of O, which drops below 10% between 20 and 30 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' From these lower  bounds up to 1 keV, the difference remains below 10% for all species considered except for  H2, for which it exceeds 10% by ∼800 eV but is only ∼14% by 1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Thus, between ∼20 eV–  1 keV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', between the ionization energy of the species considered, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 in  Appendix A), and the maximum energy of the thermal electrons in the Jovian magnetosphere  at the icy Galilean satellites, Table 1), the fits provided here can determine electron impact  ionization cross-sections within ∼10% accuracy of those determined via the more complex  DM models, which have been extensively tested, modified, and improved over the years,  demonstrating agreement with experimental data better than ∼20–35% (Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2009  and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  With the thermal electron fluxes and temperatures listed in Table 1, we use the fits for  electron impact ionization cross-sections presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1 to determine the corresponding  ionization rates in the water product atmospheres of Europa, Ganymede, and Callisto, which  are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The difference in σ(E) derived by DM2004 and by the corresponding  fit for O electron impact ionization cross-section at 20 eV is ∼32%, making the latter (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1)  in the region of Europa’s orbit not as accurate as those of the other species, for which  the difference is always < 10% (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' However, by 30 eV the difference for all species  drops to < 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The differences for electron impact ionization rates at incident electron  energies of 100 eV at Ganymede and Callisto are < 3% for all species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' We compare these  electron impact ionization rates to the analogous photoionization rates derived according  6  Figure 1: Top panel: Electron impact ionization cross-section, σ(E), for H (red lines), O  (blue lines), H2 (green lines), O2 (cyan lines), and H2O (magenta lines) as a function of  incident electron energy, E (x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The dashed colored lines are from Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2000)  (“DM2000”) for H2, O2, and H2O and from Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2004) (“DM2004”) for H and O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  and the solid colored lines are the corresponding fits (“Fit”) calculated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1 with the  coefficients from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Note the values for σ(E) begin at the ionization energies of the  species considered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 in Appendix A), hence the blank spaces below these  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Bottom panel: The dash-dotted colored lines represent the difference between σ(E)  calculated via DM2000/DM2004 and via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1, |fit−DM2000/2004|  DM2000/2004  ×100 (“Error”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  to solar activity (Huebner and Mukherjee, 2015) in Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Since the  electron temperature for the “high/cold” plasma component at Europa (Table 1) is less than  the ionization energies of the atoms and molecules considered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 in  Appendix A), we show ionization rates over a temperature range of 20–30 eV (“medium” to  “low/hot” plasma components from Bagenal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' At Europa and Ganymede, electron  impact ionization rates are greater than the photoionization rates for all species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' On the  other hand, at Callisto, where there is the most uncertainty in electron densities (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=',  Table 1), the upper bound electron impact ionization rates are greater than the upper bound  photoionization rates for all species, but the lower bound electron impact ionization rates  are less than the lower bound photoionization rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' We note, however, that the electron  impact ionization rates relate to the upstream plasma properties, and the effective electron  impact ionization strongly depends on the details of the interaction of the plasma flow with  the moons’ atmospheres and ionospheres, which cool as well as divert the plasma around the  moons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Saur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1998, 2004, Rubin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2015, Dols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  7  Energy, E[eV]  101  102  103  7  10-15  cm  H  10-17  0  H2  02  H20  10-19  100  Fit  90  DM2000  80  0054320  /DM2004  Error  Error[  102  103  101  Energy, E[eV]Table 3: Electron impact ionization cross-sections and rates in the water product atmo-  spheres of Europa (E), Ganymede (G), and Callisto (C)  Species  Satellite  Cross-Sectiona [cm−2]  Rateb [s−1]  Errorc [%]  H  E  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='07–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='11)×10−17  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='57–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='57)×10−6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='96–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='93  G  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='48×10−17  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='384–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='67)×10−7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='34  C  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='38×10−10 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='06×10−8  O  E  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='495–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='72)×10−17  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='414–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='97)×10−6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='92–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='6  G  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='03×10−16  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='719–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='88)×10−7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='36  C  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='22×10−10 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='60×10−8  H2  E  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='99–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='52)×10−17  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='34–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='20)×10−6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='64–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='86  G  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='48×10−17  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='664–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='35)×10−7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='04  C  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='58×10−10 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='01×10−8  O2  E  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='98–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='80)×10−17  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='66–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='50)×10−6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='22–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='91  G  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='22×10−16  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='155–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='49)×10−6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='12  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='77×10−9 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='64×10−7  H2O  E  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='401–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='24)×10−16  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='336–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='36)×10−5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='63–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='82  G  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='00×10−16  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='140–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='34)×10−6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='16  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='60×10−9 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='47×10−7  a Electron impact ionization cross-sections are calculated via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1 at 20–30 eV for Europa and 100 eV  for Ganymede and Callisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Note we only consider temperatures kBTe ≥ 20 eV at Europa because the  minimum temperature, kBTe = 10 eV (Table 1), is lower than the ionization energies of the species  considered (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  b Electron impact ionization rates are calculated as the product of the electron impact ionization cross-  sections and the range in thermal electron fluxes given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Note, since the minimum temperature  at Europa, kBTe = 10 eV, is lower than the ionization energies of the species considered, the lower  bound thermal electron flux is calculated according to the “medium” plasma component (kBTe = 20 eV,  ne = 158 cm−3) from Bagenal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2015), Table 5 therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  c The differences in the cross-sections derived by DM2000/DM2004 and by the corresponding fits, the  “errors,” are interpolated from that illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 1 at 20–30 eV for Europa (with the lower value  calculated at 30 eV) and 100 eV for Ganymede and Callisto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  5  Conclusion  The importance to the space physics community of data on atomic and molecular processes  driven by an ambient plasma cannot be overstated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' There are so many difficult but important  observations whose interpretation is limited by the uncertainties in the atomic and molecular  database or by the limited range of energies and species studied in the laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Therefore,  8  the combination of laboratory measurements with detailed, physically-based extrapolation  procedures, as carried out by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Becker and colleagues, will continue to be incredibly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Because the accurate DM formula used to develop useful electron impact cross sections  over a large range of energies requires a considerable understanding of atomic physics, here  we present more readily useful fits to their detailed analyses for use by the space physics  community in order to prepare for the expected data from the forthcoming observations of  plasma-atmosphere interactions at the icy Jovian satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' These are used to show the  relative importance of electron impact ionization in the icy Galilean satellites’ atmospheres  as compared to photo-ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Finally, this study can be summarized as followed:  Fits to the Deutsch–M¨ark formula for energy-dependent electron impact ionization  cross-sections have been derived for H2O, H2, O2, H, O from the species’ minimum  ionization energies up to 1 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  These cross-sections are used in tandem with electron data at the orbits of Europa,  Ganymede, and Callisto to determine the corresponding electron impact ionization  rates in these bodies’ water-product atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  At Europa and Ganymede the electron impact ionization rates are shown to exceed  the photoionization rates, whereas at Callisto, where the electron densities vary the  most, likely a result of the moon being inside or outside of the Jovian plasma sheet,  the electron impact ionization rates can be more or less than the photoionization rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  9  Appendix  A  DM Formula  The “modified” DM formula (Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 2004) derives the total energy-dependent  electron-impact ionization cross section, σ(E), of an atom as:  σ(E) =  �  n,l  πgn,lr2  n,lξn,lb(q)  n,l(u) [ln(cn,lu)/u] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  (2)  Here rn,l is the radius of maximum radial density of and ξn,l is the number of electrons in  the atomic subshell characterized by quantum numbers n and l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' gn,l is a weighting factor  originally determined from a fitting procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' u = E/En,l, where E is the incident energy  of the electrons and En,l is the ionization energy in the (n, l) subshell;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and cn,l is a constant  determined from measured cross-sections for various values of n and l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 list  values for the various terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2 for H and O atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The energy-dependent  function b(q)  n,l(u) [ln(cn,lu)/u] allows the DM formula to be applied up to keV-energy regimes,  with b(q)  n,l(u) written as the following:  b(q)  n,l(u) =  A1 − A2  1 + (u/A3)p + A2,  (3)  where A1−3 and p are constants determined from measured cross-sections for various values  of n and l, and the superscript (q) refers to the number of electrons in the (n, l) sub-  shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Tables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='4 list values for the various terms in the energy-dependent function  b(q)  n,l(u) [ln(cn,lu)/u] for H and O atoms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' We refer the reader to the review by  Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2009) for how σ(E) of atoms are used to calculate σ(E) of molecules com-  posed of those atoms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', how to estimate σ(E) of H2, O2, and H2O from σ(E) of H and  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  10  Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1: Various terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2 for electron impact ionization cross-section of H atoms  n  l  ξn,l  En,l a [eV]  rn,l a [(×10−11) m]  gn,l b  1  0  1  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='29  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='81  a En,l and rn,l are taken from Tables 2 and 4 in Desclaux (1973), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  b gn,l is determined by dividing En,l from the “reduced weighting factor” gn,lEn,l = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='20 for 1s1 in  Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2: Various terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2 for electron impact ionization cross-section of O atoms  n  l  ξn,l  En,l a [eV]  rn,l a [(×10−11) m]  gn,l b  1  0  2  563  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='684  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='124  2  0  2  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='587  2  1  4  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='79  a En,l and rn,l are taken from Tables 2 and 4 in Desclaux (1973), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  b gn,l is determined by dividing En,l from the “reduced weighting factors” gn,lEn,l = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='00, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='00, and  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='00 for 1s2, 2s2, and 2p4, respectively, in Deutsch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='3: Various terms in the energy-dependent function b(q)  n,l(u) [ln(cn,lu)/u] (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2–3)  for electron impact ionization cross-section of H atoms  n  l  q  cn,l  A1  A2  A3  p  1  0  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='87  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='95  Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='4: Various terms in the energy-dependent function b(q)  n,l(u) [ln(cn,lu)/u] (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' 2–3)  for electron impact ionization cross-section of O atoms  n  l  q  cn,l  A1  A2  A3  p  1  0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='86  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='67  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='08  2  0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='86  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='67  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='08  2  1  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='31  11  B  Photoionization Rates  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1 lists the range of photoionization rates determined by Huebner and Mukherjee  (2015) for a “quiet” Sun (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', solar minimum) – “active” Sun (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', solar maximum), which  are then scaled to the average Jovian system’s distance from the Sun, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 AU, ignoring any  possible absorption with depth into the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='1: Photoionization rates at 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='2 AU  Species  Rate [s−1]  H  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='68–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='36)×10−9  O  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='880–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='44)×10−8  H2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='00–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='25)×10−9  O2  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='73–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='36)×10−8  H2O  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='22–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='06)×10−8  Acknowledgments  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' acknowledges the support provided by NASA through the Solar System Workings  grant 80NSSC21K0152, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' acknowledges the support provided by the Swiss National Sci-  ence Foundation, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' was supported by the Swedish National Space Agency through  grant 2021-00153 and by the Swedish Research Council through grant 2017-04897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Author Contribution Statement  All authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Data Availability Statement  All data generated or analyzed during this study are included in this published article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  12  References  Addison, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Liuzzo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Arnold, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Simon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Retherford, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Molyneux, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' New constraints on Ganymede’s hydrogen corona: Analysis of Lyman-α  emissions observed by HST/STIS between 1998 and 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Planetary and Space Science,  148:35–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Bagenal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and Delamere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Flow of mass and energy in the magnetospheres of  Jupiter and Saturn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', 116:5209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Bagenal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and Dols, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The space environment of Io and Europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Journal of  Geophysical Research: Space Physics, 125(5):e2019JA027485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Bagenal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Sidrow, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Wilson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Dols, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Crary, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Steffl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=',  Delamere, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Kurth, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Plasma conditions at Europa’s  orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Icarus, 261:1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Bagenal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and Sullivan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Direct plasma measurements in the Io torus and  inner magnetosphere of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Journal of Geophysical Research, 86(A10):8447–8466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Barth, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Hord, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Pryor, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', McClintock,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Aiello, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Galileo ultraviolet spec-  trometer observations of atomic hydrogen in the atmosphere of Ganymede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Geophysical  Research Letters, 24(17):2147–2150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Boring, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Reimann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Marcan-  tonio, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Ion-induced chemistry in condensed gas solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Nuclear Instruments  and Methods in Physics Research, 218(1-3):707–711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Broadfoot, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Overview  of the Voyager ultraviolet spectrometry results through Jupiter encounter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Journal of  Geophysical Research, 86(A10):8259–8284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Brown, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Electronic sputtering of low temperature  molecular solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Nuclear Instruments and Methods in Physics Research Section B: Beam  Interactions with Materials and Atoms, 1(2-3):307–314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Brown, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Erosion and molecule formation in condensed gas films by electronic energy loss of fast  ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Nuclear Instruments and Methods in Physics Research, 198(1):1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  13  Brown, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Lanzerotti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Augustyniak, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' “Sputtering”  of ice by MeV light ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Physical Review Letters, 40(15):1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Carberry Mogan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Alday, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Vorburger, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=',  Wurz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Galli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Smith, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', and Oza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Callisto’s atmo-  sphere: First evidence for H2 and constrains on H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Journal of Geophysical Research:  Planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Cooper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Gehrels, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Energetic  Ion and Electron Irradiation of the Icy Galilean Satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Icarus, 149(1):133–159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Cunningham, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', and Osterman,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Detection of Callisto’s oxygen atmosphere with the Hubble Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Icarus, 254:178–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Desclaux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Relativistic Dirac-Fock expectation values for atoms with Z = 1 to  Z = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Atomic data and nuclear data tables, 12(4):311–406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Deutsch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Becker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Matt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and M¨ark, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Theoretical determination of  absolute electron-impact ionization cross sections of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' International Journal of  Mass Spectrometry, 197:37–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Deutsch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Becker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Probst, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Maerk, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' The semiempirical Deutsch–  M¨ark formalism: A versatile approach for the calculation of electron-impact ionization  cross sections of atoms, molecules, ions, and clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Advances in Atomic, Molecular, and  Optical Physics, 57:87–155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Deutsch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and M¨ark, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Calculation of absolute electron impact ionization cross-  section functions for single ionization of He, Ne, Ar, Kr, Xe, N and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' International journal  of mass spectrometry and ion processes, 79(3):R1–R8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Deutsch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Scheier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Becker, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Revised high energy behavior  of the Deutsch-M¨ark (DM) formula for the calculation of electron impact ionization cross  sections of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' International Journal of Mass Spectrometry, 233(1-3):13–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Dols, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Cassidy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Europa’s  atmospheric neutral escape: Importance of symmetrical O2 charge exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Icarus,  264:387–397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Feldman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' HST/STIS ultraviolet imaging of polar aurora on Ganymede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The  Astrophysical Journal, 535(2):1085.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Frank, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Ackerson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Mihalov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Observations of  plasmas in the Jovian magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Journal of Geophysical Research, 81(4):457–468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Hall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', McGrath, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and Strobel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' The far-ultraviolet  oxygen airglow of Europa and Ganymede.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The Astrophysical Journal, 499(1):475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  14  Hall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', McGrath, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content='  Detection of an oxygen atmosphere on Jupiter’s moon Europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Nature, 373(6516):677.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Huebner, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and Mukherjee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Photoionization and photodissociation rates in  solar and blackbody radiation fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Planetary and Space Science, 106:11–45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Energetic charged-particle interactions with atmospheres and surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Plasma ion-induced  molecular ejection on the Galilean satellites: Energies of ejected molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Geophysical  research letters, 10(9):892–895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Paranicas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Moore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Radiation effects on the surfaces of the Galilean satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Jupiter: The  planet, satellites and magnetosphere, pages 485–512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Planetary applications of ion  induced erosion of condensed-gas frosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Nuclear Instruments and Methods in Physics  Research, 198(1):147–157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Khurana, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Woch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Mauk,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' The configuration of Jupiter’s magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Jupiter:  The planet, satellites and magnetosphere, 1:593–616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Kivelson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Bagenal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', and Saur, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Magnetospheric interactions with satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Jupiter: The planet, satellites and  magnetosphere, pages 513–536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Krupp, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Vasyliunas, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Woch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Lagg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Khurana, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Kivelson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Mauk,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Roelof, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Williams, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Krimigis, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Frank, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and  Paterson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Dynamics of the Jovian magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' In Bagenal, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Dowling,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', and McKinnon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', editors, Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The Planet, Satellites and Magnetosphere,  volume 1, pages 617–638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Kumar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' and Hunten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' The atmospheres of Io and other satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Satellites  of Jupiter, pages 782–806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Lanzerotti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', and Augustyniak, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' On the  contribution of water products from Galilean satellites to the Jovian magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Geo-  physical Research Letters, 5(2):155–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Leblanc, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Oza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Schmidt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Cassidy, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Modolo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Chaufray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=',  and Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' On the orbital variability of Ganymede’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Icarus,  293:185–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Moore, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Mass movement and landform degradation on the icy Galilean  satellites: Results of the Galileo nominal mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Icarus, 140(2):294–312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  15  Nerney, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Io plasma torus ion composition: Voy-  ager, Galileo, and Cassini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Journal of Geophysical Research (Space Physics), 122(1):727–  744.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Neubauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' The sub-Alfv´enic interaction of the Galilean satellites with the  Jovian magnetosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Journal of Geophysical Research: Planets, 103(E9):19843–19866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Paranicas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=', Mauk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Masters, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', and Bolton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Energetic charged  particle fluxes relevant to Ganymede’s polar region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Geophysical Research Letters, page  e2022GL098077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Reimann, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Ion-induced molecular ejection from  D2O ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Surface science, 147(1):227–240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  Roth, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content='  A stable H2O atmosphere on Europa’s trailing hemisphere from HST  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Geophysical Research Letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=', Ivchenko, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content='  A sublimated water atmosphere on Ganymede  detected from Hubble Space Telescope observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
+page_content=' Nature Astronomy, pages 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content='  Probing  Ganymede’s atmosphere with HST Ly-alpha images in transit of Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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+page_content=' Europa’s far ultraviolet oxygen  aurora from a comprehensive set of HST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFIT4oBgHgl3EQf3Cv8/content/2301.11380v1.pdf'}
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diff --git a/1dE2T4oBgHgl3EQfigf6/content/tmp_files/2301.03960v1.pdf.txt b/1dE2T4oBgHgl3EQfigf6/content/tmp_files/2301.03960v1.pdf.txt
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@@ -0,0 +1,2939 @@
+The limits of human mobility traces to predict the spread of
+COVID-19
+Federico Delussu1,2, Michele Tizzoni1,3 †, and Laetitia Gauvin1,4†
+1ISI Foundation, via Chisola 5, 10126, Turin, Italy
+2Department of Applied Mathematics and Computer Science, DTU,
+Copenhagen, Denmark
+3Department of Sociology and Social Research, University of Trento, Trento, Italy
+4Institute for Research on Sustainable DevelopmentIRD, UMR 215 Prodig, 5
+cours des Humanit´es, F-93 322 Aubervilliers Cedex, France
+†these authors contributed equally to this work
+Abstract
+Mobile phone data have been widely used to model the spread of COVID-19, however,
+quantifying and comparing their predictive value across different settings is challenging.
+Their quality is affected by various factors and their relationship with epidemiological in-
+dicators varies over time. Here we adopt a model-free approach based on transfer entropy
+to quantify the relationship between mobile phone-derived mobility metrics and COVID-
+19 cases and deaths in more than 200 European subnational regions. We found that past
+knowledge of mobility does not provide statistically significant information on COVID-19
+cases or deaths in most of the regions. In the remaining ones, measures of contact rates
+were often more informative than movements in predicting the spread of the disease, while
+the most predictive metrics between mid-range and short-range movements depended on the
+region considered. We finally identify geographic and demographic factors, such as users’
+coverage and commuting patterns, that can help determine the best metric for predicting
+disease incidence in a particular location. Our approach provides epidemiologists and public
+health officials with a general framework to evaluate the usefulness of human mobility data
+in responding to epidemics.
+1
+Introduction
+The relationship between human movements and the spatial spread of infectious diseases has
+been recognized for a long time [1, 2, 3].
+Human movement has been shown to play a key
+1
+arXiv:2301.03960v1  [physics.soc-ph]  10 Jan 2023
+
+role in the dynamics of several pathogens, through two basic mechanisms: traveling infectious
+individuals may introduce a pathogen in a susceptible population, and, at the same time, human
+movement increase the contact rate between individuals, creating new opportunities for infection.
+In the past 15 years, the increasing availability of mobility data derived from mobile phones has
+fueled a large body of work aimed at identifying opportunities to use them for infectious disease
+modeling and surveillance [4, 5, 6, 7, 8, 9, 10].
+More recently, during the COVID-19 pandemic, mobile phone-derived data have been exten-
+sively harnessed to monitor the effect of non-pharmaceutical interventions (NPIs) across coun-
+tries, understand the early dynamics of COVID-19 diffusion, and forecast its spread at different
+spatial scales, from countries to cities [11, 12, 13, 14, 15, 16, 17]. By measuring human move-
+ments and combining them with phylogeography methods [18, 19], several studies shed light on
+the cryptic spread of new variants, their persistence over time and resurgence after the relaxation
+of NPIs [20, 21, 22].
+Human mobility has been shown to strongly correlate with the spread of COVID-19 during
+the early phase of the outbreak in China and in many other countries [23, 24, 25, 26, 27, 28].
+However, once COVID-19 established a foothold in a population, the relative importance of
+mobile phone-derived data to predict the epidemic dynamics on a local scale has been generally
+less understood and several studies have shown conflicting evidence about the use of mobility
+traces to model the spread of COVID-19 at later stages of the outbreak. For instance, it has been
+shown that the explanatory power of mobility metrics in relation to the case growth rate in the
+U.S., significantly declined in spring 2020, especially in rural areas [29, 30, 31]. Similar trends
+have been observed in Europe [32]. In parallel, mobile phone-derived data have been proven
+beneficial to model COVID-19 dynamics in largely populated urban areas of Western countries
+[33, 34], but less so in countries of the Global South [35].
+Several reasons have been proposed to explain the varying relationship between mobility
+metrics and epidemic indicators [29]. Mobility metrics are generally derived from raw mobile
+positioning data through complex and customized processing pipelines that can significantly
+vary across data providers [36].
+How raw data are processed, and the specific definitions of
+mobility metrics can significantly impact their interpretation with respect to epidemic variables
+[37]. Moreover, the relationship between mobility and epidemic patterns often relies on model-
+ing assumptions, typically considering linear dependencies, that may not capture the complex
+interplay of these quantities [32, 30]. Finally, mobile phone-derived metrics are generated from
+a sample of users who is generally not representative of the whole population. It is therefore of
+paramount importance to define standardized approaches that can quantify the added value of
+mobility metrics for epidemiological analysis, and make different metrics, across settings, directly
+comparable.
+Here, we extensively quantify the relationship between cell phone-derived mobility metrics and
+COVID-19 epidemiological indicators through a model-free approach, based on an information-
+theoretic measure, transfer entropy [38], adapted for small sample sizes. Leveraging granular
+2
+
+data provided by Meta that capture both users’ movements and colocation at a fine spatial scale
+[39], we measure the information flow between mobility metrics and time series of COVID-19
+incidence and deaths in four European countries, at a subnational scale, over a one year period.
+We find that the relative information added by the past knowledge of mobility metrics to the
+knowledge of the current state of COVID-19 time series is often not statistically significant.
+In statistically significant cases instead, we show that the relative information added by
+past knowledge of COVID-19 cases to the knowledge of current deaths is twice the information
+flow between past knowledge of mobility metrics and current deaths. We also show that the
+information flow of a given mobility metric to predict future COVID-19 incidence or deaths can
+be significant in one country but not in another, even if derived from the same original data
+source.
+Being a general framework, our approach provides a quantitative measure of the relative
+added explanation brought by mobile phone data to the prediction of epidemiological time series
+that does not depend on the choice of a specific forecasting model. It thus helps to better identify
+the most appropriate mobility metrics to use among those available. Our results can thus guide
+epidemiologists and public health practitioners in the evaluation of mobile phone-derived mobility
+metrics when they are interpreted as a precursor of epidemic activity.
+2
+Results
+Here, we first describe and then apply our framework to measure the information flow between
+human mobility traces and the time evolution of COVID-19 in four European countries.
+2.1
+A transfer entropy approach to link mobility behavior and COVID-
+19 epidemiology
+With the aim of quantifying the information flow from mobility-derived data to COVID-19 data,
+we first gathered a set of mobility and epidemiological indicators. Fig. 1 provides an overview
+of the datasets used in the study. In Materials and Methods, we provide a full description of all
+data sources and the data processing steps. We considered four European countries – Austria,
+France, Italy, and Spain – and their administrative subdivisions at NUTS3 level [40] which is
+the lowest, i.e. the most granular, level of the standard hierarchy of administrative regions in
+Europe (Fig. 1, leftmost column).
+In all administrative regions, we collected indicators of the COVID-19 epidemic dynamics,
+namely, the weekly and daily numbers of new COVID-19 cases and deaths over the period, from
+September 2020 until July 2021. During this period, the dynamics of COVID-19, exemplified by
+the incidence of new cases (Fig. 1, rightmost column), displayed subsequent waves, as a result of
+the complex interaction between the spread of new variants, the adoption of non-pharmaceutical
+interventions, the introduction of vaccines.
+3
+
+Figure 1:
+Summary of behavioral and epidemiological indicators.
+In each country
+under study (from top to bottom: Italy, France, Austria and Spain), we consider three different
+types of indicators: contact rates, movements (here for the sake of simplicity we only show the
+short-range movements), and COVID-19 cases. In each plot, the blue shaded area highlights
+the within-country variability, corresponding to time series in every administrative subdivision.
+The blue solid line represents the average value. All curves are normalized between 0 and 1,
+corresponding to their maximum value.
+In each country, we also collected weekly and daily time series describing movements and
+colocation patterns made available by Meta [41]. We computed contact rates from colocation
+maps (see Material and Methods and the SI for details), which measure the probability that
+two users from two locations are found in the same location at the same time [39]. Colocation
+maps were generated by Meta on a weekly basis, only. To study human movement patterns,
+we considered movement range maps provided by Meta, which report the number of users who
+moved between any two 16-level Bing tiles with an 8 hour frequency [42]. To make colocation
+and movement patterns comparable in terms of scale, we focused on short-range movements,
+i.e. movements that occurred within the same tile, and we separately considered the mid-range
+movements, i.e. movements that occur between two different tiles in the same province.
+4
+
+Country
+Contact Rate
+Movement
+Cases
+27
+2
+2
+20
+2
+DecFigure 2: Illustration of study design. We computed the transfer entropy TEX→Y to measure
+the information flow between source X (on the left) and target time series Y (right), for a given
+time lag l. In the figure example, as target time series we consider the number of COVID-19
+deaths, D(t). As source time series, we consider either mobility indicators, M s(t), M(t), CR(t),
+or COVID-19 cases C(t). Transfer entropy quantifies the amount of information that is added
+by past knowledge of mobility or cases (green and cyan bars, respectively) to current knowledge
+of deaths, with respect to the knowledge of past deaths only (blue bar). After correcting the
+TE for small sample sizes, and normalizing by the reference value represented by the blue bar,
+we finally compare the Normalized Effective Transfer Entropy of mobility and cases (rightmost
+box).
+We then processed the three datasets, starting from their raw form, to aggregate them at
+the NUTS3 resolution and create the time series: M s(t) for the short-range movements, M(t)
+for the mid-range movements and CR(t) for the contact rates. These time series were then used
+as source variables in the information-theoretic analysis. In the remainder of the paper, we will
+generally refer to CR(t), M s(t), and M(t) as mobility time series as they are all derived from
+human mobility data. We will also generally refer to the NUTS3 units as provinces, although
+their nomenclature varies across countries.
+Fig. 2 illustrates our study design based on the transfer entropy [38]. Transfer entropy is
+a metric that measures the directed statistical dependence between a source and a target time
+series and it has been applied to a wide range of research domains [43]. Here, our approach
+consists, first, in computing the transfer entropy between mobility time series, M s(t), M(t) and
+CR(t), and epidemiological time series such as the reported number of COVID-19 attributed
+deaths D(t) and cases C(t), in each administrative unit, and for different temporal lags l, using
+the definition of Shannon entropy, as described by the equations in Fig. 2.
+Intuitively, the
+transfer entropy between mobility and deaths, TEM s→D (resp. TEM→D), can be interpreted as
+5
+
+informationflow
+mobility
+p(Dt+1|Dt)
+small sample correction
+[+↓
+and normalization
+H(DD)
+cases
+p(Dt+1|D)
+-1± 1
+deaths
+H(Dt+1Dl) = Zp(Dt+1, D) 1og
+p(D+1Dthe degree of uncertainty of the reported deaths, D, at time t that is solved jointly by the time
+series of deaths and mobility trends M s (resp. M) and exceeds the current degree of uncertainty
+of D, which can be solved by D’s own past.
+It is known that transfer entropy estimates suffer in case of small sample sizes and non-
+stationarity of the source and target time series [44].
+Moreover, due to the non-parametric
+nature of the transfer entropy, values computed between different source-target time series are
+not directly comparable. To address these issues, we first adopted the definition of effective
+transfer entropy (ETE) [44]. ETE is obtained by subtracting from the original definition of TE
+a reference TE value using a shuffled version of the target time series (see Methods for details),
+thus removing spurious contributions to TE due to fluctuations observed in small sample sizes.
+Also, to address biases due to small sample sizes, we applied a Kernel Density Estimation, before
+the time series discretization that is necessary to compute the transfer entropy.
+Second, we
+normalized the effective transfer entropy by the Shannon entropy of the target variable, defining
+a normalized effective transfer entropy (NETE) [45]. We obtain a metric that is always positive
+when it is statistically significant and whose zero value indicates the absence of information
+transfer between time series. In the remainder of the article, we thus refer to the NETE between
+source X and target Y as our main quantity of interest, using the symbol NX→Y to denote it.
+To better understand the cause-effect relationship between mobility and COVID-19 deaths,
+which are encoded in the value of NM→D ,NM s→D and NCR→D, we compared them against
+the transfer entropy NC→D, where C is the time series of new COVID-19 cases. As the causal
+relationship between the number of cases and deaths is established by definition, we used the
+transfer entropy NC→D as a benchmark to evaluate the added value of mobility indicators to
+predict COVID-19 deaths. As an example, similar values of NM s→D and NC→D would suggest
+knowledge of past COVID-19 incidence encodes a similar amount of information as knowledge
+of past mobility when it comes to predicting future deaths.
+2.2
+The information flow between COVID-19 incidence and deaths
+As previously mentioned, to gauge our transfer entropy analysis framework, we first looked at
+the causal relationship between the incidence of COVID-19 cases and reported death counts. It is
+clearly expected that a major source of information that provides knowledge on future deaths is
+encoded in the time series of past case counts. We used the NETE to quantify such information
+flow.
+Fig. 3 shows the NETE between the weekly time series of COVID-19 cases and deaths in
+the four countries under study.
+In all countries, median values of NC→D increase from lags
+equal to 1 week up to a maximum of around 2-3 weeks, and then decline rapidly beyond the
+3 weeks time lag. This is in line with early estimates of the median time delay between case
+reporting and fatality, which was estimated to range between 7 and 20 days in different countries
+[46, 47]. At lag equal to 2 weeks, the mean relative explanation added by time series of cases
+with respect to deaths – that is how much of D(t) can be explained only by the past knowledge
+6
+
+Figure 3:
+Information flow between COVID-19 incidence and deaths.
+Normalized
+Effective Transfer Entropy (NETE) between COVID-19 weekly reported cases and deaths in
+the NUTS3 administrative subdivisions (provinces) of Austria, France, Italy and Spain. NETE
+is computed for lags ranging from 1 to 8 weeks, on the x-axis. Boxplots are computed on the
+distribution of NETE values of all the administrative subdivisions in each country. The horizontal
+red line marks the value NC→D = 0.
+C(t − l) – is 14% (SD=8) in Spain, 8% (SD=6) in Italy, 7% (SD=5) in Austria, and 6% (SD=5)
+in France. Boxplots computed on the distribution of administrative units in each country show
+a substantial heterogeneity of NETE across regions for lags shorter than 4 weeks. This may
+be partially explained by spatial heterogeneities in case and death reporting, and in testing
+strategies. Also, NC→D values appear to be higher in Spain, with respect to the other countries.
+A transfer entropy analysis of daily time series of COVID-19 cases and deaths displays consistent
+results (see Fig. S1), with NETE values that fall within the same range measured on a weekly
+time scale.
+These results suggest NETE estimates are robust with respect to the time scale at which
+source and target time series are compared. Moreover, it provides a reference value for NETE,
+in terms of orders of magnitude, when the existence of a causal relationship between time series
+is known.
+7
+
+0.4
+0.4
+Austria
+France
+0.3
+0.3
+ 0.2
+ 0.2
+0.1
+0.1
+0.0
+0.0
+1
+2
+3
+4
+5
+6
+1
+8
+1
+2
+3
+4
+5
+6
+7
+8
+Lag (weeks)
+Lag (weeks)
+0.41
+0.41
+Italy
+Spain
+0.3
+0.3
+ 0.2
+ 0.2
+Nc
+Nc
+0.1
+0.1
+0.0
+0.0
+1
+2
+3
+4
+5
+6
+7
+8
+1
+2
+3
+4
+5
+6
+8
+Lag (weeks)
+Lag (weeks)→ C(t)(%)
+→ D(t)(%)
+l (weeks)
+CR(t)
+M(t)
+M s(t)
+CR(t)
+M(t)
+M s(t)
+C(t)
+2
+9
+19
+3
+10
+7
+7
+79
+3
+20
+23
+5
+21
+8
+13
+69
+4
+27
+22
+9
+29
+9
+16
+46
+5
+33
+23
+10
+36
+8
+17
+18
+6
+35
+27
+10
+38
+14
+17
+7
+7
+29
+25
+11
+40
+12
+14
+4
+8
+27
+20
+11
+38
+15
+12
+8
+Table 1: Percentage of statistically significant NETE values across provinces in all
+the countries studied. This table shows the percentage of provinces, in all countries, in which
+the NETE is statistically significant (p < 0.01) for lags (l) from 2 to 8 weeks.
+2.3
+The information flow between mobility traces and COVID-19 dy-
+namics
+Having defined a benchmark of information transfer using NC→D, we measured the information
+flow between behavioral time series of mobility indicators and COVID-19 cases and deaths. Fig. 4
+summarizes the main results of our analysis. Values of NX→D, with X being either short range
+movements, mid-range movements or contact rates, were substantially smaller than NC→D in
+all countries, for any given time lag l. In particular, Fig. 4a allows to compare the distributions
+of NC→D, NCR→D, NM s→D, and NM→D, at the time lag l that maximized the median NETE
+for weekly time series, for all indicators. We found the largest median values of the normalized
+transfer entropy at l = 7 weeks for both contact rates and movements (short-range and mid-
+range). The upper quartile of the NETE distributions derived from the mobility traces generally
+fell below 5%, in all countries, while the lower quartile of NC→D was always above 5%. Also,
+the distributions of normalized transfer entropy computed from movements were much narrower
+and often including the value N = 0 within their interquartile range. Values of NM→C, shown
+in Fig. 4b, display a pattern similar to the normalized transfer entropy from the mobility time
+series to the death time series, with generally low values of NETE in all countries. Compared
+to movement time series, contact rates led generally to relatively higher values of NETE with
+both targets, cases and deaths, as shown in Fig. 4. Our result confirms the additional value
+of measuring contact rates from mobile phone data, with respect to other movement metrics
+[48]. Besides, it shows that short-range mobility within a province had often a limited predictive
+power to capture time trends of COVID-19 spread.
+To obtain a more detailed picture of the predictive power of different mobility metrics in terms
+of NETE, we computed the percentage of provinces for which mobility time series provided
+significant relative information added, with respect to the past knowledge of epidemiological
+8
+
+Figure 4:
+Information flow from mobility data to COVID-19 incidence and deaths.
+Comparison between the normalized effective transfer entropy computed from source time series
+X and target time series of reported COVID-19 deaths D (a) and cases C (b). Source time series
+are COVID-19 cases (only for deaths), contact rates, short range and mid-range movement.
+Boxplots are computed from the distribution of NETE values for a given time delay, l. In panel
+a: l= 2 weeks for cases, 7 weeks for contact rates and movement. In panel b: l= 6 weeks for
+short range and mid-range movement. The horizontal red line marks the value NX→D = 0.
+indicators only (see Tab. 1). On the one hand, our framework effectively captured the existing
+causal relationship between the time evolution of cases counts and the number of deaths, as
+the NETE between these indicators was statistically significant (p < 0.01) in about 80% of the
+provinces, at 2 weeks lag. On the other hand, we observed a statistically significant information
+transfer from mobility time series to epidemiological ones in a much smaller fraction of provinces.
+Short-range movements NETE was significant in less than 20% of provinces when considered as
+a predictor of both cases and deaths. Mid-range movement time series and contact rates were
+significant in at most 27% and 40% of provinces. This means that in most provinces, mobility
+traces did not provide any additional information to predict future COVID-19 cases or deaths,
+at any lag between 2 and 8 weeks.
+Measures of contact rate extracted from colocation maps were more suitable than movement
+9
+
+a
+C
+0.3
+CR
+Ms
+0.2
+M
+XN
+0.1
+0.0
+b
+Austria
+France
+Italy
+Spain
+CR
+0.3
+Ms
+M
+0.2
+C
+个
+XN
+0.1
+0.0
+France
+Italy
+Austria
+Spain
+Country→ C(t)(%)
+→ D(t)(%)
+l (weeks)
+CR(t)
+M(t)
+M s(t)
+CR(t)
+M(t)
+M s(t)
+C(t)
+2
+4 (1)
+4(1)
+4 (0)
+4 (1)
+5(2)
+4 (1)
+11 (6)
+3
+4 (2)
+4(2)
+4 (1)
+5 (2)
+4(1)
+4 (1)
+9 (4)
+4
+5 (2)
+4(1)
+4 (2)
+5 (2)
+4(1)
+5 (2)
+6 (3)
+5
+5 (2)
+4(1)
+5 (2)
+6 (3)
+4(1)
+5 (2)
+5 (2)
+6
+6 (2)
+4(1)
+5 (2)
+6 (3)
+4(2)
+5 (2)
+5 (2)
+7
+5 (2)
+5(1)
+5 (2)
+6 (3)
+5(2)
+6 (3)
+5 (2)
+8
+5 (3)
+5(1)
+5 (2)
+6 (3)
+5(2)
+6 (3)
+4 (1)
+Table 2: NETE results across provinces in all the countries studied. The table shows
+the average relative explanation added by source time series, with respect to past knowledge of
+the target only. Only provinces having a statistically significant NETE are considered. Numbers
+in parenthesis report the standard deviation computed over all provinces for which the NETE
+was statistically significant.
+data to capture behavioral patterns relevant to predict COVID-19 spread.
+By focusing only on those provinces where we could identify a significant information flow
+between mobility traces and COVID-19 indicators, we observe that the averaged relative expla-
+nation added by mobility data with respect to the epidemiological data ranges between 4 − 6%,
+which is about half of the averaged relative explanation added by past knowledge of cases to the
+prediction of future deaths (see Tab. 2 and Figs. S2-S9 in the SI).
+As a sensitivity analysis, we also computed the NETE on a shorter time window, between
+September 2020 and January 2021, to exclude the confounding effect of the introduction of na-
+tionwide vaccination programs. Since in those months all countries adopted mobility restrictions
+to mitigate the fall COVID-19 wave, we expect a stronger relationship between mobility and
+COVID-19 cases. Indeed, during this time frame, the information flow between movement time
+series and COVID-19 cases was consistently higher than in the full study period (see Fig. S10).
+This result indicates that, provided with time series of adequate size, the NETE can effectively
+capture the time-varying relationship between human mobility time trends and COVID-19 dy-
+namics.
+2.4
+Identifying the determinants of mobility data predictive power for
+COVID-19
+Maps of Fig. 5 highlight the spatial heterogeneity of NX→D values observed within the same
+country, Spain, for a given time lag and different source time series (see Figs. S11 - S13 for
+the maps of Austria, France, and Italy). As previously mentioned, NC→D displays higher and
+significant values in most of the country (Fig. 5a), with very few exceptions, while statistically
+10
+
+Figure 5:
+Spatial variations of normalized effective transfer entropy. Maps of NETE
+values computed for different source time series and weekly COVID-19 deaths, in the provinces
+of Spain: (a) source is COVID-19 cases at lag l=2 weeks, (b) source is contact rate at lag l=7
+weeks, (c) source is short-range movement at lag l=7 weeks. (d) source is mid-range movement
+at lag l=7 weeks. Dark grey indicates provinces with non-significant values of NETE (p > 0.01).
+Provinces in white are excluded from our sample.
+significant values of NM s→D are found only in 16 provinces out of 42 (Fig. 5c).
+To better understand the observed heterogeneity in NETE, and identify those features that
+can predict the likelihood to observe a statistically significant information transfer from mobility
+11
+
+b
+a
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Nc-→D
+NcR→D
+d
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Nms→D
+NM-Dp ≥ 0.01
+p <0.01
+precision
+0.64
+0.90
+recall
+0.95
+0.47
+f1-score
+0.77
+0.62
+(a) Movement
+p ≥ 0.01
+p<0.01
+precision
+0.71
+0.92
+recall
+0.95
+0.61
+f1-score
+0.81
+0.74
+(b) Contact rate
+Table 3: Classification performance metrics. Summary of model’s classification performance
+to predict the statistical significance of NETE at the p < 0.01 threshold when the input source
+is short-range movement (a) or contact rate (b) and target variable are COVID-19 deaths.
+to COVID-19 death counts, we resorted to a classification model. Namely, we used a random
+forest classifier to predict when the value NX→D is more likely to be statistically significant, using
+short-range movement and contact rate as source time series. We focused on these two metrics
+as they are quantities measured at the same spatial scale. Moreover, short-range movements
+represent on average 90% or more of all movements within a province (see Table S1). As input
+features to the model, we considered a set of attributes of the provinces in each country. In
+particular, we investigated the effects of population size, province area in square kilometers, the
+density of Facebook users, the number of total cumulative deaths, the ratio between the number
+of commuters traveling from or to the province, and those who live and work there, as reported
+by the census (commuting flow), and the coverage consistency, that is the correlation over time
+between the number of Facebook users sharing their location and the number of Facebook users
+taken into account to compute the colocation maps.
+The results summarized in Tab. 3 show that the model achieves a good overall performance in
+terms of precision and recall, as indicated by f1-scores generally higher than 0.6. In particular, of
+all provinces that are classified by the model as characterized by a statistically significant value of
+NETE, 90% or more display a significant transfer of information, as shown by precision values.
+On the other hand, the model’s recall is close to 0.95 when it comes to identifying provinces
+characterized by a not statistically significant NETE, therefore the model correctly identifies
+95% of those provinces where there is no actual transfer of information between mobility and
+deaths.
+To explore the importance of province features in our classification model, we examined the
+SHAP (SHapley Additive exPlanations) values associated with each, as shown in Fig. 6. SHAP
+is a method based on a game theoretic approach to explaining the output of classification mod-
+els [49]. As expected, the choice of the time lag to compute the NETE is crucial in determining
+the presence of a significant information transfer between mobility metrics and epidemiological
+indicators. Indeed, lag is ranked as the most and second most important feature explaining the
+classification, for contact rate and short-range movement, respectively. Commuting flow is the
+most important predictor of the statistical significance of NETE between short-range movements
+12
+
+(a) Movement
+(b) Contact rate
+Figure 6: SHAP plots of feature importance to predict the statistical significance of
+the NETE for all selected provinces. Color represents the feature value (blue is low and
+red is high). Panel a describes the results for NM s→D, panel b for NCR→D. The SHAP value,
+on the horizontal axis, indicates the feature importance on the model output, with larger values
+corresponding to higher relevance. Each dot represents a single observation. Features are ranked
+by importance.
+and deaths: when the number of commuters leaving or entering a province represents an impor-
+tant fraction with respect to those who remain within the province, the relationship between
+short-range mobility and COVID-19 dynamics gets weaker. However, the same feature has only
+a marginal impact on the NETE between contact rates and deaths, which suggests contact rate
+should be preferred over short-range movements to predict epidemic outcomes when a province is
+characterized by large population inflows/outflows. Province area and population size have also
+a significant impact on the information transfer between short-range movement and COVID-19
+deaths. Indeed, a larger area and population size correspond to a higher likelihood of NETE
+significance for short-range movements.
+This effect may partly explain why we observed NETE values that were statistically significant
+only in a few provinces of Austria, where spatial units were particularly small. When looking at
+the information flow between contact rates and time series of deaths, the total cumulative deaths
+represent an important explanatory variable for the classification model. Besides the analysis
+presented in Fig. 6 suggests that the coverage consistency needs to be sufficiently high in order
+to get a statistically significant transfer entropy from contact rate to deaths. In France, where in
+most provinces the coverage consistency is low and the commuting inflow and outflow are higher
+than in other countries (see Table S2), mid-range movements seem to provide a better alternative
+to contact rates and short-range movements to partially explain time trends of COVID-19 cases
+and deaths (see Fig. S14 of the SI).
+13
+
+High
+flow
+6
+Feature value
+area
+population
+cumulated deaths
+user density
+D.3D.2D.10.D
+0.1
+0.2
+0.3
+0.4
+SHAP value (impact on mpdel output)High
+lag
+cumulated deaths
+coverage consistency
+Feature value
+population
+flow
+area
+Wser density
+D.4D.3D.2D.10.D
+0.1
+0.2
+0.3
+SHAP value (impact on mpdel output)From our analysis, we thus conclude that NETE values computed using contact rates as
+source time series are less sensitive to the province’s geographic or demographic features, rather
+than to the noise of the target time series. Given good coverage, and consistency over time,
+contact rates thus represent a better epidemiological predictor of future COVID-19 deaths than
+short-range movements.
+3
+Discussion
+In this work, we have introduced a novel framework based on transfer entropy to quantify the
+amount of information that is transferred from mobile phone-derived mobility metrics to epi-
+demiological time series. Given the important role that mobility indicators have played in the
+COVID-19 pandemic, we tested our approach on mobility and epidemic time series collected
+in four European countries, between 2020 and 2021, at a subnational scale. We found that, in
+general, the relative explanation added by mobility time series to predict future epidemic trends,
+whether new cases or deaths, was relatively small, ranging between 4% and 6% on average, and
+not statistically significant in the large majority of the provinces we considered, for any mo-
+bility metric. As a comparison, these values were about half of the relative explanation added
+by past knowledge of COVID-19 incidence to predict future deaths. Our method allowed us
+to directly compare the relative explanation added by different mobile phone-derived metrics of
+mobility: short- and mid-range mobility, and contact rates. We generally found a higher informa-
+tion transfer from contact rates than movement, in line with previous studies [48], however, we
+also observed significant heterogeneities within the same country and between countries. With
+a classification model, we identified spatial features that may explain such heterogeneities. In
+provinces characterized by large populations, good coverage consistency over time, and small
+commuting in- and outflows, short-range movements can represent a useful metric to predict
+disease dynamics. Where commuting flows are large, such as in France, and Austria, mid-range
+movements, which represent less than 10% of the total movements, provided a better alternative
+to short-range ones. Our results suggest the choice of the best mobility metric to inform epi-
+demic predictions can depend on a number of different factors, even when using one single data
+provider. Moreover, our findings show that cell phone mobility metrics do not always capture
+epidemiologically-relevant behaviors and alternative data sources could be more effective for this
+aim, as, for instance, the collection of survey data [50].
+There is an emerging common understanding that mobility indicators measured from mobile
+phone data present significant gaps and do not provide a consistent picture of mobility across
+countries, and data providers [51, 52]. Previous studies have also highlighted the fact that cou-
+pling between mobility indicators and COVID-19 epidemiology is often weak, and it changes over
+time [29]. The approach we introduced here addresses the above challenges by providing a general
+framework to evaluate the quality of metrics derived from passively collected mobility traces as a
+predictor of epidemic outcomes. Our framework has the advantage of being model-free, meaning
+14
+
+that it does not depend on modeling assumptions regarding the expected relationship between
+mobility and epidemic dynamics, nor it requires any parametrization. The normalized effective
+transfer entropy we adopted is a general method. It allows us to rigorously compare different
+mobility indicators, across epidemiological settings, by measuring the relative information added
+by mobility time series to the prediction of future disease incidence. To this end, we release the
+code to reproduce our analysis between any two source and target time series (see Data and
+Code Availability). Researchers can use this tool in any epidemiological context to gauge the
+added value of a specific mobile phone-derived behavioral measure for epidemic intelligence.
+Our study comes with a number of limitations and opens new directions for future work. We
+considered mobility metrics derived from one data provider, Meta, whose user base is not rep-
+resentative of the population in the countries we considered. However, alternative data sources
+of mobility indicators in Europe with a similar breadth, such as Google or Apple, do not reach
+the same spatial granularity and provide their data only as relative changes with respect to a
+pre-pandemic baseline, thus limiting their use in a study like ours. On the other hand, movement
+and colocation maps by Meta have been extensively used in several studies, including European
+countries [53, 54, 55, 56, 57]. Here, we considered four countries with different public health
+systems, and that adopted different testing strategies. Observed differences in the predictive
+power of mobility metrics across countries may depend on the varying quality of their reporting
+systems, especially at the province level. However, all four countries belong to the European
+Union and we expect very similar standards of surveillance during the pandemic. Overall, it
+will be important to assess our findings on mobility data from other providers, and, most im-
+portantly, in countries of the non-Western world. Finally, it is important to note that transfer
+entropy measurements become more accurate as the length of the source and target time series
+increases [44]. We worked with a relatively short time series, addressing the bias due to the small
+sample by adopting the effective transfer entropy. However, we could not systematically investi-
+gate how the information transfer changed over time, performing our analysis over different time
+windows and comparing them. Future work could benefit from longer epidemic time series, over
+several years, to identify temporal changes in the information flow between human movements
+and COVID-19 dynamics.
+Measures of human mobility inferred from mobile phone data have been a critical ingredient
+to inform the public health response during the COVID-19 pandemic [58] and they will be an
+important asset in the fight against future pandemics. At the same time, their widespread use
+raises some relevant ethical concerns due to re-identification risks [59], therefore, it is fundamental
+to assess the added value of using cell phone mobility data in a given epidemic scenario and
+whether the benefits outweigh the risks. Our work provides a practical guide to identifying when
+and where mobile phone mobility metrics truly capture behavioral patterns that are relevant to
+predict disease dynamics.
+15
+
+4
+Materials and Methods
+4.1
+Epidemiological indicators
+We collected epidemiological time series in the 4 countries under study from 2 data sources.
+Daily reported cumulative COVID-19 cases were collected from the COVID-19 Data Hub [60],
+an open source aggregator of up-to-date COVID-19 statistics, at the NUTS3 level in Austria,
+France, Italy, and Spain.
+Daily reported cumulative deaths in Austria, France, and Spain were also collected from the
+COVID-19 Data Hub. For Italy, death statistics were only available on a weekly time scale from
+the public platform CovidStat (https://covid19.infn.it/iss/).
+For the analysis, we generated daily incidence time series from cumulative data by computing
+day-to-day differences. Then, we further aggregated the daily time series of deaths and cases
+into weekly ones, to perform the transfer entropy analysis on a weekly scale.
+4.2
+Mobility derived indicators
+In our study, we computed daily and weekly movement and contact rates from data provided
+by Meta through its Data for Good program [41]. Here, we first describe the raw data sources
+provided by Meta and then the data processing we applied to compute the time series for the
+transfer entropy analysis.
+4.2.1
+Raw data sources
+We collected the following datasets that were publicly released by Meta since the beginning of
+the COVID-19 pandemic, in Austria, France, Italy, and Spain:
+• Movement range maps. It reports the number of users who moved between any two
+16-level Bing tiles, with an 8-hour frequency.
+• Users’ population. It reports the number of active users in each tile with an 8-hour
+frequency. The tile resolution is 4800 x 4800 m2.
+• Colocation maps. It estimates the probability that, given any two administrative regions,
+p1 and p2, a randomly chosen user from p1 and a randomly chosen user from p2 are
+simultaneously located in the same place during a randomly chosen minute in a given week
+[39]. The dataset also reports the number of users in p1 and p2.
+• Stay put. It reports for a given administrative region the daily percentage of users staying
+put within a single location, defined at the 16-level Bing tile.
+We formalize the description of the above datasets with the notation described in Table 4:
+16
+
+Dataset name
+Xs,t
+spatial resolution
+temporal resolution
+population users
+N (pop)
+t,h
+t: tile (4800 x 4800 m2)
+h: 8 hour
+movement between tiles
+M(t1,t2),h
+(t1,t2): tile pair (600 x 600 m2)
+h: 8 hour
+colocation probability
+Pp,w
+p: province
+w : week
+colocation users
+N (coloc)
+p,w
+p: province
+w: week
+stay put
+Sr,d
+r: region
+d: day
+Table 4:
+Summary of raw data sources as time series records Xs,t, where s denotes the spatial
+resolution and t the temporal resolution.
+original data
+spatial aggregation
+temporal aggregation
+aggregated data
+name
+N (pop)
+t,h
+�(t ∈ p)
+h interpolation and mean (h ∈ w)
+N (pop)
+p,w
+province population users
+M(t1,t2),h
+� (t1, t2) ∈ p, t1 = t2
+mean (h ∈ w)
+M (within)
+p,w
+within tile province movement
+M(t1,t2),h
+� (t1, t2) ∈ p, t1 ̸= t2
+mean (h ∈ w)
+M (between)
+p,w
+between tiles province movement
+Sr,d
+∀p ∈ r
+r = p
+mean (d ∈ w)
+Sp,w
+province stay put
+Table 5: Aggregation of data sources described in Table 4, to generate our metrics of interest.
+4.2.2
+Aggregation of raw data
+We then processed the raw data sources of Table 4 to obtain a set of time series having the
+same spatiotemporal resolution, that is weekly, at the NUTS3 scale. Results of the aggregation
+process are described in Table 5. More in detail:
+• Province users population. (1) we performed a spatial aggregation by summing the
+population of tiles belonging to province p, thus obtaining a population at a (province,
+hour) level: N (pop)
+p,h
+. (2) we performed a linear interpolation of the temporal gaps that were
+present in N (pop)
+p,h
+(3) we performed a temporal aggregation by averaging in each province,
+the 8h population records within a week.
+• Within tile province movement (1) we first performed a temporal aggregation by
+averaging M(t1,t2),h for each pair (t1, t2) over a week and obtaining M(t1,t2),w (2) we then
+performed a spatial feature joining and assigned each pair (t1, t2) to the corresponding
+provinces (p1, p2) (3) from M(t1,t2),w we obtained a within tile province movement
+M (within)
+p,w
+, that is the sum of movements which occurred in the same province p and within
+the same tile.
+• Between tiles province movement in the pipeline above, from step (3) we obtain a
+between tile province movement M (between)
+p,w
+, that is the sum of movements which
+occurred in the same province p and between two different tiles, (t1, t2). By definition, the
+sum M (between)
+p,w
++ M (within)
+p,w
+represents the total volume of movements in a province, in a
+17
+
+week.
+• Province stay put (1) we performed a temporal aggregation on a weekly scale by perform-
+ing the average and obtaining Sr,w (2) we assign to each province p the regional stay-put
+time series Sr,w such that p ∈ r.
+4.2.3
+Computation of movement and contact rate
+We finally computed our metrics of interest, movement, and contact rates, as follows.
+The
+short-range movement rate is defined as:
+M s
+p,w = M (within)
+p,w
+N (pop)
+p,w
+(1)
+that is the proportion of users who moved within the same tile in a given province, in a given
+week. The mid-range movement rate is defined as:
+Mp,w = M (between)
+p,w
+N (pop)
+p,w
+(2)
+representing the proportion of users who moved between different tiles in a given province, in a
+given week. The contact rate is defined as:
+CR(t)p,w = ˆPp,w · N (pop)
+p,w
+(3)
+where ˆP denotes the colocation probability corrected by a factor that takes into account the
+overestimation of colocation probabilities due to the heterogeneous distribution of users across
+provinces and the presence of a significant fraction of static users in some periods of mobility
+restrictions [55] (see the SI for additional details).
+4.2.4
+Province sample selection
+The population of Facebook users who contribute to the generation of the movement and colo-
+cation time series varies across countries, and it changes over time. Moreover, the metrics of
+movement (short- and mid-range) and colocation, are computed from different users’ samples of
+different sizes: N (pop)
+p,w
+and N (coloc)
+p,w
+, respectively.
+In our analysis, to limit bias that may be caused by the little representativeness of the
+underlying sample of users, we selected NUTS3 regions in the 4 countries, according to the
+following criteria.
+First, we considered only regions where the sample N (pop)
+p,w
+represented at
+least 3% of the census population to guarantee we had at least 500 users in each province.
+Furthermore, we considered only those regions where the two sample sizes N (pop)
+p,w
+and N (coloc)
+p,w
+were always positively correlated over time, during the whole study period.
+We denote the
+Pearson’s correlation of weekly values of N (pop)
+p,w
+and N (coloc)
+p,w
+as coverage consistency.
+After the selection, our analysis includes 47 provinces in Austria, 51 provinces in France, 93
+provinces in Italy, and 42 provinces in Spain, for a total of 233 spatial units.
+18
+
+Given two discrete temporal signals represented as time series X and Y the Transfer Entropy
+(TE) [38] is a measure of the amount of information delivered from X to Y , defined as:
+TEXY = H(Y |Y (l)) − H(Y |Y (l), X(l)) ,
+(4)
+where X(l), Y (l) are respectively the l-lagged time series of X and Y and TEXY is formulated
+as a difference between two conditional entropy terms, where conditional entropy is expressed as
+H(a|b) = H(a, b) − H(b), and H(·) is the Shannon Entropy. Given a discrete time series S, its
+observations can be expressed as the sample {si; i = 1, .., n}, and we obtain the discrete proba-
+bility distribution p(sj). We compute the Shannon Entropy as: H(S) = �
+j p(sj) · log2(p(sj)).
+Thus TEXY can be expressed as:
+TEXY = H(Y, Y (l)) − H(Y (l)) − H(Y, Y (l), X(l)) + H(Y (l), X(l)).
+(5)
+The time series that we consider in our experiments are continuous, therefore they need to be dis-
+cretized before computing TEXY . We employ the Kernel Density Estimation (KDE) for Transfer
+Entropy estimation. KDE method evaluates the entropy terms of Eq.5 from the discretized den-
+sity estimated from each of the four features sets: {(Y, Y (l)), Y (l), (Y, Y (l), X(l)), (Y (l), X(l))}.
+KDE employs a Gaussian kernel for density estimation. Performing tests on synthetic datasets
+of different sizes, we checked this was the method the most adapted to small samples. For the
+selection of the kernel’s bandwidth, we use the Scott method [61]. The continuous density is
+then discretized with a grid obtained by an equal-width discretization of each feature’s density
+domain. We select 20 as the number of bins for each feature’s domain discretization. The dis-
+cretized density is computed with the integral of the continuous probability density functions
+over each grid cell. Concerning the implementation, for TE estimation we use the PyCausality
+Python package (https://github.com/ZacKeskin/PyCausality).
+Effective Transfer Entropy.
+We introduce the Effective Transfer Entropy (ETE) as a cor-
+rection to TE for small sample time series, as originally proposed by [44]:
+ETEXY = TEXY − 1
+Ns
+Ns
+�
+j=1
+TEX ˆ
+Yj ,
+(6)
+where the correction term is obtained by performing Ns iterations of Y shuffling, obtaining ˆYj
+and computing the average of {TEX ˆ
+Yj; j = 1, .., Ns}. In our experiments, we performed 500
+shuffling iterations.
+Normalized Transfer Entropy.
+We would like to employ TE in order to compare a set
+of input signals {Xj; j = 1, .., N} in terms of their Transfer Entropy TEXjY towards a specific
+output Y . From equation 4 we have that TEXjY is evaluated as a difference of conditional entropy
+where the first term H(Y |Y (l)) depends only on target Y . In order to ensure comparability over
+the set {TEXjY ; j = 1, .., N}, we reformulate the difference as a relative difference dividing by
+19
+
+H(Y |Y (l)). Thus the set of inputs are compared according to {TEXjY /H(Y |Y (l)); j = 1, .., N}
+and we refer to TEXY /H(Y |Y (l)) as Normalized Transfer Entropy (NTE).
+Normalized Effective Transfer Entropy.
+By combining the ETE and the NTE we can fi-
+nally introduce the Normalized Effective Transfer Entropy (NETE), which is obtained by dividing
+the ETE by the first conditional entropy term H(Y |Y (l)) as in [62]:
+NETEXY =
+TEXY −
+1
+Ns
+�Ns
+j=1 TEX ˆ
+Yj
+H(Y |Y (l))
+(7)
+In this way, the NETE accounts both for bias in small sample time series and it ensures compa-
+rability between different input sources {Xj} in terms of information transfer to different targets.
+Besides, it enables estimating the percentage of explanation value added with respect to only
+knowing the past of the time series used as a target.
+4.3
+Classification model
+The introduction of the ETE allows associating a p-value, a metric of statistical significance, to
+each NETE value computed between any pair of time series.
+In our study, we investigated a number of explanatory features to better understand why
+in some provinces the NETE could not identify a significant transfer of information between
+mobility time series and epidemiological indicators.
+More specifically, we trained a Random
+Forest classification model to predict the significance of NX→Y at the threshold of p < 0.01, in
+each province under study. The random forest was performed with 100 decision tree classifiers
+on various sub-samples of the dataset and used averaging to improve the predictive accuracy and
+control for over-fitting. The function to measure the quality of a split was the Gini impurity.
+Before applying the random forest, the data were split between training and test sets (30%). To
+compensate for the imbalance of the datasets, we applied a Synthetic Minority Oversampling
+Technique [63] on the test set.
+As input to the classification model we used a set of features that characterize each province:
+1. population size (as reported by the latest available census);
+2. area (in km2);
+3. density of Facebook users (measured as Np,w divided by area);
+4. total cumulative number of reported COVID-19 deaths during the study period;
+5. commuting flow;
+6. coverage consistency;
+20
+
+The commuting flow is defined as the ratio between the total number of daily commuters who
+travel from or to a province and the total number of commuters who work and live in that
+province.
+Commuting data were collected from the latest available census statistics in each
+country. The coverage consistency is the correlation over time between the users’ populations
+N (pop)
+p,w
+and N (coloc)
+p,w
+.
+To quantify the importance of different features in our classification model, we used their
+SHAP (SHapley Additive exPlanations) values [49]. SHAP is a method to explain model pre-
+dictions based on Shapley Values from game theory. In particular, we use TreeSHAP [64], an
+algorithm to compute SHAP values for tree ensemble models, such as the random forest classifier
+of our study.
+5
+Data and code availability
+The data and code to reproduce our analysis are available at: https://zenodo.org/record/
+7464949#.Y6L0CfxKhNg
+6
+Funding
+F.D. gratefully acknowledges support from the CRT Lagrange Fellowships in Data Science for
+Social Impact of the ISI Foundation, where this work was conducted. M.T. and L.G. acknowledge
+the Lagrange Project of the ISI Foundation funded by CRT Foundation. The funders had no
+role in the study design, decision to publish, or preparation of the manuscript.
+7
+Acknowledgements
+We gratefully acknowledge Alex Pompe for his help to understand the details of mobility data
+from Meta.
+8
+Author contributions
+FD collected data, conducted experiments, interpreted the results, made figures, and contributed
+to the writing of the paper.
+MT and LG conceived and designed the study, conducted the
+statistical analysis, interpreted the results, made figures, and wrote the paper. All authors read
+and approved the final version of the manuscript.
+9
+Competing interests
+The authors declare no competing interests.
+1
+
+Supplementary Information
+Correction to the colocation probability
+Colocation maps provided by Meta is defined as the number of colocation events over the number
+of possible events. This, by design, includes interactions between users staying within the same
+tile but not having actual contact with other users. For this reason, we estimate the contact
+rate in each province by removing the contribution due to the users staying put. We explain our
+approach to estimating such contribution in the following. Let us start by writing the original
+colocation probability P as:
+P = E
+N 2
+(8)
+where:
+• E is the number of colocation events within the province
+• N is the number of province colocation users.
+The exact formula should be P =
+E
+N(N−1) but as N is large we approximate it to 8. Let us
+denote R(c) the number of measured colocation events that are due to users who stay put only,
+then the corrected colocation probability should be written in the following way:
+ˆPp,w = E − R(c)
+N 2
+(9)
+We estimate R(c) by using the stay-put probability S, which is the probability of a user staying
+put. Let us call the tile population ratio probability distribution {ftl; t = 1, .., Tl} where T is
+the number of tiles in a province. This gives us an estimate of the contribution of the users who
+stay put to the colocation probability, as:
+R(c) =
+T
+�
+t=1
+N 2 · f 2
+t · S2.
+(10)
+So we rewrite:
+ˆPp,w = P − S2
+Tl
+�
+t=1
+f 2
+tl
+(11)
+We do not have access to the population of the tiles used for the colocation so we make an
+approximation using the population distribution given for each tile with dimensions 4800 m
+× 4800 m. As there are by definition 64 colocation tiles within a single population tile, the
+expression Eq.11 can be formulated as:
+ˆPp,w = Pp,w − S2
+p,w ·
+T
+�
+t=1
+64 ·
+�
+f (p)
+t,w
+64
+�2
+(12)
+where:
+2
+
+M s (%)
+M (%)
+Austria
+99.6 [97.9 – 100]
+0.5 [0.0 - 2.1]
+France
+91.3 [88.2 – 93.7]
+8.8 [6.3 – 11.9]
+Italy
+89.9 [86.4 – 92.8]
+10.2 [7.2 – 13.6]
+Spain
+91.5 [86.1 – 95.1]
+8.5 [4.9 – 13.9]
+Table S6: Relative proportion of mobility components in each country. Each row dis-
+plays the proportion of movements, as a percentage of the total movements within each province,
+that are represented by the short-range mobility (M s(t)) and the mid-range mobility (M(t)).
+Each table entry reports the median value and the IQR, computed over all provinces, and all
+weeks of the study period. Short-range mobility represents the large majority of movements
+within a province, in all countries.
+coverage consistency
+commuting flow
+Austria
+0.64 [0.45–0.79]
+1.05 [0.43–1.69]
+France
+0.32 [0.23–0.43]
+0.30 [0.22–0.51]
+Italy
+0.63 [0.42–0.77]
+0.21 [0.12–0.29]
+Spain
+0.86 [0.68–0.91]
+0.08 [0.05–0.10]
+Table S7: Coverage consistency and commuting flow distributions by country. Each
+table entry reports the median value and the IQR computed over all provinces, in each country,
+considered in the study.
+• f (p)
+t,w = Nt,w
+Np,w ; t ∈ p : tile t population frequency in province p.
+• Nt,w : population at (tile,week) resolution. It is obtained through mean temporal aggre-
+gation of Nt,h over the week interval denoted by w.
+• Np,w : population at (province, week) resolution.
+It is obtained through sum spatial
+aggregation of Nt,w over the tiles belonging to province p.
+• T is the number of tiles 4800 m × 4800 m
+We can introduce the quantity Qp,w as the sum of squared frequencies of the province tile
+distribution Qp,w = �
+t∈p(f (p)
+t,w)2, so that, finally:
+ˆPp,w = Pp,w − S2
+p,w · Qp,w
+64
+(13)
+References
+[1] Ira M Longini Jr.
+A mathematical model for predicting the geographic spread of new
+infectious agents. Mathematical Biosciences, 90(1-2):367–383, 1988.
+3
+
+Figure S1:
+Comparison of NETE values computed on weekly and daily time series.
+NC→D computed between time series data collected on a weekly time scale (bottom row) and a
+daily one (top row). Daily time series were available only for Austria, France and Spain.
+[2] Aidan Findlater and Isaac I Bogoch. Human mobility and the global spread of infectious
+diseases: a focus on air travel. Trends in parasitology, 34(9):772–783, 2018.
+[3] Duygu Balcan, Bruno Gon¸calves, Hao Hu, Jos´e J Ramasco, Vittoria Colizza, and Alessandro
+Vespignani. Modeling the spatial spread of infectious diseases: The GLobal Epidemic and
+Mobility computational model. Journal of Computational Science, 1(3):132–145, 2010.
+[4] Amy Wesolowski, Caroline O Buckee, Kenth Engø-Monsen, and Charlotte Jessica Eland
+Metcalf. Connecting mobility to infectious diseases: the promise and limits of mobile phone
+data. The Journal of infectious diseases, 214(suppl 4):S414–S420, 2016.
+[5] Amy Wesolowski, Nathan Eagle, Andrew J Tatem, David L Smith, Abdisalan M Noor,
+Robert W Snow, and Caroline O Buckee. Quantifying the impact of human mobility on
+malaria. Science, 338(6104):267–270, 2012.
+[6] Lorenzo Mari, Enrico Bertuzzo, Lorenzo Righetto, Renato Casagrandi, Marino Gatto, Igna-
+cio Rodriguez-Iturbe, and Andrea Rinaldo. Modelling cholera epidemics: the role of water-
+ways, human mobility and sanitation. Journal of the Royal Society Interface, 9(67):376–388,
+2012.
+4
+
+0.40
+0.40
+0.40
+Austria
+France
+Spain
+0.35
+0.35
+0.35
+0.30
+0.30
+0.30
+0.25
+0.25
+0.25
+0.20
+0.20
+→D
+0.20
+→D
+-ON
+0.15
+0.15
+0.15
+0.10
+0.10
+0.10
+0.05
+0.05
+0.05
+0.00
+0.00
+0.00
+-0.05
+-0.05
+-0.05
+14 21  28 35 42 49 56
+14 21 28 35 42 49 56
+14 21 28 35 42 49 56
+7
+Lag (days)
+Lag (days)
+Lag (days)
+0.40
+0.40
+0.40
+France
+Austria
+Spain
+0.35
+0.35
+0.35
+0.30
+0.30
+0.30
+0.25
+0.25
+0.25
+0.20
+0.20
+0.20
+D
+D
+D
+个
+个
+个
+0.15
+0.15
+0.15
+0.10
+0.10
+0.10
+0.05
+0.05
+0.05
+0.00
+0.00
+0.00
+-0.05
+-0.05
+-0.05
+2
+2
+3
+4
+6
+4
+5
+6
+7
+8
+2
+4
+7
+8
+3
+5
+6
+8
+1
+7
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+10
+
+Figure S2:
+NETE values from contact rates to deaths in Austria. Only statistically
+significant values are shown (p-value< 0.01).
+11
+
+Amsteten -
+0.025748
+0.026068
+0.035190
+Brauneu ami Inn -
+0.030781
+Bregenz -
+0.032432
+0.063635
+0.079792
+0.073301
+0.051136
+Bruck-Murzzschlag -
+0.048734
+0.047809
+Dornbim -
+0.042606
+0.048633
+0.028192
+0.037263
+0.055011
+0.036085
+0.038355
+0.036506
+E998E0'0
+0.037540
+0.030327
+0.028564
+0.040716
+Grez [Stadt] -
+0.037779
+0.076651
+0.097150
+0.055744
+Grez-Urngebung -
+0.029600
+0.064492
+0.077311
+Innshruck-Land -
+0.028612
+0.045289
+0.052419
+0.039591
+0.200
+Jerner'sdorf -
+0.035306
+0.069850
+0.113048
+0.068616
+Kirchdorf an der Krem8-
+0.065064
+0.080814
+Kitzbuhel -
+0.031631
+0.068131
+0.175
+Klagerfurt am Worthersee (Sadi) 
+0.045755
+0.073913
+0.069748
+Korreuburg -
+0.048235
+0.064034
+0.067645
+0.080225
+0.071908
+0.062741
+Kufstein ,
+0.084524
+0.078949
+0.050947
+0.036189
+0.028105
+0.040762
+0.043702
+Landeck -
+0.047941
+0.106637
+0.088543
+Leibnibz -
+0.026406
+0.125
+Leoben -
+0.029500
+Province
+Liezen -
+0.033578
+DOL'O -
+0.031302
+0.030106
+0.035957
+Midling -
+0.029218
+ETEEEO'O
+0.026258
+Neunkirchen -
+0.040286
+0.075
+Reutte -
+0.045268
+0.054860
+0.112349
+Ried in Innkreis -
+0.032220
+0.041101
+0.069890
+0.060479
+0.041664
+0.024491
+0.050
+Rahrbach -
+0.041602
+Sabzburg (Stadt) -
+0.043210
+0.072274
+0.075024
+0.073098
+0.042898
+Sazburg-Umgebung -
+0.041488
+0.032673
+0.025
+Sankt jahann im Pongau -
+0.034971
+0.068717
+SchwEz -
+0.038691
+0.091690
+0.184757
+ 0.000
+Spittal an der Drau -
+0.030718
+0.035796
+0.060162
+Steyr-Land -
+0.056514
+0.062967
+0.069762
+0.067400
+0.066843
+0.064067
+0.055595
+Urfahr-Umgebung -
+0.029842
+0.033754
+0.041654
+vbitsberg -
+0.042787
+0.043992
+0.046677
+vbcklabruck -
+0.045316
+EETEEOO
+0.087363
+0.054894
+Weiz -
+0.025742
+Webs (Stadt) -
+E660E0'0
+0.025064
+Wen -
+0.032377
+0.046296
+0.042533
+0.042002
+-1
+2
+m
+4
+6
+8Figure S3:
+NETE values from contact rates to deaths in France. Only statistically
+significant values are shown (p-value¿0.01).
+12
+
+Areyron -
+0.040657
+0.047058
+0.056892
+Bas- Rhin -
+0.033482
+0.041585
+0.028520
+BDpBA(E0
+0.031260
+Carital -
+0.024933
+0.058974
+0.043374
+Charente -
+0.025404
+TZZ8E00
+0.040474
+Creue -
+0.044353
+0.066560
+0.109015
+0.097901
+0.046231
+Cite-"or -
+0.050804
+0.083634
+0.101878
+0.112566
+0.057215
+Douhs -
+0.048890
+0.081594
+0.061416
+0.043711
+0.069546
+568E00
+Gard -
+0.035540
+0.041388
+0.028707
+0.200
+Haut-Rhin -
+0.029346
+0.032140
+0.031260
+0.059106
+TZTES00
+ZLTEEO0
+Haute-Garonne -
+0.029910
+0.039255
+0.037786
+0.036558
+Haute-Loire -
+0.034223
+EOLTEO'0
+0.175
+Haute-Marne 
+0.029728
+EZ9EE00
+Haute-Sevbie 
+0.040045
+0.077211
+0.057614
+9698E00
+0.023792
+0.151
+Haute-Seonie -
+Haute-Vienre -
+0.029789
+Hautes Pyrinees -
+0.046039
+0.052797
+0.073291
+0.087177
+0.118157
+0.097759
+0.079092
+0.125
+0.024660
+0.027809
+- 22]
+0.055111
+660600
+0.116728
+0.074720
+6580E0'0
+Province
+- Eunr
+0Z60E00
+EEL6Z00
+DOL'0
+Landes -
+0.056405
+0.091173
+0.090549
+Lair-et-Cher -
+0.027844
+5E68200
+Laire -
+0.054201
+0.085214
+0.105839
+0.095344
+0.069573
+0.075
+Laire- Atlantique -
+0.029233
+0.029794
+Lairet -
+0.025703
+0.031137
+Lat-et-Garcne -
+0.062904
+0.070896
+0.043991
+0.050
+Maine-et-Loire -
+0.034822
+0.046020
+0.048480
+Marne -
+0.025154
+Mayenne -
+0666E00
+TES9E00
+- 0.025
+0.025244
+0.049144
+0.053130
+0.042279
+Moselle -
+0.026739
+0.034717
+Nievre -
+0.022348
+0.00
+Nard -
+0.036407
+0.045561
+0.040334
+0.051478
+0.077228
+0.063907
+0.059029
+TZOEO0
+Pyrenees Alantiques -
+0.062096
+0.092766
+0.145174
+0.132909
+0.102827
+0.042085
+0.059519
+0.055424
+Savaie -
+0.034982
+0.060054
+0.057195
+Saane-et-Lnire -
+0.031605
+0.065657
+0.092507
+EOLO0
+0.038865
+0.039947
+arn-et-Garanne -
+0.045909
+8066L00
+0.109730
+0.065963
+vauckuse -
+0.026052
+L9TOE00
+Wheges -
+0.027052
+0.059475
+0.073040
+bnne -
+0.028840
+i
+2
+3
+4
+-5
+6
+1
+80Figure S4: NETE values from contact rates to deaths in Italy. Only statistically signifi-
+cant values are shown (p-value< 0.01).
+13
+
+Agigento -
+9880
+Alesaandria -
+EESZO0
+0.026729
+0.036095
+EESE00
+6006200
+SSSt00
+ZE6600
+E09ZE00
+Aeti -
+9218200
+0.037452
+Beri -
+0.024441
+E6S00
+80t9200
+81E8Z00
+ 66TZE00
+SLZOEO0
+0.047656
+886TS00
+E8L00
+0.047959
+0.051687
+0.043068
+0.032656
+E0
+0.026679
+Billa -
+ESE6E00
+00
+0.058615 
+EES00
+0.049609
+Bolpgne -
+0.031410
+0.034686
+0.025180
+Brindisi 
+0.022450
+8Tt000
+2666200
+0.040166
+0.060407
+0.081030
+Caserta -
+S6E00
+SEtOSO0
+0.054700
+EOESSO0
+LLE6500
+8L85E00
+ Catenia -
+60TE00
+0.077414
+0.112635
+0.117515
+0107535
+0.089910
+0.076614
+Caterzaro -
+0.034300
+E200
+Cornn -
+8TO00
+0.047368
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+- I iden
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+0.048102 
+ES6SE00
+Padova -
+0.040434
+0.044515
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+0.074209
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+0.056965
+0.075
+ Palerma -
+0.028142
+Parma -
+0.026819
+0.025800
+0.024981
+Perugia -
+0.042104
+ESE800
+T6ESt00
+0.044093 
+LL68E00
+EZOSZO0
+Pesaro e Uraind -
+LE00
+TZ9SE0 0
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+0.029094
+6ET00
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+0.034775
+L6SS00
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+0.040556
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+0.029401
+0T0850 0
+#S690 0
+0.056462
+88T8E00
+0.037469
+Paternza -
+0.038844
+- 0.025
+- snled
+OE00
+0.039405
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+Reggio di Calabria
+0.036117
+0.047436
+ 6E00
+0589E00
+SOtTE00
+T6TZE00
+0.046939
+Regi nleia -
+E00
+0.038054 
+0.042878
+0.037887
+0.023446
+- 0.00
+- n
+E08E600
+0.102254
+0.085088
+0.047687
+Rarma -
+0.043376
+0.052379
+0.044143
+6600
+Sracun -
+0.028196
+0.041731
+LS6E00
+0.037220
+Eranto -
+0.026919
+0.028592
+990TE00
+81 6200
+Erni -
+ETts00
+0
+0.058842
+0.054448
+9T95E00
+- auyg
+86E6E00
+EZ6t00
+9STES00
+65E00
+Tapani -
+186500
+0.069576
+0103436
+0.106727
+TEZ900
+Tentb -
+Z06SZ00
+S6S9E00
+0.047467
+0.040554
+Teva -
+T969Z00
+606E00
+0.047955
+0.052642 
+90ELt00
+- Bun
+8868200
+Varese.
+0.047314
+876 800
+0.093084
+TS95600
+0.081083
+EE9EE00
+vemezi 
+0.034065
+LE00
+OOEZEOO
+SBZTEO0
+EGEb00
+056t00
+0.031813
+8060500
+LEZ600
+0.078072
+0.104887
+0.111805
+Micenzi -
+8680200
+BELZ00
+L6TEE00
+0.033232
+Miterbo -
+TS6E00
+0.072240
+50L8600
+E9ETO
+0.087438
+ES6tE00
+1
+2
+3
+4
+5
+6
+8Figure S5:
+NETE values from contact rates to deaths in Spain.
+Only statistically
+significant values are shown (p-value< 0.01).
+14
+
+Amerfa -
+0.039917
+Aturias -
+0.031993
+Bertcekang
+0.064487
+0.049886
+0.035688
+Bizkain -
+0.024587
+0.022016
+0.042866
+0.042032
+Caritabrin -
+0.046025
+0.068889
+0.068396
+0.036596
+Castellon -
+0.048737
+0.063084
+0.051995
+0.036042
+0.200
+Ciudad Real -
+0.044180
+0.054436
+0.030186
+0.175
+Cirdaba -
+0.032927
+0.038618
+0.065027
+0.036654
+0.15
+GipuzkD8 -
+0.031801
+0.053539
+0.048215
+0.026072
+0.125
+Grenada -
+0.032374
+0.081161
+0.103030
+0.079209
+0.032532
+Province
+Huelva -
+0.033040
+0.100
+Huesch -
+0.046110
+0.031084
+0.075
+La Rinja -
+0.028844
+Lerida -
+0.033187
+0.041296
+DSIO -
+Madrid 
+0.043627
+0.037621
+ 0.025
+Murcia -
+0.045269
+0.044486
+Neverre -
+0.027031
+0.033968
+0.027792
+ 0.000
+Palencia -
+0.030504
+Pontevedra -
+6008E0'0
+Sevill -
+0.037439
+0.042436
+Earrapone -
+0.045793
+0.079908
+BzaBeJZ
+0.046435
+1
+-2
+-3
+-4
+-5
+15
+ 00Figure S6:
+NETE values from movements to deaths in Austria.
+Only statistically
+significant values are shown (p-value< 0.01).
+Figure S7: NETE values from movements to deaths in France. Only statistically signif-
+icant values are shown (p-value< 0.01).
+15
+
+Bregenz -
+0.036345
+0.066033
+Dornbim -
+0.043640
+0.067243
+ Eferding -
+0.035090
+Feldkirch -
+0.037236
+ Freistadt -
+0.038373
+0.047605
+0.048796
+0.055328
+Gmunden -
+0.031248
+Gmind -
+0.023410
+0.030299
+0.027167
+0.029017
+0.200
+ Hallein -
+0.027033
+0.034264
+0.027311
+Imst -
+0.029552
+0.175
+Innsruck-Stadt -
+0.055188
+0.080702
+0.094913
+0.085118
+0.151
+Kitzbthel -
+0.042733
+0.072762
+Krema an der Dcnau (Stadty) -
+0.036128
+0.048300
+0.055608
+0.125
+Kufstein -
+0.032149
+0.043879
+Province
+Landeck -
+0.029311
+0.040050
+0.072866
+0.061492
+DOLO-
+Liezen -
+0.025763
+0.048718
+0.034108
+0.D75
+Linz-Land -
+Rahrbach -
+0.036758
+0.044387
+0.050
+Sankt Polten (Stadt) -
+0.034767
+0.041224
+0.042029
+0.043476
+0.044951
+0.038877
+Schwaz -
+0.039763
+0.054913
+0.D25
+Scharding -
+0.032939
+0.034910
+0.042464
+0.041187
+ 0.000
+Steyr (Stadt) -
+Steyr-Land -
+0.035290
+0.035265
+0.039354
+TEamaweg -
+0.060463
+Urfahr-Umgebung -
+0.041795
+0.054897
+Waidhofen an der Ybba (Stadt] -
+0.030619
+0.037978
+0.037335
+Wien -
+0.021074
+0.022951
+Zell em See -
+0.034330
+-
+2
+1
+m
+-+
+-5
+-9
+-1
+1
+-8Haute-Garonne -
+0.039198
+Herault -
+0.035912
+0.20
+Landes -
+0.024899
+0.047341
+0.15
+g Pyrenees Aantiques -
+0.040966
+0.081955
+0.10
+Pyrenees-Orienbales -
+0.039209
+0.05
+- 2S1O.PH9n
+0.027723
+ 0.D0
+var -
+0.037235
+vendee -
+0.035500
+0.035709
+2
+1
+m
+4
+5
+1
+1
+LpFigure S8: NETE values from movements to deaths in Italy. Only statistically significant
+values are shown (p-value< 0.01).
+16
+
+Agnigento -
+0.035274
+0.060209
+0.067456
+0.063766
+0.045358
+Aosta -
+0.029482
+0.042370
+0.072343
+0.082323
+BolpgnB -
+0.030139
+0.028636
+Brescin -
+0.025976
+Brindisi -
+0.026128
+Crobne -
+0.030466
+0.040989
+0.058785
+0.075694
+0.055939
+Faggia -
+0.021785
+Grsseto -
+0.022906
+0.038876
+0.038155
+0.200
+Isernia -
+0.030238
+0.032921
+L Spezia -
+0.047417
+0.103764
+0.175
+Lecce -
+0.036225
+0.036709
+0.036007
+Livane -
+0.029951
+0.029072
+0.150
+Metera -
+0.038031
+0.043266
+0.029986
+Miland -
+0.038485
+0.060374
+0.090189
+0.111951
+0.119138
+0.076202
+0.043542
+0.125
+ Napoli -
+0.032169
+0.053370
+0.077585
+0.081536
+0.069873
+Province
+Palerrma -
+0.040487
+0.055997
+0.049803
+0.034437
+0.100
+Parmia -
+0.020899
+SEEEEOO
+0.045348
+Pescara -
+0.035626
+0.053516
+0.048144
+0.075
+Fisa -
+0.036115
+0.046651
+0.048108
+0.042774
+Ravenng -
+0.032001
+0.050158
+0.054213
+0.044289
+0.023382
+ 0.050
+Reggio nel'Emilia -
+0.036432
+Salermo -
+0.027822
+0.024835
+0.026877
+- 0.025
+Saana -
+0.021183
+Sandrio -
+0.025698
+ 0.000
+Earanto -
+0.041633
+0.042189
+0.028624
+0.027444
+0.033665
+0.035389
+Tevisn -
+0.026159
+0.020877
+vercelli -
+0.035478
+Vibo Valentia -
+0.028121
+0.044249
+0.055201
+0.037209
+0.028612
+Micerzr -
+0.022276
+0.022803
+viterbo -
+0.047049
+-T
+2-
+4
+5
+-9
+-L
+LpFigure S9: NETE values from movements to deaths in Spain. Only statistically significant
+values are shown (p-value< 0.01).
+Figure S10:
+Comparison of NETE values computed on full time series and reduced
+time series. NM→C computed between time series data collected including the vaccination
+campaign (full) and not (reduced). The reduced study period ranges from September 1, 2020 to
+January 31, 2021. The full study period extends up to July 31, 2021. We consider daily time
+series only to address biases due to small samples.
+17
+
+A Coruia -
+0.035725
+0.036865
+Balearea -
+0.068768
+0.044550
+Bercekana -
+0.039459
+0.035079
+0.030976
+Bizkain -
+0.027029
+0.028069
+Burgos -
+0.029288
+0.200
+Caritabrin -
+0.033294
+0.047135
+0.056197
+0.039901
+Castell6n -
+0.037745
+0.175
+0.030562
+0.037700
+- Bauar
+0.043924
+0.076964
+0.088080
+0.037164
+0.15
+Oceres -
+0.030211
+0.125
+Province
+0.055756
+DOL'O-
+Grenada -
+0.033007
+Guadalajara -
+0.034102
+0.036284
+0.075
+Lein -
+0.037108
+0.050
+Lugo -
+0.073411
+0.049612
+0.038789
+ 0.025
+Lerida -
+0.039123
+0.056995
+0.052213
+0.041481
+0010 -
+Palencia -
+0.049664
+0.033181
+0.034299
+0.037105
+Pontevedra 
+0.052307
+0.036459
+Segpvin -
+0.033276
+0.054284
+0.059599
+0.050507
+0.035978
+ pa
+0.047631
+0.040222
+0.037340
+valencia -
+0.057275
+0.040704
+-+
+-5
+2
+-9
+-8
+1
+LBp0.10
+0.08
+NMr-→c (reduced)
+0.06
+0.04
+0.02
+Austria
+France
+0.00
+Spain
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+NMr→c (full)Figure S11: Spatial variations of normalized effective transfer entropy. Maps of NETE
+values computed for different source time series and weekly COVID-19 deaths, in the provinces
+of Austria: (a) source is COVID-19 cases at lag l=2 weeks, (b) source is contact rate at lag l=7
+weeks, (c) source is short-range movement at lag l=7 weeks. (d) source is mid-range movement
+at lag l=7 weeks. Dark grey indicates provinces with non-significant values of NETE (p > 0.01).
+Provinces in white are excluded from our sample.
+18
+
+b
+a
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Nc→D
+NcR→D
+C
+0.25
+0.25
+0.00
+0.05
+0.10
+0.15
+0.20
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.30
+0.35
+Nms→D
+NM→DFigure S12: Spatial variations of normalized effective transfer entropy. Maps of NETE
+values computed for different source time series and weekly COVID-19 deaths, in the provinces
+of France: (a) source is COVID-19 cases at lag l=2 weeks, (b) source is contact rate at lag l=7
+weeks, (c) source is short-range movement at lag l=7 weeks. (d) source is mid-range movement
+at lag l=7 weeks. Dark grey indicates provinces with non-significant values of NETE (p > 0.01).
+Provinces in white are excluded from our sample.
+19
+
+b
+a
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Nc→D
+NcR→D
+d
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.00
+0.30
+0.35
+Nms-D
+NM→DFigure S13: Spatial variations of normalized effective transfer entropy. Maps of NETE
+values computed for different source time series and weekly COVID-19 deaths, in the provinces
+of Italy: (a) source is COVID-19 cases at lag l=2 weeks, (b) source is contact rate at lag l=7
+weeks, (c) source is short-range movement at lag l=7 weeks. (d) source is mid-range movement
+at lag l=7 weeks. Dark grey indicates provinces with non-significant values of NETE (p > 0.01).
+Provinces in white are excluded from our sample.
+20
+
+b
+a
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Nc→D
+NcR→D
+d
+C
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.00
+0.30
+0.35
+Nms-D
+NM-DFigure S14: Percentage of statistically significant NETE values, disaggregated by
+country and by mobility metric used as source variable. Target variables are: weekly
+COVID-19 cases (panel a) and weekly COVID-19 deaths (panel b).
+21
+
+b
+a
+60
+CR
+Ms
+30
+M
+40
+20
+E 20
+NETI
+10
+0
+0
+Austria
+Italy
+Austria
+Italy
+France
+Spain
+France
+Spain
\ No newline at end of file
diff --git a/1dE2T4oBgHgl3EQfigf6/content/tmp_files/load_file.txt b/1dE2T4oBgHgl3EQfigf6/content/tmp_files/load_file.txt
new file mode 100644
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+page_content=' 5  cours des Humanit´es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' F-93 322 Aubervilliers Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' France  †these authors contributed equally to this work  Abstract  Mobile phone data have been widely used to model the spread of COVID-19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' however,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  quantifying and comparing their predictive value across different settings is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Their quality is affected by various factors and their relationship with epidemiological in-  dicators varies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Here we adopt a model-free approach based on transfer entropy  to quantify the relationship between mobile phone-derived mobility metrics and COVID-  19 cases and deaths in more than 200 European subnational regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We found that past  knowledge of mobility does not provide statistically significant information on COVID-19  cases or deaths in most of the regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In the remaining ones, measures of contact rates  were often more informative than movements in predicting the spread of the disease, while  the most predictive metrics between mid-range and short-range movements depended on the  region considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We finally identify geographic and demographic factors, such as users’  coverage and commuting patterns, that can help determine the best metric for predicting  disease incidence in a particular location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Our approach provides epidemiologists and public  health officials with a general framework to evaluate the usefulness of human mobility data  in responding to epidemics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  1  Introduction  The relationship between human movements and the spatial spread of infectious diseases has  been recognized for a long time [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Human movement has been shown to play a key  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='03960v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='soc-ph]  10 Jan 2023  role in the dynamics of several pathogens, through two basic mechanisms: traveling infectious  individuals may introduce a pathogen in a susceptible population, and, at the same time, human  movement increase the contact rate between individuals, creating new opportunities for infection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In the past 15 years, the increasing availability of mobility data derived from mobile phones has  fueled a large body of work aimed at identifying opportunities to use them for infectious disease  modeling and surveillance [4, 5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  More recently, during the COVID-19 pandemic, mobile phone-derived data have been exten-  sively harnessed to monitor the effect of non-pharmaceutical interventions (NPIs) across coun-  tries, understand the early dynamics of COVID-19 diffusion, and forecast its spread at different  spatial scales, from countries to cities [11, 12, 13, 14, 15, 16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' By measuring human move-  ments and combining them with phylogeography methods [18, 19], several studies shed light on  the cryptic spread of new variants, their persistence over time and resurgence after the relaxation  of NPIs [20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Human mobility has been shown to strongly correlate with the spread of COVID-19 during  the early phase of the outbreak in China and in many other countries [23, 24, 25, 26, 27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  However, once COVID-19 established a foothold in a population, the relative importance of  mobile phone-derived data to predict the epidemic dynamics on a local scale has been generally  less understood and several studies have shown conflicting evidence about the use of mobility  traces to model the spread of COVID-19 at later stages of the outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' For instance, it has been  shown that the explanatory power of mobility metrics in relation to the case growth rate in the  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=', significantly declined in spring 2020, especially in rural areas [29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Similar trends  have been observed in Europe [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In parallel, mobile phone-derived data have been proven  beneficial to model COVID-19 dynamics in largely populated urban areas of Western countries  [33, 34], but less so in countries of the Global South [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Several reasons have been proposed to explain the varying relationship between mobility  metrics and epidemic indicators [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Mobility metrics are generally derived from raw mobile  positioning data through complex and customized processing pipelines that can significantly  vary across data providers [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  How raw data are processed, and the specific definitions of  mobility metrics can significantly impact their interpretation with respect to epidemic variables  [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Moreover, the relationship between mobility and epidemic patterns often relies on model-  ing assumptions, typically considering linear dependencies, that may not capture the complex  interplay of these quantities [32, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Finally, mobile phone-derived metrics are generated from  a sample of users who is generally not representative of the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It is therefore of  paramount importance to define standardized approaches that can quantify the added value of  mobility metrics for epidemiological analysis, and make different metrics, across settings, directly  comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Here, we extensively quantify the relationship between cell phone-derived mobility metrics and  COVID-19 epidemiological indicators through a model-free approach, based on an information-  theoretic measure, transfer entropy [38], adapted for small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Leveraging granular  2  data provided by Meta that capture both users’ movements and colocation at a fine spatial scale  [39], we measure the information flow between mobility metrics and time series of COVID-19  incidence and deaths in four European countries, at a subnational scale, over a one year period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  We find that the relative information added by the past knowledge of mobility metrics to the  knowledge of the current state of COVID-19 time series is often not statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In statistically significant cases instead, we show that the relative information added by  past knowledge of COVID-19 cases to the knowledge of current deaths is twice the information  flow between past knowledge of mobility metrics and current deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We also show that the  information flow of a given mobility metric to predict future COVID-19 incidence or deaths can  be significant in one country but not in another, even if derived from the same original data  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Being a general framework, our approach provides a quantitative measure of the relative  added explanation brought by mobile phone data to the prediction of epidemiological time series  that does not depend on the choice of a specific forecasting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It thus helps to better identify  the most appropriate mobility metrics to use among those available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Our results can thus guide  epidemiologists and public health practitioners in the evaluation of mobile phone-derived mobility  metrics when they are interpreted as a precursor of epidemic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  2  Results  Here, we first describe and then apply our framework to measure the information flow between  human mobility traces and the time evolution of COVID-19 in four European countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  A transfer entropy approach to link mobility behavior and COVID-  19 epidemiology  With the aim of quantifying the information flow from mobility-derived data to COVID-19 data,  we first gathered a set of mobility and epidemiological indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 1 provides an overview  of the datasets used in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In Materials and Methods, we provide a full description of all  data sources and the data processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We considered four European countries – Austria,  France, Italy, and Spain – and their administrative subdivisions at NUTS3 level [40] which is  the lowest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' the most granular, level of the standard hierarchy of administrative regions in  Europe (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 1, leftmost column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In all administrative regions, we collected indicators of the COVID-19 epidemic dynamics,  namely, the weekly and daily numbers of new COVID-19 cases and deaths over the period, from  September 2020 until July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' During this period, the dynamics of COVID-19, exemplified by  the incidence of new cases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 1, rightmost column), displayed subsequent waves, as a result of  the complex interaction between the spread of new variants, the adoption of non-pharmaceutical  interventions, the introduction of vaccines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  3  Figure 1:  Summary of behavioral and epidemiological indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In each country  under study (from top to bottom: Italy, France, Austria and Spain), we consider three different  types of indicators: contact rates, movements (here for the sake of simplicity we only show the  short-range movements), and COVID-19 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In each plot, the blue shaded area highlights  the within-country variability, corresponding to time series in every administrative subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  The blue solid line represents the average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' All curves are normalized between 0 and 1,  corresponding to their maximum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In each country, we also collected weekly and daily time series describing movements and  colocation patterns made available by Meta [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We computed contact rates from colocation  maps (see Material and Methods and the SI for details), which measure the probability that  two users from two locations are found in the same location at the same time [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Colocation  maps were generated by Meta on a weekly basis, only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' To study human movement patterns,  we considered movement range maps provided by Meta, which report the number of users who  moved between any two 16-level Bing tiles with an 8 hour frequency [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' To make colocation  and movement patterns comparable in terms of scale, we focused on short-range movements,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' movements that occurred within the same tile, and we separately considered the mid-range  movements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' movements that occur between two different tiles in the same province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4  Country  Contact Rate  Movement  Cases  27  2  2  20  2  DecFigure 2: Illustration of study design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We computed the transfer entropy TEX→Y to measure  the information flow between source X (on the left) and target time series Y (right), for a given  time lag l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In the figure example, as target time series we consider the number of COVID-19  deaths, D(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As source time series, we consider either mobility indicators, M s(t), M(t), CR(t),  or COVID-19 cases C(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Transfer entropy quantifies the amount of information that is added  by past knowledge of mobility or cases (green and cyan bars, respectively) to current knowledge  of deaths, with respect to the knowledge of past deaths only (blue bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' After correcting the  TE for small sample sizes, and normalizing by the reference value represented by the blue bar,  we finally compare the Normalized Effective Transfer Entropy of mobility and cases (rightmost  box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  We then processed the three datasets, starting from their raw form, to aggregate them at  the NUTS3 resolution and create the time series: M s(t) for the short-range movements, M(t)  for the mid-range movements and CR(t) for the contact rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' These time series were then used  as source variables in the information-theoretic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In the remainder of the paper, we will  generally refer to CR(t), M s(t), and M(t) as mobility time series as they are all derived from  human mobility data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We will also generally refer to the NUTS3 units as provinces, although  their nomenclature varies across countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 2 illustrates our study design based on the transfer entropy [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Transfer entropy is  a metric that measures the directed statistical dependence between a source and a target time  series and it has been applied to a wide range of research domains [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Here, our approach  consists, first, in computing the transfer entropy between mobility time series, M s(t), M(t) and  CR(t), and epidemiological time series such as the reported number of COVID-19 attributed  deaths D(t) and cases C(t), in each administrative unit, and for different temporal lags l, using  the definition of Shannon entropy, as described by the equations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Intuitively, the  transfer entropy between mobility and deaths, TEM s→D (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' TEM→D), can be interpreted as  5  informationflow  mobility  p(Dt+1|Dt)  small sample correction  [+↓  and normalization  H(DD)  cases  p(Dt+1|D)  1± 1  deaths  H(Dt+1Dl) = Zp(Dt+1, D) 1og  p(D+1Dthe degree of uncertainty of the reported deaths, D, at time t that is solved jointly by the time  series of deaths and mobility trends M s (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' M) and exceeds the current degree of uncertainty  of D, which can be solved by D’s own past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  It is known that transfer entropy estimates suffer in case of small sample sizes and non-  stationarity of the source and target time series [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Moreover, due to the non-parametric  nature of the transfer entropy, values computed between different source-target time series are  not directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' To address these issues, we first adopted the definition of effective  transfer entropy (ETE) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' ETE is obtained by subtracting from the original definition of TE  a reference TE value using a shuffled version of the target time series (see Methods for details),  thus removing spurious contributions to TE due to fluctuations observed in small sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Also, to address biases due to small sample sizes, we applied a Kernel Density Estimation, before  the time series discretization that is necessary to compute the transfer entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Second, we  normalized the effective transfer entropy by the Shannon entropy of the target variable, defining  a normalized effective transfer entropy (NETE) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We obtain a metric that is always positive  when it is statistically significant and whose zero value indicates the absence of information  transfer between time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In the remainder of the article, we thus refer to the NETE between  source X and target Y as our main quantity of interest, using the symbol NX→Y to denote it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  To better understand the cause-effect relationship between mobility and COVID-19 deaths,  which are encoded in the value of NM→D ,NM s→D and NCR→D, we compared them against  the transfer entropy NC→D, where C is the time series of new COVID-19 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As the causal  relationship between the number of cases and deaths is established by definition, we used the  transfer entropy NC→D as a benchmark to evaluate the added value of mobility indicators to  predict COVID-19 deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As an example, similar values of NM s→D and NC→D would suggest  knowledge of past COVID-19 incidence encodes a similar amount of information as knowledge  of past mobility when it comes to predicting future deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  The information flow between COVID-19 incidence and deaths  As previously mentioned, to gauge our transfer entropy analysis framework, we first looked at  the causal relationship between the incidence of COVID-19 cases and reported death counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It is  clearly expected that a major source of information that provides knowledge on future deaths is  encoded in the time series of past case counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We used the NETE to quantify such information  flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 3 shows the NETE between the weekly time series of COVID-19 cases and deaths in  the four countries under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In all countries, median values of NC→D increase from lags  equal to 1 week up to a maximum of around 2-3 weeks, and then decline rapidly beyond the  3 weeks time lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' This is in line with early estimates of the median time delay between case  reporting and fatality, which was estimated to range between 7 and 20 days in different countries  [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' At lag equal to 2 weeks, the mean relative explanation added by time series of cases  with respect to deaths – that is how much of D(t) can be explained only by the past knowledge  6  Figure 3:  Information flow between COVID-19 incidence and deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Normalized  Effective Transfer Entropy (NETE) between COVID-19 weekly reported cases and deaths in  the NUTS3 administrative subdivisions (provinces) of Austria, France, Italy and Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' NETE  is computed for lags ranging from 1 to 8 weeks, on the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Boxplots are computed on the  distribution of NETE values of all the administrative subdivisions in each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The horizontal  red line marks the value NC→D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  C(t − l) – is 14% (SD=8) in Spain, 8% (SD=6) in Italy, 7% (SD=5) in Austria, and 6% (SD=5)  in France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Boxplots computed on the distribution of administrative units in each country show  a substantial heterogeneity of NETE across regions for lags shorter than 4 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' This may  be partially explained by spatial heterogeneities in case and death reporting, and in testing  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Also, NC→D values appear to be higher in Spain, with respect to the other countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  A transfer entropy analysis of daily time series of COVID-19 cases and deaths displays consistent  results (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' S1), with NETE values that fall within the same range measured on a weekly  time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  These results suggest NETE estimates are robust with respect to the time scale at which  source and target time series are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Moreover, it provides a reference value for NETE,  in terms of orders of magnitude, when the existence of a causal relationship between time series  is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4  Austria  France  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='0  1  2  3  4  5  6  1  8  1  2  3  4  5  6  7  8  Lag (weeks)  Lag (weeks)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='41  Italy  Spain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  Nc  Nc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='0  1  2  3  4  5  6  7  8  1  2  3  4  5  6  8  Lag (weeks)  Lag (weeks)→ C(t)(%)  → D(t)(%)  l (weeks)  CR(t)  M(t)  M s(t)  CR(t)  M(t)  M s(t)  C(t)  2  9  19  3  10  7  7  79  3  20  23  5  21  8  13  69  4  27  22  9  29  9  16  46  5  33  23  10  36  8  17  18  6  35  27  10  38  14  17  7  7  29  25  11  40  12  14  4  8  27  20  11  38  15  12  8  Table 1: Percentage of statistically significant NETE values across provinces in all  the countries studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' This table shows the percentage of provinces, in all countries, in which  the NETE is statistically significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01) for lags (l) from 2 to 8 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  The information flow between mobility traces and COVID-19 dy-  namics  Having defined a benchmark of information transfer using NC→D, we measured the information  flow between behavioral time series of mobility indicators and COVID-19 cases and deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 4  summarizes the main results of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Values of NX→D, with X being either short range  movements, mid-range movements or contact rates, were substantially smaller than NC→D in  all countries, for any given time lag l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In particular, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 4a allows to compare the distributions  of NC→D, NCR→D, NM s→D, and NM→D, at the time lag l that maximized the median NETE  for weekly time series, for all indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We found the largest median values of the normalized  transfer entropy at l = 7 weeks for both contact rates and movements (short-range and mid-  range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The upper quartile of the NETE distributions derived from the mobility traces generally  fell below 5%, in all countries, while the lower quartile of NC→D was always above 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Also,  the distributions of normalized transfer entropy computed from movements were much narrower  and often including the value N = 0 within their interquartile range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Values of NM→C, shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 4b, display a pattern similar to the normalized transfer entropy from the mobility time  series to the death time series, with generally low values of NETE in all countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Compared  to movement time series, contact rates led generally to relatively higher values of NETE with  both targets, cases and deaths, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Our result confirms the additional value  of measuring contact rates from mobile phone data, with respect to other movement metrics  [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Besides, it shows that short-range mobility within a province had often a limited predictive  power to capture time trends of COVID-19 spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  To obtain a more detailed picture of the predictive power of different mobility metrics in terms  of NETE, we computed the percentage of provinces for which mobility time series provided  significant relative information added, with respect to the past knowledge of epidemiological  8  Figure 4:  Information flow from mobility data to COVID-19 incidence and deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Comparison between the normalized effective transfer entropy computed from source time series  X and target time series of reported COVID-19 deaths D (a) and cases C (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Source time series  are COVID-19 cases (only for deaths), contact rates, short range and mid-range movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Boxplots are computed from the distribution of NETE values for a given time delay, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In panel  a: l= 2 weeks for cases, 7 weeks for contact rates and movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In panel b: l= 6 weeks for  short range and mid-range movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The horizontal red line marks the value NX→D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  indicators only (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' On the one hand, our framework effectively captured the existing  causal relationship between the time evolution of cases counts and the number of deaths, as  the NETE between these indicators was statistically significant (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01) in about 80% of the  provinces, at 2 weeks lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' On the other hand, we observed a statistically significant information  transfer from mobility time series to epidemiological ones in a much smaller fraction of provinces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Short-range movements NETE was significant in less than 20% of provinces when considered as  a predictor of both cases and deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Mid-range movement time series and contact rates were  significant in at most 27% and 40% of provinces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' This means that in most provinces, mobility  traces did not provide any additional information to predict future COVID-19 cases or deaths,  at any lag between 2 and 8 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Measures of contact rate extracted from colocation maps were more suitable than movement  9  a  C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  CR  Ms  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  M  XN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='0  b  Austria  France  Italy  Spain  CR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  Ms  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  C  个  XN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='France  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='Italy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='Austria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='Spain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='Country→ C(t)(%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='→ D(t)(%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='l (weeks)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='M(t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='M s(t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='6 (3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='5(2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='6 (3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4 (1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='Table 2: NETE results across provinces in all the countries studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The table shows  the average relative explanation added by source time series, with respect to past knowledge of  the target only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Only provinces having a statistically significant NETE are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Numbers  in parenthesis report the standard deviation computed over all provinces for which the NETE  was statistically significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  data to capture behavioral patterns relevant to predict COVID-19 spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  By focusing only on those provinces where we could identify a significant information flow  between mobility traces and COVID-19 indicators, we observe that the averaged relative expla-  nation added by mobility data with respect to the epidemiological data ranges between 4 − 6%,  which is about half of the averaged relative explanation added by past knowledge of cases to the  prediction of future deaths (see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 2 and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' S2-S9 in the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  As a sensitivity analysis, we also computed the NETE on a shorter time window, between  September 2020 and January 2021, to exclude the confounding effect of the introduction of na-  tionwide vaccination programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Since in those months all countries adopted mobility restrictions  to mitigate the fall COVID-19 wave, we expect a stronger relationship between mobility and  COVID-19 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Indeed, during this time frame, the information flow between movement time  series and COVID-19 cases was consistently higher than in the full study period (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  This result indicates that, provided with time series of adequate size, the NETE can effectively  capture the time-varying relationship between human mobility time trends and COVID-19 dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4  Identifying the determinants of mobility data predictive power for  COVID-19  Maps of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 5 highlight the spatial heterogeneity of NX→D values observed within the same  country, Spain, for a given time lag and different source time series (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' S11 - S13 for  the maps of Austria, France, and Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As previously mentioned, NC→D displays higher and  significant values in most of the country (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 5a), with very few exceptions, while statistically  10  Figure 5:  Spatial variations of normalized effective transfer entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Maps of NETE  values computed for different source time series and weekly COVID-19 deaths, in the provinces  of Spain: (a) source is COVID-19 cases at lag l=2 weeks, (b) source is contact rate at lag l=7  weeks, (c) source is short-range movement at lag l=7 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' (d) source is mid-range movement  at lag l=7 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Dark grey indicates provinces with non-significant values of NETE (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Provinces in white are excluded from our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  significant values of NM s→D are found only in 16 provinces out of 42 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 5c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  To better understand the observed heterogeneity in NETE, and identify those features that  can predict the likelihood to observe a statistically significant information transfer from mobility  11  b  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='35  Nc-→D  NcR→D  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='35  Nms→D  NM-Dp ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01  p <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01  precision  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='90  recall  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='47  f1-score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='62  (a) Movement  p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01  p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01  precision  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='92  recall  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='61  f1-score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='74  (b) Contact rate  Table 3: Classification performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Summary of model’s classification performance  to predict the statistical significance of NETE at the p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01 threshold when the input source  is short-range movement (a) or contact rate (b) and target variable are COVID-19 deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  to COVID-19 death counts, we resorted to a classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Namely, we used a random  forest classifier to predict when the value NX→D is more likely to be statistically significant, using  short-range movement and contact rate as source time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We focused on these two metrics  as they are quantities measured at the same spatial scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Moreover, short-range movements  represent on average 90% or more of all movements within a province (see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As input  features to the model, we considered a set of attributes of the provinces in each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In  particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' we investigated the effects of population size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' province area in square kilometers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' the  density of Facebook users,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' the number of total cumulative deaths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' the ratio between the number  of commuters traveling from or to the province,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' and those who live and work there,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' as reported  by the census (commuting flow),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' and the coverage consistency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' that is the correlation over time  between the number of Facebook users sharing their location and the number of Facebook users  taken into account to compute the colocation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  The results summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 3 show that the model achieves a good overall performance in  terms of precision and recall, as indicated by f1-scores generally higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In particular, of  all provinces that are classified by the model as characterized by a statistically significant value of  NETE, 90% or more display a significant transfer of information, as shown by precision values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  On the other hand, the model’s recall is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='95 when it comes to identifying provinces  characterized by a not statistically significant NETE, therefore the model correctly identifies  95% of those provinces where there is no actual transfer of information between mobility and  deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  To explore the importance of province features in our classification model, we examined the  SHAP (SHapley Additive exPlanations) values associated with each, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' SHAP  is a method based on a game theoretic approach to explaining the output of classification mod-  els [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As expected, the choice of the time lag to compute the NETE is crucial in determining  the presence of a significant information transfer between mobility metrics and epidemiological  indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Indeed, lag is ranked as the most and second most important feature explaining the  classification, for contact rate and short-range movement, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Commuting flow is the  most important predictor of the statistical significance of NETE between short-range movements  12  (a) Movement  (b) Contact rate  Figure 6: SHAP plots of feature importance to predict the statistical significance of  the NETE for all selected provinces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Color represents the feature value (blue is low and  red is high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Panel a describes the results for NM s→D, panel b for NCR→D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The SHAP value,  on the horizontal axis, indicates the feature importance on the model output, with larger values  corresponding to higher relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Each dot represents a single observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Features are ranked  by importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  and deaths: when the number of commuters leaving or entering a province represents an impor-  tant fraction with respect to those who remain within the province, the relationship between  short-range mobility and COVID-19 dynamics gets weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' However, the same feature has only  a marginal impact on the NETE between contact rates and deaths, which suggests contact rate  should be preferred over short-range movements to predict epidemic outcomes when a province is  characterized by large population inflows/outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Province area and population size have also  a significant impact on the information transfer between short-range movement and COVID-19  deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Indeed, a larger area and population size correspond to a higher likelihood of NETE  significance for short-range movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  This effect may partly explain why we observed NETE values that were statistically significant  only in a few provinces of Austria, where spatial units were particularly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' When looking at  the information flow between contact rates and time series of deaths, the total cumulative deaths  represent an important explanatory variable for the classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Besides the analysis  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' 6 suggests that the coverage consistency needs to be sufficiently high in order  to get a statistically significant transfer entropy from contact rate to deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In France, where in  most provinces the coverage consistency is low and the commuting inflow and outflow are higher  than in other countries (see Table S2), mid-range movements seem to provide a better alternative  to contact rates and short-range movements to partially explain time trends of COVID-19 cases  and deaths (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' S14 of the SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  13  High  flow  6  Feature value  area  population  cumulated deaths  user density  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4  SHAP value (impact on mpdel output)High  lag  cumulated deaths  coverage consistency  Feature value  population  flow  area  Wser density  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  SHAP value (impact on mpdel output)From our analysis, we thus conclude that NETE values computed using contact rates as  source time series are less sensitive to the province’s geographic or demographic features, rather  than to the noise of the target time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Given good coverage, and consistency over time,  contact rates thus represent a better epidemiological predictor of future COVID-19 deaths than  short-range movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  3  Discussion  In this work, we have introduced a novel framework based on transfer entropy to quantify the  amount of information that is transferred from mobile phone-derived mobility metrics to epi-  demiological time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Given the important role that mobility indicators have played in the  COVID-19 pandemic, we tested our approach on mobility and epidemic time series collected  in four European countries, between 2020 and 2021, at a subnational scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We found that, in  general, the relative explanation added by mobility time series to predict future epidemic trends,  whether new cases or deaths, was relatively small, ranging between 4% and 6% on average, and  not statistically significant in the large majority of the provinces we considered, for any mo-  bility metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As a comparison, these values were about half of the relative explanation added  by past knowledge of COVID-19 incidence to predict future deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Our method allowed us  to directly compare the relative explanation added by different mobile phone-derived metrics of  mobility: short- and mid-range mobility, and contact rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We generally found a higher informa-  tion transfer from contact rates than movement, in line with previous studies [48], however, we  also observed significant heterogeneities within the same country and between countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' With  a classification model, we identified spatial features that may explain such heterogeneities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In  provinces characterized by large populations, good coverage consistency over time, and small  commuting in- and outflows, short-range movements can represent a useful metric to predict  disease dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Where commuting flows are large, such as in France, and Austria, mid-range  movements, which represent less than 10% of the total movements, provided a better alternative  to short-range ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Our results suggest the choice of the best mobility metric to inform epi-  demic predictions can depend on a number of different factors, even when using one single data  provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Moreover, our findings show that cell phone mobility metrics do not always capture  epidemiologically-relevant behaviors and alternative data sources could be more effective for this  aim, as, for instance, the collection of survey data [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  There is an emerging common understanding that mobility indicators measured from mobile  phone data present significant gaps and do not provide a consistent picture of mobility across  countries, and data providers [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Previous studies have also highlighted the fact that cou-  pling between mobility indicators and COVID-19 epidemiology is often weak, and it changes over  time [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The approach we introduced here addresses the above challenges by providing a general  framework to evaluate the quality of metrics derived from passively collected mobility traces as a  predictor of epidemic outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Our framework has the advantage of being model-free, meaning  14  that it does not depend on modeling assumptions regarding the expected relationship between  mobility and epidemic dynamics, nor it requires any parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The normalized effective  transfer entropy we adopted is a general method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It allows us to rigorously compare different  mobility indicators, across epidemiological settings, by measuring the relative information added  by mobility time series to the prediction of future disease incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' To this end, we release the  code to reproduce our analysis between any two source and target time series (see Data and  Code Availability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Researchers can use this tool in any epidemiological context to gauge the  added value of a specific mobile phone-derived behavioral measure for epidemic intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Our study comes with a number of limitations and opens new directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We  considered mobility metrics derived from one data provider, Meta, whose user base is not rep-  resentative of the population in the countries we considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' However, alternative data sources  of mobility indicators in Europe with a similar breadth, such as Google or Apple, do not reach  the same spatial granularity and provide their data only as relative changes with respect to a  pre-pandemic baseline, thus limiting their use in a study like ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' On the other hand, movement  and colocation maps by Meta have been extensively used in several studies, including European  countries [53, 54, 55, 56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Here, we considered four countries with different public health  systems, and that adopted different testing strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Observed differences in the predictive  power of mobility metrics across countries may depend on the varying quality of their reporting  systems, especially at the province level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' However, all four countries belong to the European  Union and we expect very similar standards of surveillance during the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Overall, it  will be important to assess our findings on mobility data from other providers, and, most im-  portantly, in countries of the non-Western world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Finally, it is important to note that transfer  entropy measurements become more accurate as the length of the source and target time series  increases [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We worked with a relatively short time series, addressing the bias due to the small  sample by adopting the effective transfer entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' However, we could not systematically investi-  gate how the information transfer changed over time, performing our analysis over different time  windows and comparing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Future work could benefit from longer epidemic time series, over  several years, to identify temporal changes in the information flow between human movements  and COVID-19 dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Measures of human mobility inferred from mobile phone data have been a critical ingredient  to inform the public health response during the COVID-19 pandemic [58] and they will be an  important asset in the fight against future pandemics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' At the same time, their widespread use  raises some relevant ethical concerns due to re-identification risks [59], therefore, it is fundamental  to assess the added value of using cell phone mobility data in a given epidemic scenario and  whether the benefits outweigh the risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Our work provides a practical guide to identifying when  and where mobile phone mobility metrics truly capture behavioral patterns that are relevant to  predict disease dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  15  4  Materials and Methods  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  Epidemiological indicators  We collected epidemiological time series in the 4 countries under study from 2 data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Daily reported cumulative COVID-19 cases were collected from the COVID-19 Data Hub [60],  an open source aggregator of up-to-date COVID-19 statistics, at the NUTS3 level in Austria,  France, Italy, and Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Daily reported cumulative deaths in Austria, France, and Spain were also collected from the  COVID-19 Data Hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' For Italy, death statistics were only available on a weekly time scale from  the public platform CovidStat (https://covid19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='it/iss/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  For the analysis, we generated daily incidence time series from cumulative data by computing  day-to-day differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Then, we further aggregated the daily time series of deaths and cases  into weekly ones, to perform the transfer entropy analysis on a weekly scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  Mobility derived indicators  In our study, we computed daily and weekly movement and contact rates from data provided  by Meta through its Data for Good program [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Here, we first describe the raw data sources  provided by Meta and then the data processing we applied to compute the time series for the  transfer entropy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1  Raw data sources  We collected the following datasets that were publicly released by Meta since the beginning of  the COVID-19 pandemic, in Austria, France, Italy, and Spain:  Movement range maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It reports the number of users who moved between any two  16-level Bing tiles, with an 8-hour frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Users’ population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It reports the number of active users in each tile with an 8-hour  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The tile resolution is 4800 x 4800 m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Colocation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It estimates the probability that, given any two administrative regions,  p1 and p2, a randomly chosen user from p1 and a randomly chosen user from p2 are  simultaneously located in the same place during a randomly chosen minute in a given week  [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The dataset also reports the number of users in p1 and p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Stay put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It reports for a given administrative region the daily percentage of users staying  put within a single location, defined at the 16-level Bing tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  We formalize the description of the above datasets with the notation described in Table 4:  16  Dataset name  Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='t  spatial resolution  temporal resolution  population users  N (pop)  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='h  t: tile (4800 x 4800 m2)  h: 8 hour  movement between tiles  M(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='t2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='h  (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='t2): tile pair (600 x 600 m2)  h: 8 hour  colocation probability  Pp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='w  p: province  w : week  colocation users  N (coloc)  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='w  p: province  w: week  stay put  Sr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='d  r: region  d: day  Table 4:  Summary of raw data sources as time series records Xs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' where s denotes the spatial  resolution and t the temporal resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  original data  spatial aggregation  temporal aggregation  aggregated data  name  N (pop)  t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='h  �(t ∈ p)  h interpolation and mean (h ∈ w)  N (pop)  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='w  province population users  M(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='t2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='h  � (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' t2) ∈ p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' t1 = t2  mean (h ∈ w)  M (within)  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='w  within tile province movement  M(t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='t2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='h  � (t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' t2) ∈ p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' t1 ̸= t2  mean (h ∈ w)  M (between)  p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='w  between tiles province movement  Sr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='d  ∀p ∈ r  r = p  mean (d ∈ w)  Sp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='w  province stay put  Table 5: Aggregation of data sources described in Table 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' to generate our metrics of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2  Aggregation of raw data  We then processed the raw data sources of Table 4 to obtain a set of time series having the  same spatiotemporal resolution, that is weekly, at the NUTS3 scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Results of the aggregation  process are described in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' More in detail:  Province users population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' (1) we performed a spatial aggregation by summing the  population of tiles belonging to province p, thus obtaining a population at a (province,  hour) level: N (pop)  p,h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' (2) we performed a linear interpolation of the temporal gaps that were  present in N (pop)  p,h  (3) we performed a temporal aggregation by averaging in each province,  the 8h population records within a week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Within tile province movement (1) we first performed a temporal aggregation by  averaging M(t1,t2),h for each pair (t1, t2) over a week and obtaining M(t1,t2),w (2) we then  performed a spatial feature joining and assigned each pair (t1, t2) to the corresponding  provinces (p1, p2) (3) from M(t1,t2),w we obtained a within tile province movement  M (within)  p,w  , that is the sum of movements which occurred in the same province p and within  the same tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Between tiles province movement in the pipeline above, from step (3) we obtain a  between tile province movement M (between)  p,w  , that is the sum of movements which  occurred in the same province p and between two different tiles, (t1, t2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' By definition, the  sum M (between)  p,w  + M (within)  p,w  represents the total volume of movements in a province, in a  17  week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Province stay put (1) we performed a temporal aggregation on a weekly scale by perform-  ing the average and obtaining Sr,w (2) we assign to each province p the regional stay-put  time series Sr,w such that p ∈ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  Computation of movement and contact rate  We finally computed our metrics of interest, movement, and contact rates, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  The  short-range movement rate is defined as:  M s  p,w = M (within)  p,w  N (pop)  p,w  (1)  that is the proportion of users who moved within the same tile in a given province, in a given  week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The mid-range movement rate is defined as:  Mp,w = M (between)  p,w  N (pop)  p,w  (2)  representing the proportion of users who moved between different tiles in a given province, in a  given week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The contact rate is defined as:  CR(t)p,w = ˆPp,w · N (pop)  p,w  (3)  where ˆP denotes the colocation probability corrected by a factor that takes into account the  overestimation of colocation probabilities due to the heterogeneous distribution of users across  provinces and the presence of a significant fraction of static users in some periods of mobility  restrictions [55] (see the SI for additional details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4  Province sample selection  The population of Facebook users who contribute to the generation of the movement and colo-  cation time series varies across countries, and it changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Moreover, the metrics of  movement (short- and mid-range) and colocation, are computed from different users’ samples of  different sizes: N (pop)  p,w  and N (coloc)  p,w  , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In our analysis, to limit bias that may be caused by the little representativeness of the  underlying sample of users, we selected NUTS3 regions in the 4 countries, according to the  following criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  First, we considered only regions where the sample N (pop)  p,w  represented at  least 3% of the census population to guarantee we had at least 500 users in each province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Furthermore, we considered only those regions where the two sample sizes N (pop)  p,w  and N (coloc)  p,w  were always positively correlated over time, during the whole study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  We denote the  Pearson’s correlation of weekly values of N (pop)  p,w  and N (coloc)  p,w  as coverage consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  After the selection, our analysis includes 47 provinces in Austria, 51 provinces in France, 93  provinces in Italy, and 42 provinces in Spain, for a total of 233 spatial units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  18  Given two discrete temporal signals represented as time series X and Y the Transfer Entropy  (TE) [38] is a measure of the amount of information delivered from X to Y , defined as:  TEXY = H(Y |Y (l)) − H(Y |Y (l), X(l)) ,  (4)  where X(l), Y (l) are respectively the l-lagged time series of X and Y and TEXY is formulated  as a difference between two conditional entropy terms, where conditional entropy is expressed as  H(a|b) = H(a, b) − H(b), and H(·) is the Shannon Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Given a discrete time series S, its  observations can be expressed as the sample {si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='., n}, and we obtain the discrete proba-  bility distribution p(sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We compute the Shannon Entropy as: H(S) = �  j p(sj) · log2(p(sj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Thus TEXY can be expressed as:  TEXY = H(Y, Y (l)) − H(Y (l)) − H(Y, Y (l), X(l)) + H(Y (l), X(l)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  (5)  The time series that we consider in our experiments are continuous, therefore they need to be dis-  cretized before computing TEXY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We employ the Kernel Density Estimation (KDE) for Transfer  Entropy estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' KDE method evaluates the entropy terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='5 from the discretized den-  sity estimated from each of the four features sets: {(Y, Y (l)), Y (l), (Y, Y (l), X(l)), (Y (l), X(l))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  KDE employs a Gaussian kernel for density estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Performing tests on synthetic datasets  of different sizes, we checked this was the method the most adapted to small samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' For the  selection of the kernel’s bandwidth, we use the Scott method [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The continuous density is  then discretized with a grid obtained by an equal-width discretization of each feature’s density  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We select 20 as the number of bins for each feature’s domain discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The dis-  cretized density is computed with the integral of the continuous probability density functions  over each grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Concerning the implementation, for TE estimation we use the PyCausality  Python package (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='com/ZacKeskin/PyCausality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Effective Transfer Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  We introduce the Effective Transfer Entropy (ETE) as a cor-  rection to TE for small sample time series, as originally proposed by [44]:  ETEXY = TEXY − 1  Ns  Ns  �  j=1  TEX ˆ  Yj ,  (6)  where the correction term is obtained by performing Ns iterations of Y shuffling, obtaining ˆYj  and computing the average of {TEX ˆ  Yj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='., Ns}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In our experiments, we performed 500  shuffling iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Normalized Transfer Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  We would like to employ TE in order to compare a set  of input signals {Xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='., N} in terms of their Transfer Entropy TEXjY towards a specific  output Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' From equation 4 we have that TEXjY is evaluated as a difference of conditional entropy  where the first term H(Y |Y (l)) depends only on target Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In order to ensure comparability over  the set {TEXjY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='., N}, we reformulate the difference as a relative difference dividing by  19  H(Y |Y (l)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Thus the set of inputs are compared according to {TEXjY /H(Y |Y (l));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='., N}  and we refer to TEXY /H(Y |Y (l)) as Normalized Transfer Entropy (NTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Normalized Effective Transfer Entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  By combining the ETE and the NTE we can fi-  nally introduce the Normalized Effective Transfer Entropy (NETE), which is obtained by dividing  the ETE by the first conditional entropy term H(Y |Y (l)) as in [62]:  NETEXY =  TEXY −  1  Ns  �Ns  j=1 TEX ˆ  Yj  H(Y |Y (l))  (7)  In this way, the NETE accounts both for bias in small sample time series and it ensures compa-  rability between different input sources {Xj} in terms of information transfer to different targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Besides, it enables estimating the percentage of explanation value added with respect to only  knowing the past of the time series used as a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3  Classification model  The introduction of the ETE allows associating a p-value, a metric of statistical significance, to  each NETE value computed between any pair of time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  In our study, we investigated a number of explanatory features to better understand why  in some provinces the NETE could not identify a significant transfer of information between  mobility time series and epidemiological indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  More specifically, we trained a Random  Forest classification model to predict the significance of NX→Y at the threshold of p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01, in  each province under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The random forest was performed with 100 decision tree classifiers  on various sub-samples of the dataset and used averaging to improve the predictive accuracy and  control for over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The function to measure the quality of a split was the Gini impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Before applying the random forest, the data were split between training and test sets (30%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' To  compensate for the imbalance of the datasets, we applied a Synthetic Minority Oversampling  Technique [63] on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  As input to the classification model we used a set of features that characterize each province:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' population size (as reported by the latest available census);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' area (in km2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' density of Facebook users (measured as Np,w divided by area);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' total cumulative number of reported COVID-19 deaths during the study period;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' commuting flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' coverage consistency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  20  The commuting flow is defined as the ratio between the total number of daily commuters who  travel from or to a province and the total number of commuters who work and live in that  province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Commuting data were collected from the latest available census statistics in each  country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The coverage consistency is the correlation over time between the users’ populations  N (pop)  p,w  and N (coloc)  p,w  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  To quantify the importance of different features in our classification model, we used their  SHAP (SHapley Additive exPlanations) values [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' SHAP is a method to explain model pre-  dictions based on Shapley Values from game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' In particular, we use TreeSHAP [64], an  algorithm to compute SHAP values for tree ensemble models, such as the random forest classifier  of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  5  Data and code availability  The data and code to reproduce our analysis are available at: https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='org/record/  7464949#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='Y6L0CfxKhNg  6  Funding  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' gratefully acknowledges support from the CRT Lagrange Fellowships in Data Science for  Social Impact of the ISI Foundation, where this work was conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' acknowledge  the Lagrange Project of the ISI Foundation funded by CRT Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The funders had no  role in the study design, decision to publish, or preparation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  7  Acknowledgements  We gratefully acknowledge Alex Pompe for his help to understand the details of mobility data  from Meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  8  Author contributions  FD collected data, conducted experiments, interpreted the results, made figures, and contributed  to the writing of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  MT and LG conceived and designed the study, conducted the  statistical analysis, interpreted the results, made figures, and wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' All authors read  and approved the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  9  Competing interests  The authors declare no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  1  Supplementary Information  Correction to the colocation probability  Colocation maps provided by Meta is defined as the number of colocation events over the number  of possible events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' This, by design, includes interactions between users staying within the same  tile but not having actual contact with other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' For this reason, we estimate the contact  rate in each province by removing the contribution due to the users staying put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' We explain our  approach to estimating such contribution in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Let us start by writing the original  colocation probability P as:  P = E  N 2  (8)  where:  E is the number of colocation events within the province  N is the number of province colocation users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  The exact formula should be P =  E  N(N−1) but as N is large we approximate it to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Let us  denote R(c) the number of measured colocation events that are due to users who stay put only,  then the corrected colocation probability should be written in the following way:  ˆPp,w = E − R(c)  N 2  (9)  We estimate R(c) by using the stay-put probability S, which is the probability of a user staying  put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Let us call the tile population ratio probability distribution {ftl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='., Tl} where T is  the number of tiles in a province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' This gives us an estimate of the contribution of the users who  stay put to the colocation probability, as:  R(c) =  T  �  t=1  N 2 · f 2  t · S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  (10)  So we rewrite:  ˆPp,w = P − S2  Tl  �  t=1  f 2  tl  (11)  We do not have access to the population of the tiles used for the colocation so we make an  approximation using the population distribution given for each tile with dimensions 4800 m  × 4800 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' As there are by definition 64 colocation tiles within a single population tile, the  expression Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='11 can be formulated as:  ˆPp,w = Pp,w − S2  p,w ·  T  �  t=1  64 ·  �  f (p)  t,w  64  �2  (12)  where:  2  M s (%)  M (%)  Austria  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='6 [97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='9 – 100]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='5 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='0 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1]  France  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3 [88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2 – 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='7]  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='8 [6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='3 – 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='9]  Italy  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='9 [86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='4 – 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='8]  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2 [7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='2 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='6]  Spain  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='5 [86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1 – 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='1]  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='5 [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='9 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='9]  Table S6: Relative proportion of mobility components in each country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Each row dis-  plays the proportion of movements, as a percentage of the total movements within each province,  that are represented by the short-range mobility (M s(t)) and the mid-range mobility (M(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Each table entry reports the median value and the IQR, computed over all provinces, and all  weeks of the study period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Short-range mobility represents the large majority of movements  within a province, in all countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  coverage consistency  commuting flow  Austria  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='64 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='45–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='79]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='05 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='43–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='69]  France  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='32 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='23–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='43]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='30 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='22–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='51]  Italy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='63 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='42–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='77]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='21 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='12–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='29]  Spain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='86 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='68–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='91]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='08 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='05–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='10]  Table S7: Coverage consistency and commuting flow distributions by country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Each  table entry reports the median value and the IQR computed over all provinces, in each country,  considered in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  f (p)  t,w = Nt,w  Np,w ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' t ∈ p : tile t population frequency in province p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Nt,w : population at (tile,week) resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' It is obtained through mean temporal aggre-  gation of Nt,h over the week interval denoted by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Np,w : population at (province, week) resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  It is obtained through sum spatial  aggregation of Nt,w over the tiles belonging to province p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  T is the number of tiles 4800 m × 4800 m  We can introduce the quantity Qp,w as the sum of squared frequencies of the province tile  distribution Qp,w = �  t∈p(f (p)  t,w)2, so that, finally:  ˆPp,w = Pp,w − S2  p,w · Qp,w  64  (13)  References  [1] Ira M Longini Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  A mathematical model for predicting the geographic spread of new  infectious agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Mathematical Biosciences, 90(1-2):367–383, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  3  Figure S1:  Comparison of NETE values computed on weekly and daily time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  NC→D computed between time series data collected on a weekly time scale (bottom row) and a  daily one (top row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Daily time series were available only for Austria, France and Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  [2] Aidan Findlater and Isaac I Bogoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Human mobility and the global spread of infectious  diseases: a focus on air travel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Trends in parasitology, 34(9):772–783, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  [3] Duygu Balcan, Bruno Gon¸calves, Hao Hu, Jos´e J Ramasco, Vittoria Colizza, and Alessandro  Vespignani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Modeling the spatial spread of infectious diseases: The GLobal Epidemic and  Mobility computational model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Journal of Computational Science, 1(3):132–145, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  [4] Amy Wesolowski, Caroline O Buckee, Kenth Engø-Monsen, and Charlotte Jessica Eland  Metcalf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Connecting mobility to infectious diseases: the promise and limits of mobile phone  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' The Journal of infectious diseases, 214(suppl 4):S414–S420, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  [5] Amy Wesolowski, Nathan Eagle, Andrew J Tatem, David L Smith, Abdisalan M Noor,  Robert W Snow, and Caroline O Buckee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Quantifying the impact of human mobility on  malaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Science, 338(6104):267–270, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  [6] Lorenzo Mari, Enrico Bertuzzo, Lorenzo Righetto, Renato Casagrandi, Marino Gatto, Igna-  cio Rodriguez-Iturbe, and Andrea Rinaldo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Modelling cholera epidemics: the role of water-  ways, human mobility and sanitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Journal of the Royal Society Interface, 9(67):376–388,  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='40  Austria  France  Spain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='20  →D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='20  →D  ON  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content=' Domestic and international mobility trends in the united kingdom  during the covid-19 pandemic: an analysis of facebook data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='35  Nms-D  NM→DFigure S13: Spatial variations of normalized effective transfer entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Maps of NETE  values computed for different source time series and weekly COVID-19 deaths, in the provinces  of Italy: (a) source is COVID-19 cases at lag l=2 weeks, (b) source is contact rate at lag l=7  weeks, (c) source is short-range movement at lag l=7 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' (d) source is mid-range movement  at lag l=7 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Dark grey indicates provinces with non-significant values of NETE (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  Provinces in white are excluded from our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  20  b  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='35  Nc→D  NcR→D  d  C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='35  Nms-D  NM-DFigure S14: Percentage of statistically significant NETE values, disaggregated by  country and by mobility metric used as source variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content=' Target variables are: weekly  COVID-19 cases (panel a) and weekly COVID-19 deaths (panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
+page_content='  21  b  a  60  CR  Ms  30  M  40  20  E 20  NETI  10  0  0  Austria  Italy  Austria  Italy  France  Spain  France  Spain' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dE2T4oBgHgl3EQfigf6/content/2301.03960v1.pdf'}
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+arXiv:2301.00272v1  [hep-ph]  31 Dec 2022
+Quark spectral functions from spectra of mesons and vice versa
+V. ˇSauli1, ∗
+1Department of Theoretical Physics, Institute of Nuclear Physics Rez near Prague, CAS, Czech Republic
+Within the QCD functional formalism, having the approximations controlled by physical masses
+and decays of pseudoscalar mesons, we extract spectral function of quarks from which the meson
+are composed. We choose the pion for the case of light quarks and ηc(N) for the extraction of
+charm quark spectral function.
+For this purpose we solved improved ladder-rainbow truncation
+of the spectral Dyson-Schwinger equations for quarks coupled to Bethe-Salpeter equation for the
+pion and the pseudoscalar charmonia. We begin with indefinite gauge fixing method for class of
+covariant linear gauges and search for its optimal value in given fixed truncation of Dyson-Schwinger
+equations. All kernels are represented self-consistently by known or extrapolated solutions known
+form lattice or QCD DSEs solutions.
+We require the formalism gives us the spectral functions
+with arbitrarily high numerical accuracy, while providing known experimental properties of mesons
+simultaneously. , we found that the ladder rainbow approximation can serve for this purpose when
+the Yennie gauge is employed. Properties of such spectral functions are shown and its connection
+with confinement is discussed.
+PACS numbers: 11.10.St, 11.15.Tk
+I.
+INTRODUCTION
+QCD is a rigid part the Standard Model already for half century and passed many nontrivial tests when compared to
+the experiment. The knowledge of correlation functions at time-like momentum region is crucial for the first principle
+determination of hadronic resonances and understanding of production of hadrons [1, 2]. Lattice theory is formulated
+in the Euclidean space where it is also solved, however the analytical continuation to the timelike Minkowski subspace
+represents is quite often an ill defined numerical problem.
+A complementary and very attractive approach is the spectral functional formalism, where the analytical contin-
+uation is performed at very beginning and the set of Dyson-Schwinger equations is solved for spectral function in
+Minkowski space. Such method is appreciated quite recently [1, 3–7] and includes the topics of spectral renormal-
+ization - primary or secondary subtractions technique performed at the timelike momentum scale. A Yang-Mills
+sector of SU(3) gauge theory was considered in [8, 9] bringing a new insights in the conventional Landau gauge. In
+order to get agreement with lattice data, the importance of transverse vertices in pure gluodynamics was shown [4].
+A meaningful comparison to recent lattice data was missing when the first spectral DSEs study [10] has appeared.
+Nowadays, the transverse QCD vertices are known to be very important in the quark as well as in the gluon sector in
+the Landau gauge and they are responsible for a large enhancement (suppression) of the propagator (proper selfener-
+gies) in infrared domain. How to incorporate transverse vertices in spectral quark sector was only suggested in [1] for
+the case of quark-photon vertex but not yet implemented in practice. The purpose of presented paper is not a jump
+to bandwagon or chasing the train of DSEs scheduled in the Euclidean space [11, 12], but to push theory of spectral
+DSEs in its own direction.
+Since the relativity is less urgent for mutual interaction of heavy quarks Q = c, b inside heavy mesons, nonrelativistic
+quantum mechanic was widely used to describe quarkonia and their transitions ( for a review see [16]) instead. History
+tell us, that in addition to perturbative Coulomb “one gluon exchange” potential, the linear rising potential have been
+proposed to explain spectra of excited quarkonia [17]. If fine tuned and ignoring quark content mixing and ignoring
+resonant character of excited states, such models reasonably describe static spectra of strangeonia [18] as well. In
+lattice QCD, a confining potential for a static quark-antiquark pair are computed with Wilson loops. This technique
+lies aside of quark-antiquark scattering kernel used in the DSE/BSEs heavy quarkonia studies [13–15]). To match
+the two different approaches -the DSEs and Wilsonian static quark potential together, is longstanding desire but
+unfinished story (for attempts see [19]).
+Actually, to the author best knowledge, there is not known truncation of DSES, which naturally offer the interaction
+kernels, which is consistent with string picture of confinement. The string-like interaction is either introduced by hand
+∗Electronic address: sauli@ujf.cas.cz
+
+2
+[14, 15] or even completely avoided by the use of auxiliary entire function [13]. In all cases, the approximations made
+turns to be odd from perspective of spectral quark functions.
+Before presenting the details of truncation, which complies with the existence of quark spectral function, let us
+mention here the so called hindered transitions , which were measured at various channels [20–22]. The large discrep-
+ancy between of measured rate with nonrelativistic theory prediction were usually attributed to missing relativistic
+corrections. To explain the quarkonia and their transitions, a very recent treatments based either on BSE/DSEs
+formalism, nonrelativistic quantum mechanic or other techniques [23–32] still represent very different approaches with
+not completely clear connection to QCD. To this point, a systematically improvable truncation of DSEs with a clear
+bridge to analytic properties of S-matrix could be a reliable candidate. Notably the formalism of spectral DSEs we
+present here, leads to the dispersion relation for hadronic form factors (including the hindered transitions as well).
+The main aims of presented work is twofold: we solve the spectral quark DSE and extract thus information on the
+quark spectral function. Simultaneously, within the obtained quark propagators we solve the BSE for mesons and
+check the solution against the experimental data. We employ the calculation scheme, which gives us solution with
+desired analytical properties for physical meson from the very beginning. We expect the ladder-rainbow approximation
+gives the first estimate of spectra for both light and heavy mesons. For this purpose we leave popular Landau gauge
+and extrapolate known spectral solution obtained recently [4] into other linear gauges. Enchantingly, it turns out the
+pion and heavy quarkonium (pseudoscalar charmonia) can be described within the same kernel. A correctly obtained
+meson properties thus control reliability of obtained quark spectral functions. An on shell peak is washout specifically
+and anomalous branch point is generated, which naturally explains a large contribution of seemingly unphysical
+gauge term in considered approximation. We discuss the limitation methods as well as we suggest future prospects
+and directions where the method can provide useful results.
+II.
+TRUNCATION OF SDES SYSTEM FOR THE PION
+For purpose of completeness we write down all necessary equations here. The appropriate quark DSE for the quark
+propagator S can be written in the following way
+S−1(q, µ) = A(µ) ̸ q − Bq(µ) − [ΣR(q) − ΣR(µ)] ,
+ΣR(q) = i4
+3
+�
+dDk
+(2π)D γµS(k)ΓνGµν(k − q) ,
+(2.1)
+where for the product of the quark-gluon vertex Γ and the gluon propagator G we take
+ΓνGµν(p) = γνN(ξ)
+�
+−gµν + pµpν
+p2
+� �
+do
+ρT (o)
+p2 − o + iǫ − ξg2
+p2
+pµpν
+p2
+,
+(2.2)
+where ρT is the gluon spectral function obtained with Landau gauge in the paper [4].
+We adopt a conventional renormalization condition and we take ℜA(µ) = 1 ℜB(µ) = 300MeV at the timelike
+subtracting point µ2 = 0.5GeV 2, noting that the imaginary parts of functions A, B, which completely characterize
+the quark propagator
+S−1(p) ≠ pA(p) − B(p)
+(2.3)
+do note take arbitrary value at the timelike renormalization point, but they are matter of numerical search. The
+letter R stands for the fact that the quark selfenergy was (or can be) regularized before the secondary subtractive
+renormalization takes its place. Proper regularization is more crucial and unavoidable step in gluon sector of DSEs
+in order to prevent violation of gauge invariance by inappropriate numeric.
+The last term in the Eq. (2.2) is aforementioned gauge term appearing in the product with the gauge coupling
+g2. The prefactor N(ξ) nonlinearly depends on the coupling as well as on the gauge parameter ξ. The lattice data
+for the gluon propagator are known only for ξ gauge parameter ξ being smaller then 0.5 [33, 34] in QCD without
+quark. Solving a Nielsen identities the solution of truncated system of Yang-Mills DSEs has been obtained in [35].
+Both studies show the gluon propagator is suppressed gradually in the infrared domain when ξ is getting larger. We
+exploit this and perform a very simple extrapolation of our DSE kernel into other gauges.
+The overall prefactor N(ξ) the Eq. (2.2) is varied and represents the only change when we extrapolate to nontrivial
+value of the gauge parameter ξ. Gauge term is not getting dressed, due to the unbroken gauge invariance in QCD and
+ξ appears linearly in our LRA employed. The extrapolation to other gauges is implicit since the ratio of the transverse
+and gauge term is adjusted by the solution of Bethe-Salpeter equation for the pion. Hence we can estimate the gauge
+only at the end of (quite nontrivial) search of solution. Nevertheless, for the first time we provide the first naive guess
+
+3
+0
+0.5
+1
+1.5
+ [GeV]
+0
+2
+4
+6
+ σ v,s   [GeV
+-2,GeV
+-1] 
+u,d - s
+u,d - v
+charm v
+charm s
+u,d
+c
+FIG. 1: Quark spectral functions, solid line stand for the functions σv, , dashed line for σs plotted against the energy . The
+left two blobs are for light quarks, the one on the right for the charm quark.
+————————————————————————————-
+ξ ≃ 3, which is based on comparison with solutions of Yang-Mills system [35] performed for various gauge parameters.
+Notably, the agreement of both solutions [1] and [35] with lattice data [36] in Landau gauge in the infrared domain
+is enough for our rough estimate of the gauge parameter. Further approximation we describe in the next text could
+not be crucial since the meson physics is not sensitive to UV tails.
+The gluonic spectral function is a continuous function starting to be nonzero at p2 = 0, showing the violation of
+passivity bellow several GeV . Here however, in order to reduce number of numerical integration in our pioneering
+study we use a simplified (UV finite) fit for the gluonic spectral function
+ρT (o) = −δ(o − m2
+g) + δ(o − Λ2)
+(2.4)
+with mg = 0.6GeV and Λ = 2GeV .
+Thus we have estimated the gauge only at the end of (quite nontrivial) search of solution for the pion BSE which
+produces the correct pion mass mπ = 140MeV . The gauge choice for which the pion properties are obtained most
+effectively corresponds numerically with the following rate
+g2ξ
+N(ξ) = 3
+(2.5)
+while for the absolute value we get 4N(ξ)
+3(4π)2 = 16
+3 , ( 4g2ξ
+3(4π)2 = 16).
+It is tentative to identify our gauge with a Yennie gauge. Our guess is certainly naive, since based on comparison
+with different truncation of DSEs for the Yang-Mills system [35] performed for various gauge parameters, including
+the Landau gauge as well.
+The agreement of both solutions [1] and [35] with lattice data [36] in Landau gauge
+in the infrared domain is only approximate, also a further approximation represented by Eq. (2.4) calls for better
+identification of the gauge parameter if needed for any future purpose. As we have checked, the realistic picture can
+be obtained for the other rates (2.5), providing the range of our estimate ξ = 3 ± 1. Let us stress for clarity that
+extreme cases like Landau gauge ξ = 0 or complete neglections of the first term N(ξ) = 0 do not lead to satisfactory
+picture at all.
+As the name indicates, the spectral DSE is nothing else but rewritten original DSE in a way that it can be solved
+for the two quark spectral functions σv(o) and σs(o) , rather then for the propagator in momentum space. The quark
+propagator then can be calculated through its spectral representation:
+S(p) =
+� ∞
+0
+do̸ pσv(o) + σs(o)
+p2 − o + iǫ
+(2.6)
+in the entire complex plane of square momentum p2 ( real value p2 > 0 correspond with Minkowski space domain in
+our metric choice).
+The solution of spectral DSE is described in details in series of papers [1, 3, 4] and the only change is improved
+numeric for purpose of presented paper. We use two Gaussian integrator, the first is adjusted to fit the dominant peak
+
+4
+0
+0.5
+1
+1.5
+2
+2.5
+o
+1/2  [GeV]
+0
+0.5
+1
+1.5
+2
+DLSF
+charm v
+charm s
+u,d -v
+u,d -s
+FIG. 2: Dimensionless quark spectral functions oσv(o) (solid line) and
+�
+(o)σs(o) (dashed line) for the light and charm quark.
+At larger (smaller) energy scale the broad peak for the charm flavor (u,d) quark spectral function develops.
+————————————————————————————-
+in the quark spectral function, the second one has been suitably mapped to the rest of infinite interval of spectral
+variable o. The deviation from assumed analyticity σ2 as established in [3, 4], it can be arbitrarily minimized when
+approaching the correct values of imaginary parts of the functions A(µ) and B(µ) at renormalized point µ. The value
+σ2 = 10−6 can be achieved easily and seems to be limited by numeric rather then systematics.
+The BSE for the pion has been solved for the dominant BS vertex component by eigenvalue method in complex
+momentum space. Such single component approximation is working well not only for the ground state [44] but with
+a slight modification for the excited states as well [15]. The BSE involves product of the scalar functions Sv(k +
+P/2)Sv(k −P/2) and Ss(k +P/2)Ss(k −P/2) evaluated at complex momenta (kE is real Euclidean momentum, while
+the total momentum PE = (im, 0) in rest of the pion), which we evaluate within the use the spectral representation
+2.6). Since the shape of spectral functions is difficult to control (at least at this stage), we do not use some numerical
+fit and implement additional integration over the spectral representation to determine products SiSi i. As usually,
+numerical codes either for BSE and DSE are available for public [37]. An alternative way to solve BSE in Minkowski
+space are known, till now used simplified systems (e.g. for constituent quark models[38–40]). The method could be
+necessary if one evaluates the resonant hadronic form factor [1], however as the BSE is converted into more dimensional
+integro-differential equations, we prefer to solve BSE defined in the complex Euclidean space for purpose of presented
+paper.
+The resulting spectral functions for the the light quarks are shown in the figure 1. We work in the izospin limit
+and ignore electromagnetic interaction, thus the spectral function for the up quark is identical to the d quark one.
+According to broad shapes of both functions σv,s, they describe confined objects- the light quark excitations. The
+quarks continuously change colors inside hadrons by exchanging of gluons, hence a width of the main peak can be
+interpreted as the inverse of mean time τu,d ≃ 0.2GeV −1, which the quark of given flavor spent with a given color.
+Similarly to quark weak decays, they do not represent observable, we avoid the name “decay width” in this context.
+Attentive reader has surely noticed that the on-shell delta functions are absent in the spectrum. Consequently the
+thresholds vanishes at evaluated form factors, which is in expected accordance with Wilsonian area law. Such behavior
+is intuitively expected, and in fact it has been mimic in [41–43] by the introduction of certain infrared cutoff in the
+Feynman(Schwinger) parameter in various evaluations of hadronic form factors.
+III.
+ηc(N) QUARKONIA
+The same QCD kernel that govern interaction between quark-antiquark in the light meson is used to calculate the
+heavy quarkonia. However as the interaction is not flavor universal- it constitute by the quark-gluon vertex as well,
+a changes, e.g. softening of the interaction is expected.
+It can be done by a change of effective mass parameters which now takes rescaled values by the factor r = 0.721,
+more precisely they take the values
+mc
+g = 0.433 GeV ; mc
+Λ = 1.442 GeV ;
+(3.1)
+while the dimensionless couplings like g; ξ do not change their values. We renormalize such that ℜAc(µ) = 1 and
+
+5
+BSE EXP.
+2980 2980
+3442 3638
+4150 3810
+4720
+–
+TABLE I: Comparison with PDG data (second column) and calculated spectrum.
+ReBc(µ) = r1.3GeV . The search gives for the the imaginary parts ℑAc(µ) = 0.113 and ℑBc(µ) = 0.106rGeV at
+renormalization point µ2 = r20.5GeV 2.
+The kernel is not further tuned however as the excited states ranges over the relatively large scale, small further
+change we need is to incorporate the total momentum into the kernel. Elsewhere more important diamond diagrams
+(the diagrams with interupted quark horizontal lines by gluon lines) should contribute to the kernel with substantially
+small effect (note M(ηc) ≃ M(J/ψ) Instead of evaluating these complicated diagrams we mimic their small effect and
+insert the following prefactor
+fη =
+1
+√
+2
+�
+1 + M(ηc(2))2
+P 2
+.
+(3.2)
+This formally leaves the BSE for ηc(2) completely identical to the pion case, while the couplings are softened by few
+percentage for higher excited states.
+Thus as expected for charmonium, the two mutually opposite poles of the kernel are getting closer, which suppress
+the metric term when comparing to the pion. Nevertheless, the entire effect on the charm quark spectral function is
+very the same as for the light quark. The resulting charm quark spectral functions are added into the fig. 1. Since
+the spectral function are dimensinfull object, we introduce the dimensionless quantity
+�
+(o)σs(o) and oσv(o) for a
+better comparison of spectral function of different flavors. These object are compared in the figure 2. The on-shell
+singularity is washout to a broad peak and heavy free quark excitation does not exists at all. A picture of confinement
+that emerge in spectral framework of DSEs is very the same for the light as well for the heavy quarks. A scalar string
+interaction governed by a linear potential is not actually needed at any quark sector.
+The spectra of bottomonia can be obtained by a similar fashion, however our two mutually beating poles turns
+to be cruel approximation at bottomonium scale and slightly more honest approximation is required. We plan to
+perform more comprehensive study of BB system in the future.
+The obtained masses are tabled in the Tab I further predicted and not yet observed states we only list here:
+5436,6186,7030MeV,... Recal, those above the first excitation all they lie above open charm threshold and they
+become broad resonance and our predicted values ignores coupling to D mesons completely. Interestingly, not the
+sure of mass but the mass itself is linear in principal value N , not supporting the string/Regge trajectory at all.
+Furthermore, the vertex functions are not orthogonal in sense two states with different N can be produced in single
+photon annihilation of e+e− (this is a bit free extension of quantum mechanical orthogonality, it obviously relies on
+the formula for normalization of BSE). We also show our ultimate numerical search for eigenvalue λ and the deviation
+σ2 in the figure 3. A single point shown in this figure costed one day of work of recent single processor. Even working
+with multiprocessor machines the reader can imagine the time consumed before the truncation of DSE/BSE has been
+established.
+IV.
+CONCLUSION
+Using indefinite gauge fixing we have solved coupled set of spectral quark Dyson-Schwinger equations and Bethe-
+Salpeter equation for the pion and we have extended the method to the heavy quarks sector represented by pseudoscalar
+charmonia.
+Facing the resulting spectral functions we get simple picture of confinement of the light as well as the heavy quarks:
+quarks are never on-shell inside the hadrons, the inverse of quark propagator never gets zero for a real momenta. The
+sharp singularity is completely wash out due to the imaginary part which is gradually growing from the anomalous
+thresholds- the zero momenta.
+Notably, the interaction of heavy quarks in quarkonium is far from conventional
+historical wisdom: it does not lead color coulomb plus linear potential in the nonrelativistic limit.
+For the pion case the solution presented here has been already obtained for kindred model, albeit the renormalization
+and the kernels slightly differ numerically. That description of both - the light and heavy meson systems is possible
+
+6
+3000
+3500
+4000
+4500
+5000
+M [MeV]
+1e-06
+0.0001
+0.01
+1
+ σ
+2 :
+ λ :
+ η c (3)
+ η c (2)
+ η c (4)
+ η c (1)
+FIG. 3: The eigenvalue λ and the numerical error σ from the solution of BSE the solution for the ground state and the the
+first three excited state of pseudoscalar charmonium. The definitions can be find in [15], the bound states are for λ → 1 σ → 0
+when satisfied simultaneously for the meson mass M.
+————————————————————————————-
+within DSEs/BSEs formalism is not surprising fact [13]. However the use of almost identical kernel used for the
+charmonium and for the pion case is astonishing. Notably, the interaction of heavy quarks in quarkonium is far from
+conventional historical wisdom: it does not lead to static color coulomb plus linear potential in the nonrelativistic
+limit. The gauge term, which is not contribution for on-shell scattering fermions at all, turns to be important part.The
+form of kernels suggest that our gauge choice is very close to the Yennie gauge ξ = 3, known because of cancellation of
+infrared infinities in perturbation theory in this gauge. However here it comes out due to its perspective in truncation
+convergence of QCD gap equations.
+Obviously, using of spectral representation can be seen as heavy hammer tool for calculation of form factor for
+spacelike argument. There, the convenient calculation within the use of Euclidean metric works sufficiently irrespective
+of analytical property of the kernels. We also expect no big improvements when Isgur-Wise functions [45, 46] are
+calculated within the use of presented formalism as well. There are likely other quantities insensitive to the issue of
+confinement especially if vertices and quarks lines lie outside the timelike domain of momenta. On the other side, the
+methodology of calculation of form factor at resonant region is a challenge where spectral DSEs will take their correct
+place. Within the truncation presented here one can get the celebrated dispersion relation form [48] for Vacuum
+Hadron Polarization as well as one can enjoy the resulting dispersion relation for the electromagnetic meson form
+factor [1].
+At last but at not least we could stress again the reasoning and strategy of our indefinite gauge method. Obviously,
+If the quark SR exist in a given gauge, it is natural to expect that it exists in some other gauges as well. However to
+reach the resulting SR with the simultaneous reliable solution for meson spectra requires very different effort when
+one goes from one gauge choice to another one (here we the kernels are QCD vertices itself and they are not crippled
+presence by ad hoc auxiliar functions ). It is well known that LRA -Γµ = gT γµ- with the lattice gluon propagator
+obtained in Landau gauge, does not provide a good starting point for the calculation of mesons.
+That such LRA does not receive a proper strength one can see also from DSE solution alone. It has been checked
+that decreasing the fixing parameters, one gradually observe the growth of the peak in the quark spectral function
+and the dirac delta function is formed after passing through the critical coupling ξg2 with a lattice gluon propagator
+matched to the transverse (or metric tensor) part of interaction. At this critical point one gets non-confining (NC)
+propagator solution of familiar form
+SNC(p) =
+R
+̸ p − mp
++
+� ∞
+th
+doσv(o) ̸ p + σs(o)
+(p2 − o + iǫ)
+(4.1)
+with two continuous spectral functions σv,s being nonzero only from the threshold (being identical to the fermion pole
+mass mp, if one allows nontrivial gluon spectral function). Such solution typically arise at non-confining theory like
+QED, being preserved for not large coupling in toy quantum field models [47]. With the value R ≃ 0.75 the authors
+
+7
+of [6] obtained such solution for fermion propagator within LRA and lattice Landau gauge gluon data.
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+
diff --git a/2tAyT4oBgHgl3EQfb_cv/content/tmp_files/load_file.txt b/2tAyT4oBgHgl3EQfb_cv/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,506 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf,len=505
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='00272v1  [hep-ph]  31 Dec 2022  Quark spectral functions from spectra of mesons and vice versa  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' ˇSauli1, ∗  1Department of Theoretical Physics, Institute of Nuclear Physics Rez near Prague, CAS, Czech Republic  Within the QCD functional formalism, having the approximations controlled by physical masses  and decays of pseudoscalar mesons, we extract spectral function of quarks from which the meson  are composed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We choose the pion for the case of light quarks and ηc(N) for the extraction of  charm quark spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  For this purpose we solved improved ladder-rainbow truncation  of the spectral Dyson-Schwinger equations for quarks coupled to Bethe-Salpeter equation for the  pion and the pseudoscalar charmonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We begin with indefinite gauge fixing method for class of  covariant linear gauges and search for its optimal value in given fixed truncation of Dyson-Schwinger  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' All kernels are represented self-consistently by known or extrapolated solutions known  form lattice or QCD DSEs solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  We require the formalism gives us the spectral functions  with arbitrarily high numerical accuracy, while providing known experimental properties of mesons  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' , we found that the ladder rainbow approximation can serve for this purpose when  the Yennie gauge is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Properties of such spectral functions are shown and its connection  with confinement is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  PACS numbers: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='St, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='Tk  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  INTRODUCTION  QCD is a rigid part the Standard Model already for half century and passed many nontrivial tests when compared to  the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The knowledge of correlation functions at time-like momentum region is crucial for the first principle  determination of hadronic resonances and understanding of production of hadrons [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Lattice theory is formulated  in the Euclidean space where it is also solved, however the analytical continuation to the timelike Minkowski subspace  represents is quite often an ill defined numerical problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  A complementary and very attractive approach is the spectral functional formalism, where the analytical contin-  uation is performed at very beginning and the set of Dyson-Schwinger equations is solved for spectral function in  Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Such method is appreciated quite recently [1, 3–7] and includes the topics of spectral renormal-  ization - primary or secondary subtractions technique performed at the timelike momentum scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' A Yang-Mills  sector of SU(3) gauge theory was considered in [8, 9] bringing a new insights in the conventional Landau gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' In  order to get agreement with lattice data, the importance of transverse vertices in pure gluodynamics was shown [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  A meaningful comparison to recent lattice data was missing when the first spectral DSEs study [10] has appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Nowadays, the transverse QCD vertices are known to be very important in the quark as well as in the gluon sector in  the Landau gauge and they are responsible for a large enhancement (suppression) of the propagator (proper selfener-  gies) in infrared domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' How to incorporate transverse vertices in spectral quark sector was only suggested in [1] for  the case of quark-photon vertex but not yet implemented in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The purpose of presented paper is not a jump  to bandwagon or chasing the train of DSEs scheduled in the Euclidean space [11, 12], but to push theory of spectral  DSEs in its own direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Since the relativity is less urgent for mutual interaction of heavy quarks Q = c, b inside heavy mesons, nonrelativistic  quantum mechanic was widely used to describe quarkonia and their transitions ( for a review see [16]) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' History  tell us, that in addition to perturbative Coulomb “one gluon exchange” potential, the linear rising potential have been  proposed to explain spectra of excited quarkonia [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' If fine tuned and ignoring quark content mixing and ignoring  resonant character of excited states, such models reasonably describe static spectra of strangeonia [18] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' In  lattice QCD, a confining potential for a static quark-antiquark pair are computed with Wilson loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' This technique  lies aside of quark-antiquark scattering kernel used in the DSE/BSEs heavy quarkonia studies [13–15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' To match  the two different approaches -the DSEs and Wilsonian static quark potential together, is longstanding desire but  unfinished story (for attempts see [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Actually, to the author best knowledge, there is not known truncation of DSES, which naturally offer the interaction  kernels, which is consistent with string picture of confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The string-like interaction is either introduced by hand  ∗Electronic address: sauli@ujf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='cas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='cz  2  [14, 15] or even completely avoided by the use of auxiliary entire function [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' In all cases, the approximations made  turns to be odd from perspective of spectral quark functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Before presenting the details of truncation, which complies with the existence of quark spectral function, let us  mention here the so called hindered transitions , which were measured at various channels [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The large discrep-  ancy between of measured rate with nonrelativistic theory prediction were usually attributed to missing relativistic  corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' To explain the quarkonia and their transitions, a very recent treatments based either on BSE/DSEs  formalism, nonrelativistic quantum mechanic or other techniques [23–32] still represent very different approaches with  not completely clear connection to QCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' To this point, a systematically improvable truncation of DSEs with a clear  bridge to analytic properties of S-matrix could be a reliable candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Notably the formalism of spectral DSEs we  present here, leads to the dispersion relation for hadronic form factors (including the hindered transitions as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The main aims of presented work is twofold: we solve the spectral quark DSE and extract thus information on the  quark spectral function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Simultaneously, within the obtained quark propagators we solve the BSE for mesons and  check the solution against the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We employ the calculation scheme, which gives us solution with  desired analytical properties for physical meson from the very beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We expect the ladder-rainbow approximation  gives the first estimate of spectra for both light and heavy mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' For this purpose we leave popular Landau gauge  and extrapolate known spectral solution obtained recently [4] into other linear gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Enchantingly, it turns out the  pion and heavy quarkonium (pseudoscalar charmonia) can be described within the same kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' A correctly obtained  meson properties thus control reliability of obtained quark spectral functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' An on shell peak is washout specifically  and anomalous branch point is generated, which naturally explains a large contribution of seemingly unphysical  gauge term in considered approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We discuss the limitation methods as well as we suggest future prospects  and directions where the method can provide useful results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  TRUNCATION OF SDES SYSTEM FOR THE PION  For purpose of completeness we write down all necessary equations here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The appropriate quark DSE for the quark  propagator S can be written in the following way  S−1(q, µ) = A(µ) ̸ q − Bq(µ) − [ΣR(q) − ΣR(µ)] ,  ΣR(q) = i4  3  �  dDk  (2π)D γµS(k)ΓνGµν(k − q) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='1)  where for the product of the quark-gluon vertex Γ and the gluon propagator G we take  ΓνGµν(p) = γνN(ξ)  �  −gµν + pµpν  p2  � �  do  ρT (o)  p2 − o + iǫ − ξg2  p2  pµpν  p2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='2)  where ρT is the gluon spectral function obtained with Landau gauge in the paper [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  We adopt a conventional renormalization condition and we take ℜA(µ) = 1 ℜB(µ) = 300MeV at the timelike  subtracting point µ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5GeV 2, noting that the imaginary parts of functions A, B, which completely characterize  the quark propagator  S−1(p) ≠ pA(p) − B(p)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='3)  do note take arbitrary value at the timelike renormalization point, but they are matter of numerical search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The  letter R stands for the fact that the quark selfenergy was (or can be) regularized before the secondary subtractive  renormalization takes its place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Proper regularization is more crucial and unavoidable step in gluon sector of DSEs  in order to prevent violation of gauge invariance by inappropriate numeric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The last term in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='2) is aforementioned gauge term appearing in the product with the gauge coupling  g2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The prefactor N(ξ) nonlinearly depends on the coupling as well as on the gauge parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The lattice data  for the gluon propagator are known only for ξ gauge parameter ξ being smaller then 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5 [33, 34] in QCD without  quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Solving a Nielsen identities the solution of truncated system of Yang-Mills DSEs has been obtained in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Both studies show the gluon propagator is suppressed gradually in the infrared domain when ξ is getting larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We  exploit this and perform a very simple extrapolation of our DSE kernel into other gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The overall prefactor N(ξ) the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='2) is varied and represents the only change when we extrapolate to nontrivial  value of the gauge parameter ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Gauge term is not getting dressed, due to the unbroken gauge invariance in QCD and  ξ appears linearly in our LRA employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The extrapolation to other gauges is implicit since the ratio of the transverse  and gauge term is adjusted by the solution of Bethe-Salpeter equation for the pion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Hence we can estimate the gauge  only at the end of (quite nontrivial) search of solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Nevertheless, for the first time we provide the first naive guess  3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5  [GeV]  0  2  4  6  σ v,s   [GeV  2,GeV  1]  u,d - s  u,d - v  charm v  charm s  u,d  c  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' 1: Quark spectral functions, solid line stand for the functions σv, , dashed line for σs plotted against the energy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The  left two blobs are for light quarks, the one on the right for the charm quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  ————————————————————————————-  ξ ≃ 3, which is based on comparison with solutions of Yang-Mills system [35] performed for various gauge parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Notably, the agreement of both solutions [1] and [35] with lattice data [36] in Landau gauge in the infrared domain  is enough for our rough estimate of the gauge parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Further approximation we describe in the next text could  not be crucial since the meson physics is not sensitive to UV tails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The gluonic spectral function is a continuous function starting to be nonzero at p2 = 0, showing the violation of  passivity bellow several GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Here however, in order to reduce number of numerical integration in our pioneering  study we use a simplified (UV finite) fit for the gluonic spectral function  ρT (o) = −δ(o − m2  g) + δ(o − Λ2)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='4)  with mg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='6GeV and Λ = 2GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Thus we have estimated the gauge only at the end of (quite nontrivial) search of solution for the pion BSE which  produces the correct pion mass mπ = 140MeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The gauge choice for which the pion properties are obtained most  effectively corresponds numerically with the following rate  g2ξ  N(ξ) = 3  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5)  while for the absolute value we get 4N(ξ)  3(4π)2 = 16  3 , ( 4g2ξ  3(4π)2 = 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  It is tentative to identify our gauge with a Yennie gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Our guess is certainly naive, since based on comparison  with different truncation of DSEs for the Yang-Mills system [35] performed for various gauge parameters, including  the Landau gauge as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The agreement of both solutions [1] and [35] with lattice data [36] in Landau gauge  in the infrared domain is only approximate, also a further approximation represented by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='4) calls for better  identification of the gauge parameter if needed for any future purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' As we have checked, the realistic picture can  be obtained for the other rates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5), providing the range of our estimate ξ = 3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Let us stress for clarity that  extreme cases like Landau gauge ξ = 0 or complete neglections of the first term N(ξ) = 0 do not lead to satisfactory  picture at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  As the name indicates, the spectral DSE is nothing else but rewritten original DSE in a way that it can be solved  for the two quark spectral functions σv(o) and σs(o) , rather then for the propagator in momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The quark  propagator then can be calculated through its spectral representation:  S(p) =  � ∞  0  do̸ pσv(o) + σs(o)  p2 − o + iǫ  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='6)  in the entire complex plane of square momentum p2 ( real value p2 > 0 correspond with Minkowski space domain in  our metric choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The solution of spectral DSE is described in details in series of papers [1, 3, 4] and the only change is improved  numeric for purpose of presented paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We use two Gaussian integrator, the first is adjusted to fit the dominant peak  4  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5  o  1/2  [GeV]  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5  2  DLSF  charm v  charm s  u,d -v  u,d -s  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' 2: Dimensionless quark spectral functions oσv(o) (solid line) and  �  (o)σs(o) (dashed line) for the light and charm quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  At larger (smaller) energy scale the broad peak for the charm flavor (u,d) quark spectral function develops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  ————————————————————————————-  in the quark spectral function, the second one has been suitably mapped to the rest of infinite interval of spectral  variable o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The deviation from assumed analyticity σ2 as established in [3, 4], it can be arbitrarily minimized when  approaching the correct values of imaginary parts of the functions A(µ) and B(µ) at renormalized point µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The value  σ2 = 10−6 can be achieved easily and seems to be limited by numeric rather then systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The BSE for the pion has been solved for the dominant BS vertex component by eigenvalue method in complex  momentum space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Such single component approximation is working well not only for the ground state [44] but with  a slight modification for the excited states as well [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The BSE involves product of the scalar functions Sv(k +  P/2)Sv(k −P/2) and Ss(k +P/2)Ss(k −P/2) evaluated at complex momenta (kE is real Euclidean momentum, while  the total momentum PE = (im, 0) in rest of the pion), which we evaluate within the use the spectral representation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Since the shape of spectral functions is difficult to control (at least at this stage), we do not use some numerical  fit and implement additional integration over the spectral representation to determine products SiSi i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' As usually,  numerical codes either for BSE and DSE are available for public [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' An alternative way to solve BSE in Minkowski  space are known, till now used simplified systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' for constituent quark models[38–40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The method could be  necessary if one evaluates the resonant hadronic form factor [1], however as the BSE is converted into more dimensional  integro-differential equations, we prefer to solve BSE defined in the complex Euclidean space for purpose of presented  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The resulting spectral functions for the the light quarks are shown in the figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We work in the izospin limit  and ignore electromagnetic interaction, thus the spectral function for the up quark is identical to the d quark one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  According to broad shapes of both functions σv,s, they describe confined objects- the light quark excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The  quarks continuously change colors inside hadrons by exchanging of gluons, hence a width of the main peak can be  interpreted as the inverse of mean time τu,d ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='2GeV −1, which the quark of given flavor spent with a given color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Similarly to quark weak decays, they do not represent observable, we avoid the name “decay width” in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Attentive reader has surely noticed that the on-shell delta functions are absent in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Consequently the  thresholds vanishes at evaluated form factors, which is in expected accordance with Wilsonian area law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Such behavior  is intuitively expected, and in fact it has been mimic in [41–43] by the introduction of certain infrared cutoff in the  Feynman(Schwinger) parameter in various evaluations of hadronic form factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  ηc(N) QUARKONIA  The same QCD kernel that govern interaction between quark-antiquark in the light meson is used to calculate the  heavy quarkonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' However as the interaction is not flavor universal- it constitute by the quark-gluon vertex as well,  a changes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' softening of the interaction is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  It can be done by a change of effective mass parameters which now takes rescaled values by the factor r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='721,  more precisely they take the values  mc  g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='433 GeV ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' mc  Λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='442 GeV ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='1)  while the dimensionless couplings like g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' ξ do not change their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We renormalize such that ℜAc(µ) = 1 and  5  BSE EXP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  2980 2980  3442 3638  4150 3810  4720  –  TABLE I: Comparison with PDG data (second column) and calculated spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  ReBc(µ) = r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='3GeV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The search gives for the the imaginary parts ℑAc(µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='113 and ℑBc(µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='106rGeV at  renormalization point µ2 = r20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='5GeV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The kernel is not further tuned however as the excited states ranges over the relatively large scale, small further  change we need is to incorporate the total momentum into the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Elsewhere more important diamond diagrams  (the diagrams with interupted quark horizontal lines by gluon lines) should contribute to the kernel with substantially  small effect (note M(ηc) ≃ M(J/ψ) Instead of evaluating these complicated diagrams we mimic their small effect and  insert the following prefactor  fη =  1  √  2  �  1 + M(ηc(2))2  P 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='2)  This formally leaves the BSE for ηc(2) completely identical to the pion case, while the couplings are softened by few  percentage for higher excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Thus as expected for charmonium, the two mutually opposite poles of the kernel are getting closer, which suppress  the metric term when comparing to the pion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Nevertheless, the entire effect on the charm quark spectral function is  very the same as for the light quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The resulting charm quark spectral functions are added into the fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Since  the spectral function are dimensinfull object, we introduce the dimensionless quantity  �  (o)σs(o) and oσv(o) for a  better comparison of spectral function of different flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' These object are compared in the figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The on-shell  singularity is washout to a broad peak and heavy free quark excitation does not exists at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' A picture of confinement  that emerge in spectral framework of DSEs is very the same for the light as well for the heavy quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' A scalar string  interaction governed by a linear potential is not actually needed at any quark sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The spectra of bottomonia can be obtained by a similar fashion, however our two mutually beating poles turns  to be cruel approximation at bottomonium scale and slightly more honest approximation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We plan to  perform more comprehensive study of BB system in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  The obtained masses are tabled in the Tab I further predicted and not yet observed states we only list here:  5436,6186,7030MeV,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Recal, those above the first excitation all they lie above open charm threshold and they  become broad resonance and our predicted values ignores coupling to D mesons completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Interestingly, not the  sure of mass but the mass itself is linear in principal value N , not supporting the string/Regge trajectory at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Furthermore, the vertex functions are not orthogonal in sense two states with different N can be produced in single  photon annihilation of e+e− (this is a bit free extension of quantum mechanical orthogonality, it obviously relies on  the formula for normalization of BSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We also show our ultimate numerical search for eigenvalue λ and the deviation  σ2 in the figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' A single point shown in this figure costed one day of work of recent single processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Even working  with multiprocessor machines the reader can imagine the time consumed before the truncation of DSE/BSE has been  established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  CONCLUSION  Using indefinite gauge fixing we have solved coupled set of spectral quark Dyson-Schwinger equations and Bethe-  Salpeter equation for the pion and we have extended the method to the heavy quarks sector represented by pseudoscalar  charmonia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Facing the resulting spectral functions we get simple picture of confinement of the light as well as the heavy quarks:  quarks are never on-shell inside the hadrons, the inverse of quark propagator never gets zero for a real momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The  sharp singularity is completely wash out due to the imaginary part which is gradually growing from the anomalous  thresholds- the zero momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Notably, the interaction of heavy quarks in quarkonium is far from conventional  historical wisdom: it does not lead color coulomb plus linear potential in the nonrelativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  For the pion case the solution presented here has been already obtained for kindred model, albeit the renormalization  and the kernels slightly differ numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' That description of both - the light and heavy meson systems is possible  6  3000  3500  4000  4500  5000  M [MeV]  1e-06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='01  1  σ  2 :  λ :  η c (3)  η c (2)  η c (4)  η c (1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' 3: The eigenvalue λ and the numerical error σ from the solution of BSE the solution for the ground state and the the  first three excited state of pseudoscalar charmonium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The definitions can be find in [15], the bound states are for λ → 1 σ → 0  when satisfied simultaneously for the meson mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  ————————————————————————————-  within DSEs/BSEs formalism is not surprising fact [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' However the use of almost identical kernel used for the  charmonium and for the pion case is astonishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Notably, the interaction of heavy quarks in quarkonium is far from  conventional historical wisdom: it does not lead to static color coulomb plus linear potential in the nonrelativistic  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' The gauge term, which is not contribution for on-shell scattering fermions at all, turns to be important part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='The  form of kernels suggest that our gauge choice is very close to the Yennie gauge ξ = 3, known because of cancellation of  infrared infinities in perturbation theory in this gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' However here it comes out due to its perspective in truncation  convergence of QCD gap equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  Obviously, using of spectral representation can be seen as heavy hammer tool for calculation of form factor for  spacelike argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' There, the convenient calculation within the use of Euclidean metric works sufficiently irrespective  of analytical property of the kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' We also expect no big improvements when Isgur-Wise functions [45, 46] are  calculated within the use of presented formalism as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' There are likely other quantities insensitive to the issue of  confinement especially if vertices and quarks lines lie outside the timelike domain of momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' On the other side, the  methodology of calculation of form factor at resonant region is a challenge where spectral DSEs will take their correct  place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Within the truncation presented here one can get the celebrated dispersion relation form [48] for Vacuum  Hadron Polarization as well as one can enjoy the resulting dispersion relation for the electromagnetic meson form  factor [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  At last but at not least we could stress again the reasoning and strategy of our indefinite gauge method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Obviously,  If the quark SR exist in a given gauge, it is natural to expect that it exists in some other gauges as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' However to  reach the resulting SR with the simultaneous reliable solution for meson spectra requires very different effort when  one goes from one gauge choice to another one (here we the kernels are QCD vertices itself and they are not crippled  presence by ad hoc auxiliar functions ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' It is well known that LRA -Γµ = gT γµ- with the lattice gluon propagator  obtained in Landau gauge, does not provide a good starting point for the calculation of mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  That such LRA does not receive a proper strength one can see also from DSE solution alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' It has been checked  that decreasing the fixing parameters, one gradually observe the growth of the peak in the quark spectral function  and the dirac delta function is formed after passing through the critical coupling ξg2 with a lattice gluon propagator  matched to the transverse (or metric tensor) part of interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' At this critical point one gets non-confining (NC)  propagator solution of familiar form  SNC(p) =  R  ̸ p − mp  +  � ∞  th  doσv(o) ̸ p + σs(o)  (p2 − o + iǫ)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='1)  with two continuous spectral functions σv,s being nonzero only from the threshold (being identical to the fermion pole  mass mp, if one allows nontrivial gluon spectral function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Such solution typically arise at non-confining theory like  QED, being preserved for not large coupling in toy quantum field models [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' With the value R ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='75 the authors  7  of [6] obtained such solution for fermion propagator within LRA and lattice Landau gauge gluon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Sauli, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' D 106, 3, 034030 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Sauli, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' D 1021, 014049 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Sauli, Few Body Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' 61 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Sauli, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='D 106, 9, 094022 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='  [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' Krein, Few Body Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' D 101,1 014027 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' Bhatnagar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Gebrehana, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content='  [27] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Babiarz, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Goncalves, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Pasechnik, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Sch¨afer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Soni, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Joshi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Shah, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Chauhan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Pandya, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' Oliveira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' Cardoso, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Oliveira, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content=' e-Print:  1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='06737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content='A Ivanov , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Liptaj, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+page_content='  [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Fischer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
+page_content=' Nickel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAyT4oBgHgl3EQfb_cv/content/2301.00272v1.pdf'}
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+1
+Process Variation-Aware Compact Model of Strip
+Waveguides for Photonic Circuit Simulation
+Aneek James, Anthony Rizzo, Yuyang Wang, Asher Novick, Songli Wang, Robert Parsons, Kaylx Jang,
+Maarten Hattink, and Keren Bergman
+Abstract—We report a novel process variation-aware compact
+model of strip waveguides that is suitable for circuit-level sim-
+ulation of waveguide-based process design kit (PDK) elements.
+The model is shown to describe both loss and—using a novel
+expression for the thermo-optic effect in high index contrast
+materials—the thermo-optic behavior of strip waveguides. A
+novel group extraction method enables modeling the effective
+index’s (neff) sensitivity to local process variations without the
+presumption of variation source. Use of Euler-bend Mach-
+Zehnder interferometers (MZIs) fabricated in a 300 mm wafer
+run allow model parameter extraction at widths up to 2.5 µm
+(highly multi-mode) with strong suppression of higher-order
+mode excitation. Experimental results prove the reported model
+can self-consistently describe waveguide phase, loss, and thermo-
+optic behavior across all measured devices over an unprecedented
+range of optical bandwidth, waveguide widths, and temperatures.
+Index Terms—Silicon photonics, compact modeling, process
+variation.
+I. INTRODUCTION
+S
+ILICON photonics (SiPh) has seen explosive growth in
+demand as a technology platform, driven by its adoption
+in data centers (DC), high performance computing (HPC) [1]–
+[3], quantum computing [4]–[8], and radio-frequency com-
+munication systems [9]–[11]. SiPh’s rapid rise and matura-
+tion has been enabled by its ability to leverage decades of
+research in the complementary metal–oxide–semiconductor
+(CMOS) industry, drastically reducing the typical research and
+development (R&D) costs associated with new semiconductor
+technologies [12]–[14]. SiPh, however, has not yet been able
+to mimic CMOS yield prediction tools for evaluating photonic
+integrated circuits (PICs). Yield is a ubiquitous metric used
+across semiconductor manufacturing, with improvements in
+yield being strongly correlated with reductions in the time
+and costs associated with PIC design cycles [15]–[17]. The
+need for predictive yield models can be mitigated to some
+This work was supported in part by the U.S. Advanced Research Projects
+Agency–Energy under ENLITENED Grant DE-AR000843 and in part by
+the U.S. Defense Advanced Research Projects Agency under PIPES Grant
+HR00111920014.
+A. James, Y. Wang, A. Novick, S. Wang, R. Parsons, K. Jang, M. Hattink,
+and K. Bergman are with the Department of Electrical Engineering, Columbia
+University, New York, NY 10027, USA. (Corresponding author: Aneek James,
+e-mail: aej2149@columbia.edu).
+A. Rizzo is with the Air Force Research Laboratory Information Directorate,
+Rome, NY 13441, USA.
+© 2023 IEEE. Personal use of this material is permitted. Permission from
+IEEE must be obtained for all other uses, in any current or future media,
+including reprinting/republishing this material for advertising or promotional
+purposes, creating new collective works, for resale or redistribution to servers
+or lists, or reuse of any copyrighted component of this work in other works.
+TABLE I
+FEATURES FOR MODELING STRIP WAVEGUIDE PERFORMANCE. THE
+MODEL IN THIS WORK DESCRIBES PHASE, LOSS AND THERMAL BEHAVIOR
+EFFECTS OVER A BROAD RANGE OF WAVELENGTHS AND WAVEGUIDE
+GEOMETRIES.
+Model Features
+[25]
+[26]
+This Work
+Wavelength [nm]
+1550
+1520–1570
+1450–1650
+Nominal Width Range [nm]
+480
+480–500
+400–2500
+Considered Variation Sources
+w,t
+w,t
+Arbitrary
+Statistical Parameter Variations
+
+
+
+Waveguide Scattering Losses
+
+
+
+Thermo-optic Effect
+
+
+
+w - Waveguide Width Variations
+t - Waveguide Thickness Variations
+degree by designing variation-robust devices [18] or PICs
+such that performance variations can be tolerated or cor-
+rected for post fabrication [19], [20]. In each of these cases,
+however, quantitative yield data cannot be determined prior
+to fabrication—an obstacle that will be exacerbated as the
+number of components per PIC in silicon is projected to
+scale well into the millions within the next decade [21].
+Circuit designers also need tools to optimize system-level
+performance through device-level design choices [22]. To meet
+rising circuit design complexity, commercial foundries must
+develop process design kits (PDKs) that include compact
+models that are both parameterized over a wide range of
+relevant design and environmental variables and describe all
+important device figures of merit [23], [24]. It is essential
+that strip waveguides in particular—a critical component of
+most SiPh circuits—are accurately modeled according to their
+expected fabricated performance.
+Broadly speaking, there are three ways to construct compact
+models: (i) look up table-based models, obtained directly from
+measurements or device simulations, (ii) models based on
+empirical fit functions, and (iii) physics-based models [23].
+Most previously reported work falls under the look-up table-
+based category [25]–[29]. These models can be parameterized
+using look-up tables (LUTs), where interpolation is used to
+predict the performance of designs not explicitly defined in
+the table. Ensuring that LUT models are accurate over a wide
+range of input parameters, however, requires measuring all
+waveguide figures of merit for every combination of input
+parameters; a task that scales exponentially with the number of
+modeled independent variables. Prior demonstrations methods
+also require the explicit connection of the measured effective
+and group index variations to a predefined number of process
+variation sources, introducing the possibility of error if any
+arXiv:2301.01689v1  [physics.optics]  4 Jan 2023
+
+2
+Fig. 1.
+a, Example electric field profile taken from Lumerical MODE. b,
+Simulated (scatter) and modeled (dashed) effective index vs wavelength for
+several waveguide widths. Each waveguide was simulated with a thickness of
+220 nm.
+systemic deviations exist between the simulation configuration
+and the realities of the fabrication process.
+In this paper, we report to the best of our knowledge, the first
+geometry-parameterized compact model of strip waveguides
+that can capture device performance over a wide range of
+wavelengths and waveguide geometries (see Table I). Using a
+novel derivation of the thermo-optic effect that is accurate for
+high-index contrast waveguides, we demonstrate our model’s
+ability to describe both scattering loss and the thermo-optic
+effect as a function of both design and statistical parameters.
+A novel group-extraction-based method allows the characteri-
+zation of process variations without presumption of a source or
+its associated sensitivity. This extraction methodology is used
+to construct a model from dozens of geometric variations of
+Mach-Zehnder Interferometers (MZIs) fabricated in a 300 mm
+commercial foundry. These use of Euler bends in these MZIs
+permits the characterization of wide waveguide performance
+with minimal higher-order mode excitation. Experimental re-
+sults validate the model’s accuracy in describing the phase,
+loss, and thermo-optic performance across the entire wafer.
+The model is also implemented in Verilog-A to demonstrate
+compatibility with electronic-photonic co-simulation environ-
+ments [30]–[32]. This work represents a key step toward
+the modeling of waveguide-based PDK components, enabling
+true-to-measurement circuit simulation at massive integration
+densities.
+II. PHYSICS-AWARE MODEL DEVELOPMENT
+Because the mode condition of an optical waveguide
+is
+described
+via
+a
+transcendental
+equation,
+completely
+generalized analytical solutions for the effective index (neff)
+are impossible to derive [33]. We therefore propose, as
+discussed in [34], finding a behavioral model that accurately
+captures its dependence on all design parameters over the
+relevant ranges of interest. In this section, we develop
+dependency models for the design parameters available.
+Fig. 2. a-c, Plot of simulated (scatter) and modeled (dashed) neff parameters
+�
+∂2neff/∂λ2, ∂neff/∂λ, neff,0
+�
+vs waveguide width (respectively). These
+values were for a waveguide with a thickness of 220 nm at a wavelength
+of 1550 nm. d, Comparison of the model (dashed) and simulated (scatter)
+neff vs waveguide width for different thicknesses. Simulated at 1550 nm.
+The semi-physical nature of the model is then leveraged
+to describe both the scattering loss and the thermo-optic
+coefficient. Process variations, whether of a design parameter
+or not, will be covered in Section IV.
+A. Wavelength Dependence
+The wavelength dependence of the waveguide neff is first
+considered. The neff of several silicon-on-insulator (SOI)
+waveguide geometries were simulated in Lumerical MODE
+(Fig. 1a). From the results, it is shown that the wavelength
+dependence over the S-, C-, and L-bands for all geometries is
+well-approximated by a second-order Taylor expansion for a
+wide range of waveguide widths sufficiently above the cutoff
+condition (Fig. 1b):
+neff, model(λ) =
+2
+�
+i=0
+1
+i!
+∂ineff
+∂λi
+����
+λ=λ0
+(λ − λ0)i.
+(1)
+B. Geometric Dependence
+As the Taylor expansion only captures the wavelength-
+dependence, it is clear that the fitting parameters ∂2neff/∂λ2,
+∂neff/∂λ and ∂0neff/∂λ0 (hereafter referred to as neff,0) are
+responsible for capturing the dependence on waveguide geom-
+etry. With respect to width, all three fitting parameters were
+previously found in [35] to be well described by the following
+behavioral model:
+∂ineff
+∂λi (w) = pi0 · w2 + pi1w + pi2
+w2 + pi3w + pi4
+,
+(2)
+for a total of fifteen model parameters. To verify correctness
+of the model, all three parameters were fitted to the simu-
+lation data with (1)-(2) using ordinary least squares (OLS)
+
+a
+Thickness
+Width
+2.8
+2.6
+neff
+2.4
+2.2
+1450
+1500
+1550
+1600
+1650
+Wavelength [nm]
+400 nm
+580 nm
+ 760 nm
+940 nm215 nm
+235 nm
+255 nm
+275 nm
+295 nm3
+Fig. 3. Comparison between modeled (dashed) and simulated (scatter) neff
+for higher order modes and the fundamental TM mode. All waveguides were
+simulated with a thickness of 220 nm.
+regression. The model was able to match all three parameters
+over the entire range of the width sweep (Fig. 2a-c). The
+close matching of the modeled and extracted Taylor parameters
+means that our modification of (2) still preserves its ability to
+match the behavior of effective index as a function of wave-
+length. By extension, these three Taylor parameters allow for
+a robust description of neff as a function of waveguide width
+(Fig. 2d). The data also demonstrates this agreement is not
+unique to any particular waveguide thickness, with different
+thicknesses producing different sub-parameter fits. Finally, it
+should be noted that both the numerator and denominator in
+(2) are polynomials of equal order. Our model consequently
+predicts that, for a given wavelength, the effective index will
+asymptotically approach a constant value as w approaches
+infinity. The value that the model approaches as w tends
+towards infinity can be interpreted as the equivalent neff of
+an infinite slab of the same thickness:
+lim
+w→∞ neff(λ, w) = nslab(λ).
+(3)
+In this way, our behavioral model can elegantly capture all
+significant features of effective index for the design parameters
+of interest. The model’s accuracy holds true for higher order
+modes as well, provided that they are sufficiently far away
+from their respective waveguide cutoff condition (Fig. 3).
+C. Scattering Loss
+Scattering loss due to sidewall roughness (SWR) can be a
+significant source of loss in most reported waveguide designs,
+making it critical for designers to accurately model [36]. In this
+section, we demonstrate our model’s ability to capture SWR
+loss as a function of waveguide geometry. It was first noted
+in [37] that the traditional Payne and Lacey model of SWR-
+induced loss [38], [39] was found to be identical in behavior to
+the derivative of the effective index with respect to waveguide
+width:
+αSWR(λ, w) = R ∂
+∂w [neff(λ, w)] ,
+(4)
+where R is a proportionality constant. As our model can
+describe neff as a function of width, a closed-form repre-
+sentation of ∂neff/∂w can be exactly derived. This equation
+Fig. 4.
+a, Graphical representation of a waveguide simulated with some
+sidewall roughness. The inset is a magnified view of the waveguide to clarify
+the definition of σrms. b, Scattering losses estimated from FDTD compared
+to the fit using our model based on Lumerical MODE data.
+can then be fitted to measured waveguide loss data to extract
+the proportionality constant. We validate this by fitting (4)
+to the scattering loss of a 7 µm long SOI waveguide with
+some SWR wall roughness in Lumerical 3D-FDTD (Fig. 4a).
+The roughness Root Mean Square (RMS) and correlation
+length were arbitrarily chosen to be σrms
+= 5 nm and
+Lcorr = 1 µm respectively. These parameters were then used to
+generate a random, anisotropic SWR on the waveguide walls
+[40]. Propagation losses were simulated for waveguide widths
+ranging from 450 nm to 850 nm. The results of the fitting are
+shown in Fig. 4b, with our model closely matching trend of
+the scattering loss behavior extracted from FDTD simulations.
+D. Thermo-Optic Effect
+Our model can also completely describe the thermo-optic
+coefficient of an arbitrary waveguide geometry without the
+need for any thermal measurements. The thermo-optic co-
+efficient of a waveguide mode most importantly requires
+knowledge of the confinement factor, which is the fraction of
+a mode’s power confined within each constituent waveguide
+material. Kawakami showed in [41] that for a waveguide
+made up of N materials, each with with an index nk and a
+confinement factor Γk:
+N
+�
+k
+Γkn2
+k = ngneff
+(5a)
+�
+k
+Γk = 1,
+(5b)
+where (5b) is derived from noting that the sum of all con-
+finement factors must equal unity due to power conservation.
+
+2.5
+n
+2.0
+TEO
+TE1
+TE2
+TMO
+0.5
+1.0
+1.5
+2.0
+WG Width [um]20m
+Width
+SOl Waveguide4
+A closed-form of the confinement factor for a two-material
+waveguide (e.g. SOI wires) can then be derived:
+Γcore = ngneff − n2
+clad
+n2
+core − n2
+clad
+(6a)
+Γclad = n2
+core − ngneff
+n2
+core − n2
+clad
+,
+(6b)
+where Γcore is the power contained in the waveguide core and
+Γclad is the power contained in the cladding.
+Next, we must obtain an expression that describes the
+thermo-optic effect on neff in terms of the confinement factor.
+A common approximation of the thermo-optic coefficient of
+neff is
+∂neff
+∂T
+≈ Γ1
+∂n1
+∂T + Γ2
+∂n2
+∂T + . . . ,
+(7)
+where δ represents a small perturbation in the values, Γn is
+the confinement of the mode within material n and ∂nn/∂T
+is the thermo-optic coefficient of material n [42]. Though
+this equation is widely used [43]–[45] and may be accurate
+in certain scenarios, to the authors’ knowledge it has never
+been demonstrated to be a generally accurate approximation.
+We therefore start from first principles and consider a general
+perturbation of the wave equation [46]:
+δ
+�
+β2
+eff
+�
+= Γcore
+ω2
+c2 δ
+�
+n2
+core
+�
++ Γclad
+ω2
+c2 δ
+�
+n2
+clad
+�
+,
+(8)
+where βeff is the effective wavenumber, Γcore is the con-
+finement in the waveguide core, Γclad is the confinement in
+the waveguide cladding, and ncore and nclad are the core and
+cladding indices respectively. Carrying this operation through
+and combining with (1) (see Appendix A for details) yields:
+neff(λ, w, T) ≈ neff,T0(λ, w) + ∂neff
+∂T (T − T0)
+(9a)
+∂neff
+∂T
+= Γcore
+ncore
+neff, T0
+∂ncore
+∂T
++ Γclad
+nclad
+neff, T0
+∂nclad
+∂T
+,
+(9b)
+where neff,T0 is the neff at some reference temperature T0. The
+key addition to (9) compared to prior literature is the scaling
+of each thermo-optic term by ratio between the material and
+effective indices. As the index contrast between the core and
+cladding decreases, our model will approach the (7). Thus
+it is clear that our model will outperform (7) in accuracy
+when describing high index contrast materials, such as the
+SOI waveguide geometries prevalent in SiPh.
+With these expressions, our confinement factor and the
+thermo-optic coefficient models can be validated. The simu-
+lated confinement factor is compared to our model prediction
+at 1550 nm in Fig. 5a. The optical properties of silicon and
+silicon dioxide used in our model were taken directly from
+[47]. There was a near perfect agreement between the modeled
+and simulated confinement factor, showing that the general
+behavior of confinement factor is captured by our model
+(Fig. 5a). The modeled thermo-optic coefficient is validated
+by simulating how the neff of a SOI waveguide varies with
+temperature using Lumerical MODE (Fig. 5b). Silicon was
+assumed to have a thermo-optic coefficient of 1.9 × 10−4 K−1
+[48] and SiO2 was assumed to have a thermo-optic coefficient
+of 1 × 10−5 K−1 [49]. The model and simulations show
+Fig. 5.
+a Modeled (dashed) and simulated (scatter) confinement factor vs
+waveguide width for different thicknesses. b, Comparison between simulated
+(scatter), previously reported model (dotted, Eq. (7)) and our work (dashed
+line, Eq. (9)) describing neff vs Temperature of a 480 x 220 nm waveguide.
+exceptional agreement from 300 - 1200 K, despite the fact that
+our model does not require any data from thermal simulations
+or measurements. As predicted, the previously reported model
+of the thermo-optic effect (7) significantly under-predicts the
+expected change in neff. It should be noted that in real devices,
+waveguide geometry itself is a function of T due to thermal
+expansion. This can be accounted for by modeling w as a func-
+tion of T. Experimental results in Section V-C, however, show
+that assuming a constant width geometry provides sufficient
+accuracy.
+Having a model of the thermo-optic effect that is accurate
+over a wide range of conditions like this one holds a great
+deal of potential to enable more robust design exploration,
+such as evaluating photonic waveguide heater designs [50],
+[51], characterizing self-heating in micro-resonators [52], or
+studying the effect of ambient temperature fluctuations in a
+system.
+E. Parameter Extraction
+The practical utility of a compact model is greatly deter-
+mined by the associated parameter extraction procedure to
+connect the model to a given foundry process. This is particu-
+larly important when developing statistical models, as accurate
+parameter extraction is the only way to guarantee that process
+variations are accurately reflected in the model. A popular
+solution is to leverage the phase-sensitivity of interferometric
+optical filters—such as Mach-Zehnder interferometers (MZIs),
+microresonators, or arrayed waveguide gratings (AWGs)—to
+monitor process variations across a wafer. Regardless of the
+chosen device, a shared difficulty lies in accurately guessing
+what particular interference fringe position corresponds to a
+particular fringe order [25], [26], [53]. Our method is based on
+
+a
+0.80
+0.75
+210 nm
+220 nm
+230 nm
+215 nm
+225 nm
+0.4
+0.6
+0.8
+1.0
+WG Width [um]
+b
+2.7
+neff
+2.6
+400
+600
+800
+1000
+1200
+Temperature [K]
+This Work - Previous Model
+ Simulated5
+the curve-fitting method presented in [25] and [54], with some
+additional steps described to include waveguide dispersion as
+an extracted parameter.
+The first step in parameter extraction is to characterize
+the group index (ng) of a fabricated interferometer from
+a wavelength sweep of the device. To enable this, (1) is
+rearranged into a more suitable form:
+neff(λ) = 1
+2
+∂2neff
+∂λ2 λ2 + Bλ + C
+(10a)
+B = ∂neff
+∂λ − ∂2neff
+∂λ2 λ0
+(10b)
+C = 1
+2
+∂2neff
+∂λ2 λ2
+0 − ∂neff
+∂λ λ0 + neff,0,
+(10c)
+where B and C are fitting parameters that aggregate the 1st and
+0th order terms from (1) respectively. Following the procedure
+described in [54], it is first noted that the fringe condition of
+an inteferometric device is described by
+φ = 2π
+λ neff(λ)L = 2πm,
+(11)
+where φ is the phase difference between the interferometry
+arms, L is the path length of the interferometer, λ is a partic-
+ular fringe wavelength, and m is an integer corresponding to
+the particular fringe order. To extract our model parameters,
+a wavelength sweep of the interferometric device is required.
+Once this is performed, a peak finding algorithm can be used
+to detect the wavelength of all detected fringes. A function that
+relates the relative fringe locations to the ng of the waveguide
+is now required. This can be done by defining a continuous
+function that will yield an integer value at each of the detected
+fringe locations. Let m0 represent the particular fringe order
+corresponding to an arbitrarily chosen reference fringe located
+at λ0. The fringe order m of any other fringe can be defined
+relative to this reference as
+m = m0 +
+� λ
+λ0
+dm
+dλ dλ = m0 + ngL ·
+� 1
+λ − 1
+λ0
+�
+.
+(12)
+This continuous function now allows us to redefine the mea-
+sured fringes into a form suitable for parameter extraction.
+A reference fringe variable n is now defined by letting
+m = (m0 + n). Inserting this back into (12) produces:
+n = ngL ·
+� 1
+λn
+− 1
+λ0
+�
+,
+(13)
+where each relative fringe n is located at an associated
+wavelength λn. Using (13), the ng of the measured device
+is now directly related to the measured fringe locations. This
+fitting equation must now be extended to our specific model
+parameters. The ng of a waveguide is defined to be
+ng = neff − λ∂neff
+∂λ .
+(14)
+Combining with (10a) yields an expression for ng in terms of
+our compact model:
+ng = C − 1
+2
+∂2neff
+∂λ2 λ2.
+(15)
+Fig. 6. a, Captured spectrum of simulated MZI used for parameter extraction.
+The waveguide mode was simulated in Lumerical MODE, and then exported
+to a MZI waveguide simulation block in Lumerical INTERCONNECT. b,
+Linear Regression of fringe wavelengths to extract the ng performed on the
+detected fringes from a. c, Possible neff solutions (black, dashed), along with
+the actual solution (red), determined by the ng extracted in b.
+By inserting (15) back into (13), we can derive an OLS
+regression-compatible expression:
+n = CΛC − ∂2neff
+∂λ2 ΛS
+(16a)
+ΛC = L ·
+� 1
+λn
+− 1
+λ0
+�
+(16b)
+ΛS = L
+2 ·
+�
+λn − λ2
+n
+λ0
+�
+,
+(16c)
+where [ΛC, ΛS] are explanatory variables. Performing an OLS
+regression between n and [ΛC, ΛS] gives us two of our three
+fitting parameters in (10). Finally, B can be calculated by
+combining equations (11) and (10a):
+B = m
+L − 1
+2
+∂2neff
+∂λ2 λm − C
+λm
+,
+(17)
+where the only uncertainty is what fringe order m corresponds
+to each measured fringe λm. Once B is determined from (17),
+(10b) and (10c) can be used to determine the original fitting
+parameters in (1). It should be noted that each detected fringe
+(m, λm) location will yield very small variations in the B value
+due to resolution-based uncertainty in the exact value for λm.
+For a best guess, all values Bm taken from each measured
+fringe λm should be averaged together.
+To validate this method under ideal conditions, an MZI
+constructed using 480 nm x 220 nm waveguides is simulated in
+Lumerical INTERCONNECT. To ensure accuracy, the wave-
+guide’s neff was first simulated in MODE and then exported
+to a MODE Waveguide element in INTERCONNECT. As the
+full-width half-maximum (FWHM) of the MZI does not affect
+the extracted neff, the waveguides were arbitrarily assumed to
+have a 2.5 dB/cm loss and the coupling coefficient was chosen
+to ensure critical coupling. The spectrum of the simulated MZI
+
+a
+Power [dBm]
+20
+b
+10
+C
+2.5
+neff
+2.4
+2.3
+1500
+1550
+1600
+1650
+Wavelength [nm]6
+is shown in Fig. 6a. Fringe locations were extracted using
+a peak finding algorithm. The fringe located closest to the
+center of the sweep was arbitrarily chosen as n = 0. Using
+(16), OLS regression found ∂2neff/∂λ2 = −0.136 µm−2 and
+C = 3.9215 (Fig. 6b). From here, the family of solutions for
+neff is plotted in Fig. 6c. Each particular solution corresponds
+to a different guess on the fringe orders detected, e.g. m0 = 52
+vs. m0 = 53. The separation between each neff solution plotted
+in 6c is determined by the free-spectral range (FSR) of the
+interferometer, with a larger FSR corresponding more widely
+separated solutions.
+To determine the correct fringe order of the reference we
+use the fact that, from the simulations performed in Section
+II, we know the waveguide geometry has an neff of 2.411 at
+the reference fringe location. In Section III we explain how
+to increase the accuracy of this estimation to avoid errors
+introduced by this simulation. From this, the reference fringe
+order is found to be m0 ≈ 114.03. Since fringe orders must
+be integer numbers, the result is rounded to the nearest integer
+114. By combining (10a)-(10c), the original fitting coefficients
+are found to be ∂neff/∂λ = −1.078 µm−1 and neff,0 = 2.411.
+To evaluate accuracy of our extraction, we define the relative
+error between the extracted and simulated neff’s σerror by:
+σerror =
+��
+(neff, model − neff, sim)2 dλ
+�
+n2
+eff, simdλ
+,
+(18)
+where neff, sim is the effective index from the MODE simula-
+tion, used as a reference to quantify our method’s accuracy,
+and neff, model is the result from applying our extraction method
+to the simulated MZI. Upon evaluation, the total relative error
+was found to be 0.017%. Since the order of the reference
+fringe is correct, the remaining model error is attributed to
+inaccuracies in the initial regression fit using (16a)-(16c).
+III. MORE ROBUST neff EXTRACTION UNDER PROCESS
+VARIABILITY
+The reliability of the extraction is highly sensitive to the
+guessed value of the reference fringe order. For the example
+in Section II-E, we used a priori knowledge of the neff at the
+reference fringe to estimate its order. Therefore, any deviation
+between the assumed and actual waveguide dimensions risks
+introducing error. By noting that the initial order estimate
+rounded to the nearest integer, we can use (11) to define
+a boundary beyond which our fringe order guess will be
+incorrect [25]:
+|∆m| = |neff, actual − neff, guess| ≤ λm0
+2L .
+(19)
+We can see that, to raise confidence in the guessed fringe order,
+either the accuracy of our neff guess must be increased or the
+interferometric path length must be decreased. As explained in
+Section II-E, our extraction method begins by directly extract-
+ing the ng of a given interferometer via optical sweep. Process
+variations will therefore appear as variations in the extracted
+values for ∂2neff/∂λ2 and C. By measuring several devices
+of the same drawn width across the all measured dies, wafers,
+and lots, the influence of the random width and thickness
+variations can be eliminated by averaging their extracted fitting
+Fig. 7. a, Plot of the ng error function for one sample. The error function
+shows a minimum at roughly 491.5 nm, which closely agrees with the actual
+waveguide width of 490 nm. b, Convergence of the etch bias estimate for
+different numbers of samples averaged.
+parameters. As the sample size becomes sufficiently large—
+with the necessary sample size being a function of the severity
+of the process variations—any remaining deviation between
+the nominal and averaged parameters will be the result of a
+systemic etch biases on the waveguide width. We therefore
+propose estimating this etch bias by creating a preliminary neff
+model based on the results of a photonic mode solver, such as
+Lumerical MODE. Using this model, an equivalent waveguide
+width can be found by minimizing the error function
+min
+w
+��
+[ng, model(w, λ) − ng, meas(λ)]2 dλ
+�
+n2g, meas(λ)dλ
+,
+(20)
+where ng, meas is the extracted model of ng using the averaged
+extracted parameters and ng, model is the simulation-based,
+width-dependent a priori model of neff. The neff of our
+equivalent waveguide width can then be plugged into the a pri-
+ori model to provide a more accurate fringe order estimate.
+In this way, we can increase the accuracy of our guessed
+effective index, regardless of whether the modeled waveguide
+composition is accurate to the virtual device composition.
+We now discuss the robustness of this optimization routine
+in the presence of other systemic non-idealities and its ability
+to perform etch bias correction. To do this, we need a ’ground
+truth’ value for neff, which we obtain by simulating all the
+non-idealities in Lumerical MODE. Subsequently we perform
+the parameter extraction using Lumerical INTERCONNECT.
+By comparing the extracted neff to the known simulated
+value for neff, we can directly evaluate the robustness of our
+methodology.
+A. Statistical Geometric Variation
+To test the extraction procedure’s accuracy under process
+variations, a simulation of 100 random variations on the
+
+a
+2
+Actual
+Guess
+Error Function
+!
+460
+480
+500
+520
+540
+Guessed Widths [nm]
+b
+4 nm
+26 nm
+Etch Bias [nm]
+Estimated
+11 nm
+33 nm
+15
+18 nm
+40 nm
+10
+20
+40
+60
+80
+100
+Number of Samples7
+Fig. 8. Plot of mean error in neff over the simulation bandwidth per simulated
+device. Each FSR was simulated with 100 random deviations from the target
+waveguide geometries. Both width and thickness were assumed to have a
+3σ = 5 nm.
+waveguide geometry was run. The nominal waveguide di-
+mensions were assumed to be 480 x 220 nm. To simulate
+systemic variations, each waveguide was arbitrarily assumed
+to have an etch bias of +10 nm. Random fluctuations were
+simulated by subjecting each device to a normally distributed
+variation of 3σ = 5 nm on both the waveguide width and
+thickness, as this value is consistent with the worst-case
+reported values for geometric variations [25]–[27]. Each mode
+profile was then exported to INTERCONNECT and simulated
+with interferometer FSRs ranging from 4 - 40 nm to investigate
+the effect this had on the extraction error. The resulting error
+function for one of these samples, with a ground truth width of
+490nm, is shown in Fig. 7a. We see the convergence behavior
+of the etch bias estimate evolves as a function of device sample
+size increases for several FSR designs in Fig. 7b. It can be seen
+that all FSR designs can yield at least an estimated etch bias
+within 2 nm of the actual value, indicating the utility of our
+etch bias correction.
+Fig. 8 shows the relationship between the average, per sam-
+ple error and the interferometer FSR. The error is measured
+in three scenarios: i.) a ‘na¨ıve’ case, where the fringe order is
+estimated assuming no etch bias; ii.) where the fringe order is
+estimated through our etch bias prediction methodology, based
+on 30 measured samples; and iii.) where the exact neff from
+simulations is used to determine the actual fringe orders. The
+last scenario, that produced an average per sample error of
+roughly 0.017% represents an error floor for the first two.
+This error floor is completely determined by errors in the
+initial ng regression, as well as any fundamental limitations
+in our chosen behavioral model. As the FSR is increased,
+the average per sample error in both cases improves steadily
+until it reaches the aforementioned floor. This is consistent
+with (19), indicating that a larger FSR corresponds to a wider
+margin of error for the fringe order estimate. For both the na¨ıve
+and bias compensation methods, there is a critical FSR value
+beyond which the fringe order is correctly estimated for all
+samples. It is clear, however, that estimating the presence of
+any etch biases drastically improves the fringe order accuracy,
+reaching the error floor for a much smaller FSR than when
+using the na¨ıve method.
+Fig. 9.
+a, ng relative error vs simulated sidewall angle. b, Comparison
+between the simulated (scatter) and estimated (dashed) neff for different
+sidewall angles.
+B. Sidewall Angle
+We now consider how the parameter extraction behaves
+when used for waveguides with some sidewall angle. Up
+to this point, our simulations assumed the waveguides to
+have no sidewall angle. Real waveguides, however, typically
+deviate from this ideal [55]. To study how our bias correction
+behaves under these conditions, a SOI waveguide with the
+same nominal (480 x 220 nm) design as before was simulated
+with a series of sidewall angles from 85 to 90 degrees as
+this is a range typical of foundries [25], [56]. As only the
+aggregate behavior is being studied, width and thickness
+variations were not included. As seen in Fig. 9a, the minimum
+of the error function optimized in the etch bias estimation step
+remain roughly constant for all considered sidewall angles.
+This results in very accurate predictions of the effective index
+from our model, even though the fundamental geometry is
+different. We interpret this as our optimization routine is
+picking an ‘equivalent’ waveguide width that matches the
+extracted ng profile. This equivalent width always seems to
+result in a waveguide design with a similar confinement factor
+and effective index—and therefore behavior—as seen in Fig.
+9b.
+C. Material Variation
+This method for increasing the accuracy of the guessed neff
+relies on the assumption that the material properties of the
+fabricated waveguides generally match the assumed material
+properties used in the simulation data used to construct the
+model. In practice, however, there can be a great deal of
+deviation between the assumed and actual optical properties of
+the waveguide materials. As a workaround, the authors suggest
+extracting and building a model based around the dispersion
+of the waveguide ∂2neff/∂λ2, as this waveguide parameter
+
+Naive
+0.75
+Bias Correction
+neff Error [%]
+Exact
+0.50
+0.25
+0.00
+10
+20
+30
+40
+Target FSR [nm]Naive
+Bias Correction
+0 =90.0
+0 =87.5
+0 =85.08
+Fig. 10. a, Illustration of measured reticles on a custom 300 mm wafer, with a blown-up microscopic image of a die with 135 MZIs. b, Nominal neff and
+ng model extracted from device measurements. c, Width-based model extraction for each die tested. d, Total model parameter µm variance σ explained vs
+number of principal components included.e, Plot of the width-independent subparameters for neff,0, ∂neff/∂λ0,and ∂2neff/∂λ2
+0 vs V .
+can be extracted exactly from measurements. The nominal
+model of ∂2neff/∂λ2 can then replace ng in (20) to estimate
+the width of the measured device. This width can then be
+used in conjunction with simulation data to assign it an neff
+guess. Though the limits of such a technique are unclear to the
+authors, experimental results in Section V demonstrate to be
+effective enough for describing the neff, loss, and thermo-optic
+effect for all measured device performance.
+IV. EXTRACTING LOCAL PARAMETER VARIATIONS
+Process variations (e.g. thickness variation, cladding and
+core index variations) will appear in our model as varia-
+tions in the fifteen model parameters that comprise Eq. (1).
+Capturing these variations requires the ability to extract their
+value locally, which cannot be done just by looking at the
+performance of any individual device. It is commonly assumed
+in prior literature that most process parameters slowly vary
+across the entire wafer [25]. This assumption implies that
+the values of the parameters comprising our model also
+vary slowly across the wafer. The authors therefore propose
+analyzing the performance of several waveguide width designs
+in close proximity to each other to locally extract all of the
+fifteen model parameters. Each local extraction serves as the
+observations of each model parameter that are tracked across
+the entire wafer.
+The simplest way to create a statistical model is to treat each
+of fifteen sub-parameters as independent statistical variables.
+This is not ideal, however, as each additional variable drasti-
+cally increases the number of required iterations for accurate
+Monte Carlo simulations. To minimize model complexity, we
+would like to represent each sub-parameter as a linear function
+of an ensemble of variables:
+pni = pni,avg. + ⃗s · ⃗V .
+(21)
+⃗V is the vector of variables that represent the process varia-
+tions. Minimizing model complexity would be the equivalent
+of minimizing the size of ⃗V . ⃗s describes the corresponding
+sensitivities of a given parameter to each element in ⃗V . To
+minimize the size of ⃗V , we leverage the fact that each extracted
+model parameter will be strongly correlated to one another.
+This is because the variations in each model parameter share
+common origins such as wafer thickness, annealing time,
+etc. We therefore propose using principal component analysis
+(PCA), a technique for transforming a number of possibly
+correlated variables into a smaller number of uncorrelated
+variables (i.e principal components) [57], to minimize model
+complexity. The chosen principal components are then the
+variables that make up ⃗V . The chosen principal components
+are then the variables that make up ⃗V . The number of
+components in ⃗V is flexible (see Appendix B for details).
+Since our waveguide geometry is primarily a function of two
+process variables—waveguide width and thickness—we use
+only the first principal component to preserve its physical
+interpretation. The result is a model of effective index as a
+function of width and our process variations–∆w, representing
+width variations and an additional variable we will call V ,
+representing an aggregate of other process variations, including
+thickness variation:
+neff,model(w + ∆w, V ).
+(22)
+This full model of neff is then used in the local optimization
+and re-extraction of each measured device. The cost function
+
+b
+d
+2.75
+%1
+100
+2
+neff
+2.50
+Explained [
+15
+7
+17
+73
+24
+26
+75
+4.25
+4.00
+6
+3.75
+357
+9111315
+1000
+2000
+Number of Components
+Width [um]
+c
+e
+17
+24
+26
+2
+5
+13
+0.25
+re/Heue
+aneff/a2
+-0.75
+.00
+.25
+neff,0
+2.50
+2.25
+2 0.4
+2 0.4
+-0.5
+0.4
+2 0.4
+2
+-1.0
+0.0
+0.5
+2 0.4 2 0.4 2 0.4
+WG Width [um]
+V9
+Fig. 11. a, Measured vs modeled optical spectrum of a 480 nm waveguide MZI with ∆L = 100 µm. b, (i) Histogram of extracted V data along with its
+associated Gaussian distribution (red, dashed) overlaid on top. (ii) Spatial map of the average value for V per measured die across the wafer. c, Mean and
+standard deviation of ∆w, neff, and ng.
+is defined as the sum of the relative neff and ng errors to match
+both the measured fringe locations and FSR respectively.
+Thus, we can employ a two-stage direct statistical com-
+pact model extraction procedure [24]. In the first stage, we
+use group extraction to obtain the complete set of fifteen
+parameters for a uniform device. In a second step, a subset
+of model parameters are re-extracted for each member of a
+large ensemble of devices measurements. This approach will
+be the most accurate representation of how device performance
+varies across the wafer without any presumption of variation
+source, statistical distribution, correlation, and the resulting
+model sensitivity to the variation. An inherent strength of
+this approach over others is that it is potentially useful for
+modeling other waveguide geometries as well. This potential
+is due to the model designer having the option of picking
+the number of principal components based on a physical
+assumption on the key process variables or optimize the
+percentage of explained parameter variance (see Appendix B).
+While further investigation would be required to confirm this,
+the methodology’s flexibility holds a great deal of promise.
+V. EXPERIMENTAL DEMONSTRATION
+We measured 7 reticles, each with 135 MZIs consisting
+of 27 different waveguide widths (w) from 400 nm to 2500
+nm and 5 different arm length delays (∆L) from 100 nm to
+500 nm, fabricated on a custom 300 mm full wafer through
+AIM Photonics (Fig. 10a). All 135 MZI were measured on
+reticle 2 while a smaller subset of 30 MZIs were measured
+on each of the remaining reticles, totaling 315 measured
+devices. Devices with the same waveguide width are placed
+adjacently to minimize the impact of local process variations
+on device performance. All MZIs have a nominal waveguide
+height of 220 nm, and grating couplers designed for quasi-TE
+polarization are utilized for optical I/O. The two arms of each
+MZIs consist of symmetric waveguide bends to mitigate the
+impact of bending on the ng. For devices with waveguides
+beyond the single-mode cutoff width, Euler bends are used to
+maintain single mode operation and high mode isolation [18].
+A tunable laser was swept from 1450–1610 nm at a 10 pm
+resolution to characterize the transmission spectrum of each
+MZI.
+A. Nominal Extraction
+A nominal model neff is created by averaging the extracted
+parameters for all measured devices as shown in Fig. 10b. We
+apply the extraction method described in Section II-E to every
+collected transmission spectrum. A preliminary model is built
+using the simulation data described in Section II to estimate
+the expected device FSR for each waveguide width variation.
+This estimated FSR is then fed into a peak finding algorithm
+to extract the ng parameters, and then estimate fringe orders—
+and, therefore the neff—of each measured device. As the
+measured ng deviated a great deal from the simulated values,
+the technique described in Section III-C is employed where a
+preliminary model based on ∂2neff/∂λ2 is created and used
+to estimate waveguide geometry to estimate the fringe order.
+All three Taylor-expansion parameters are then derived using
+(10a)-(10c), and then averaged across for each width variation
+across the entire wafer to create a nominal experimental model.
+The extraction is then repeated locally for devices that are
+close in proximity to one another to extract local values for
+the model’s sub-parameters (Fig. 10c).
+Fig. 10d shows that using (31) this first principal component
+can explain 62.7% of all variance in the sub-parameter values
+across the wafer. The authors determined that due to the clear
+connection between the V and the three model parameters
+
+a
+0
+100
+(i)
+(ii)
+Power [dBm]
+100
+μAw
+MVo
+C
+50
+50
+-100
+Measurement
+Model
+0.1
+(!)
+(iv)
+1500
+1550
+1600
+yauo
+2.50
+Wavelength [nm]
+b
+2.25
+(i)
+(ii)
+0.0
+-0.38
+(v)
+(vi)
+4.25
+0.010
+-0.20
+0.07
+Ong
+4.00
+0.005
+-1
+0.47
+3.75
+0.30
+0.75
+2
+1
+1
+2
+-2
+0
+Width [um]
+Width [um]
+V10
+Fig. 12. a, Measured thermo-optic response of a measured MZI device. b,
+Extracted value of ∂Tchip/∂Theater vs waveguide width using (9).
+that determine device behavior as w → ∞, this principal
+component was likely capturing width-independent sources of
+variance such as thickness variations (Fig. 10e). The authors
+will now show that this provides a model robust enough for
+capturing statistical behavior while preserving the goal for
+clear physical interpretation.
+B. Statistical Extraction
+The sum of the relative neff and ng errors is optimized using
+the Nelder-Mead algorithm [58]. The drawn waveguide width
+and the extracted V from the local extraction performed in
+Fig. 10c are used as the initial guesses for w and V . Despite
+the limited sample size of collected data, we can already see
+several notable preliminary statistical trends in ∆w as a result
+of our model. The model of neff extracted from each local
+optimization was found to result in a close agreement between
+the measured and modeled MZI performance. The extracted
+values for V exhibits the intended physical behavior of process
+parameter that varies slowly across the wafer (Fig. 11b). Local
+optimization yielded a total, average intra-die, and average
+local device standard deviation σV of 0.603, 0.386, and 0.167
+respectively, showing a correlation between device proximity
+and their extracted V values. The mean values of V for die
+both (i) in close proximity to each other and (ii) equidistant
+from the center of the wafer tend to be similar in value, as
+shown in the inset of Fig. 11b.
+Decoupling the process variations of V from the width
+variations ∆w enables extraction of width-dependent systemic
+effects, as shown in Fig. 11c(i). Our method estimates that
+waveguide widths with smaller mean errors also tend to have
+smaller σ∆w (Fig. 11c(ii)). This carries over as an explanation
+Fig. 13. a, Microscopic image of a die with waveguide spiral test structures
+for measuring width-dependent loss. Inset: magnified image showing three of
+the test structures. b, Propagation loss measurement and fit data for a 440 nm
+waveguide.
+for why for w = 2µm, neff varies more than for w = 1.2µm,
+allowing insight on what waveguide geometry best minimizes
+both σneff and σng. This sort of process insight for circuit
+designers is only possible due to the group benchmarking of
+all device performance within a localized area.
+C. Thermo-optic Effect Model Validation
+To validate the thermo-optic effect model developed in
+Section II-D, we re-characterized the MZI transmission spectra
+from a single die of the chip shown in Fig. 10a. The thermal
+characterization was performed by adhering a Thorlabs TLK-
+H polyimide heater to the side of the chip stage. The heater
+was controlled by a Thorlabs TC200 Temperature Controller to
+set the heater temperature. Thermal paste was applied between
+the chip and the chip stage to minimize thermal resistance
+between the chip and the heater. The thermal response of one
+of the tested MZI is shown in Fig. 12. The fringe closest to
+1550 nm is tracked at each temperature step and plotted against
+temperature to extract ∂λ/∂T. This value is then compared to
+our predicted value for ∂λ/∂T gained by taking the derivative
+of λ in (11) with respect to temperature
+∂λ
+∂Tchip
+=
+∆L
+m
+∂neff
+∂Tchip
+∂Tchip
+Theater
+1 − ∆L
+m
+∂neff,model
+∂λ
+,
+(23)
+where m is the order of the tracked fringe, ∆L is the path
+length difference between the two arms, and ∂Tchip/∂Theater
+represents the heat transfer efficiency from the heater to the
+chip itself. This last term is included as the authors only know
+
+0
+a
+Power [dBm]
+10
+31.8 °C
+20
+36.8 °C
+41.7°C
+0.073 nm/K
+-30
+44.9 °C
+1527
+1528
+1529
+1530
+1531
+1532
+Wavelength [nm]
+b
+0.90
+0.85
+0.80
+500
+1000
+1500
+2000
+Width [nm]a
+ COLUMBIA UNIVERSITY
+DARPA
+IN THE CITY OF NEW YORI
+ch Lightwave Research Laborato
+1CM
+Power [dBm]
+-10
+-15
+α = 1.916 dB/cm
+2
+4
+Spiral Length [cm]
+400 um11
+Fig. 14.
+a, Plot of measured (scatter) and modeled (line) propagation loss
+vs ∂neff/∂w. Slope of fit represents R in (4), while the intercept represents
+non-SWR loss. b, Plot of measured (scatter) and modeled (line) propagation
+loss vs waveguide width.
+the temperature of the resistive heater rather than the chip
+temperature itself. We know ∂Tchip/∂Theater ≤ 1 as heater
+cannot raise the temperature of the chip to a value higher than
+its own. The extracted parameters for ∆w and V are used in
+calculating (23).
+On average, the measured thermo-optic effect was found to
+be 0.91× our model’s (9) prediction. This value was found to
+be independent of waveguide geometry with the exception of
+400 nm (Fig. 12b). This error for 400 nm is assumed to be be-
+cause this width is close enough to the cutoff condition for our
+model to lose accuracy. In contrast, the measured thermo-optic
+effect was 1.21× the previously reported model’s prediction.
+This implies that either the chip’s change in temperature is
+greater than the heater’s or the previously reported model is
+incorrect. The change in temperature of the PIC can only ever
+be smaller than the heater’s temperature delta, making the old
+model’s prediction clearly nonphysical.
+D. Scattering Loss Model Validation
+To validate the scattering loss model, we measured a die
+with 25 spiral loss structures consisting of 5 different wave-
+guide widths (w) from 400 nm to 500 nm and 5 different spiral
+lengths (∆L) from 1 cm to 5 cm, fabricated on a custom 300
+mm full wafer through AIM Photonics (Fig. 13). Again, all
+spiral structures have a nominal waveguide height of 220 nm,
+and grating couplers designed for quasi-TE polarization are
+utilized for optical I/O. The losses of each spiral length were
+recorded, and then fit to a linear equation. The slope of the
+this fit was taken to be the propagation loss associated with
+each waveguide width. The results of our model fit are shown
+in Fig. 14. The model built in Section V-B was used to build
+a model of ∂neff/∂w. Fitting our modeled ∂neff/∂w to the
+measured propagation loss yields proportionality constant of
+R = 6.206 × 10−8 cm and (Fig. 14a). The intercept of the loss
+Fig. 15. a, Cadence Virtuoso schematic of the MZI test circuit. All circuit
+models were written in Verilog-A. The optical stimulus is provided by
+a continuous-wave (CW) Laser and detected with a photodetector (PD).
+b, Comparison of measured and simulated performance for an MZI with
+w = 2 µm and ∆L = 100 µm.
+fit is interpreted as the aggregate non-SWR loss, with a value
+of 0.901 dB. Fig. 14b shows the excellent agreement between
+our model and the data, predicting the similar propagation
+losses of both the 440 nm and 460 nm waveguides. As
+mentioned in Section V-B, both 400 and 420 nm waveguide
+widths are likely near the cutoff condition. Since (4) is only
+valid sufficiently far away from this condition, those data
+points are not included in the plot.
+E. Verilog-A Implementation
+To demonstrate its compatibility with electronic-photonic
+co-simulation, the circuit model was implemented in Verilog-
+A within Cadence Virtuoso (Fig. 15a). As Verilog-A does not
+inherently support optical signals, some compatibility code as
+well as a small library of photonic device models were built
+based upon on previously reported demonstrations [32], [59],
+[60].
+VI. CONCLUSION
+In summary, we have demonstrated a novel compact model
+that can greatly expand the accuracy of circuit-level simulation
+capabilities of silicon PICs. In contrast to prior work that
+focused on providing metrology information that could be use
+to fabrication engineers [26], [61], [62], we present this PDK
+model as a tool suitable for true-to-measurement circuit simu-
+lation and optimization. By leveraging this underlying physical
+behavior and locally extracting process variations by perform-
+ing group extraction, we have demonstrated a framework for
+building a model of neff that is entirely driven by measurement
+data. This model was shown to accurately describing the phase,
+
+a
+Loss [dB/cm]
+R = 6.206e-08 cm
+1.9
+αnon-SwWR = 0.901 dB
+1.8
+1.7
+2.2
+2.3
+2.4
+2.5
+2.6
+2.7
+neff/Ow [um-1]
+b
+Loss [dB/cm]
+1.9
+1.8
+1.7
+440
+450
+460
+470
+480
+490
+500
+Width [nm]a
+ght<0:3
+50:50 Coupler
+50:50 Coupler
+Waveguides
+eftLig1<0:3>
+rightLig1<0:3>
+leftLig1<0:3>
+rightLig1<:3>
+leftLig2<0:3>
+coupler
+coupler
+rightLig2<0:3>
+eftLig2<0:3>
+rightLig2<0:3>
+inLight<0:3tatic_waveguide_R2_d1outLight<0:3>
+CW Laser
+lastightout<0:3
+PD
+b
+Transmission [dB]
+-20
+Measured
+VerilogA
+1520
+1530
+1540
+1550
+1560
+Wavelength [nm]12
+loss, and thermo-optic behavior of the measured integrated
+waveguides over 4× the optical bandwidth and over 80×
+the range of waveguides widths reported in prior work. We
+envision that the advancement over prior demonstrations this
+work represents can support the development of waveguide-
+based PDK components and enable the robust optimization of
+next generation PICs.
+VII. ACKNOWLEDGEMENTS
+This work was supported in part by the U.S. Advanced
+Research Projects Agency–Energy under ENLITENED Grant
+DE-AR000843 and in part by the U.S. Defense Advanced Re-
+search Projects Agency under PIPES Grant HR00111920014.
+The authors thank AIM Photonics for chip fabrication.
+APPENDIX
+A. Derivation of Thermo-Optic Model
+Starting from (8) from [46], the effect of a thermal pertur-
+bation on the effective index is investigated. Carrying out this
+perturbation and following the chain rule yields:
+2β ∂β
+∂T = 2Γcore
+ω2
+c2 ncore
+∂ncore
+∂T
++ 2Γclad
+ω2
+c2 nclad
+∂nclad
+∂T
+.
+(24)
+Noting that β = ωneff/c and inserting above, the relationship
+simplifies to (25). Combining with (1) yields:
+neff = neff, T0 + ∂neff
+∂T (T − T0)
+(25a)
+∂neff
+∂T
+= Γcore
+ncore
+neff
+∂ncore
+∂T
++ Γclad
+nclad
+neff
+∂nclad
+∂T
+.
+(25b)
+It is noted that neff appears on both sides of the equation.
+Multiplying both sides by effective index yields a quadratic
+equation whose solution is:
+neff = neff, T0
+2
++ 1
+2
+�
+n2
+eff, T0 + 4n
+′(T − T0)
+(26a)
+n
+′ = Γcorencore
+∂ncore
+∂T
++ Γcladnclad
+∂nclad
+∂T
+.
+(26b)
+The expression can be simplified by noting that n2
+eff, T0 ≫ 4n
+′
+for typical values for the thermo-optic coefficients. Under-
+standing this, it is clear that the behavior of the square root
+term is approximately linear. The 1st order Taylor expansion
+of the square root term is:
+neff, T0 + 1
+2
+4n
+′
+�
+n2
+eff, T0 + 4n
+′(T − T0)
+(T − T0).
+(27)
+Noting again that n2
+eff, T0 ≫ 4n
+′, (27) simplifies to:
+neff, T0 +
+2n
+′
+neff, T0
+(T − T0).
+(28)
+Replacing the square root term in (26) with this expression
+and simplifying will then yield (9).
+B. Principal Component Analysis
+To start, we form a matrix X our of our list of local sub-
+parameter extractions, where each column represents a model
+parameter and each row is an observation of said parameter:
+X =
+�
+������
+∂0neff
+∂λ0
+0,1
+∂0neff
+∂λ0
+1,1
+· · ·
+∂2neff
+∂λ2
+4,1
+∂0neff
+∂λ0
+0,2
+∂0neff
+∂λ0
+1,2
+· · ·
+∂2neff
+∂λ2
+4,2
+...
+...
+...
+...
+∂0neff
+∂λ0
+0,n
+∂0neff
+∂λ0
+1,n
+· · ·
+∂2neff
+∂λ2
+4,n
+�
+������
+.
+(29)
+A covariance matrix S is then created from X and find its
+eigenvectors:
+S =
+�
+����
+⃗v0
+⃗v1
+...
+⃗vn
+�
+����
+�
+����
+λ0
+0
+· · ·
+0
+0
+λ1
+· · ·
+0
+...
+...
+...
+...
+0
+0
+· · ·
+λn
+�
+����
+�
+����
+⃗v0
+⃗v1
+...
+⃗vn
+�
+����
+−1
+,
+(30)
+where
+[v0, v1, · · · , vn]
+lists
+the
+eigenvectors
+and
+[λ0, λ1, · · · , λn]
+are
+their
+associated
+eigenvalues.
+The
+eigenvectors of the correlation matrix represent the directions
+of the axes where there is the most variance (i.e. the most
+information). Each eigenvalue λi is proportional to how much
+variance is captured by its associated principal component
+vi. Picking the eigenvectors with the largest eigenvalues
+allows us to reduce data dimensionality at the expense of
+some accuracy. The percentage of variability explained by a
+principal component is calculated as
+�M
+i=0 λi
+�N
+i=0 λi
+,
+(31)
+where λi is the eigenvalue for each eigenvector, M is the
+number of principal components the designer has chosen
+to include, and N is the maximum number of principal
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+R. Baets, “Silicon microring resonators,” Laser & Photonics Reviews,
+vol. 6, no. 1, pp. 47–73, 2012.
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+and N. G. Tarr, “Birefringence control using stress engineering in
+silicon-on-insulator (soi) waveguides,” Journal of Lightwave Technology,
+vol. 23, no. 3, pp. 1308–1318, 2005.
+[57] H. Abdi and L. J. Williams, “Principal component analysis,” Wiley
+interdisciplinary reviews: computational statistics, vol. 2, no. 4, pp. 433–
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+no. 7, p. 2928, 2009.
+[59] E. Kononov, “Modeling photonic links in verilog-a,” Ph.D. dissertation,
+Massachusetts Institute of Technology, 2013.
+[60] J. C. Leu, “Integrated silicon photonic circuit simulation,” Ph.D. disser-
+tation, Massachusetts Institute of Technology, 2018.
+[61] Y. Xing, M. Wang, A. Ruocco, J. Geessels, U. Khan, and W. Bogaerts,
+“Compact silicon photonics circuit to extract multiple parameters
+for process control monitoring,” OSA Continuum, vol. 3, no. 2, pp.
+379–390, Feb 2020. [Online]. Available: https://opg.optica.org/osac/
+abstract.cfm?URI=osac-3-2-379
+[62] D. S. Boning, S. I. El-Henawy, and Z. Zhang, “Variation-aware methods
+and models for silicon photonic design-for-manufacturability,” Journal
+of Lightwave Technology, vol. 40, no. 6, pp. 1776–1783, 2022.
+Aneek James received his B.S. in Electrical and Electronics Engineering from
+the University of Georgia, Athens, GA in 2017 and his M.S., and M.Phil., in
+Electrical Engineering from Columbia University, New York, NY in 2019
+and 2021, respectively. He is working as a Ph.D. candidate in Electrical
+Engineering in the Lightwave Research Laboratory under Professor Keren
+Bergman. His research interests include modeling fabrication variations in
+silicon photonic devices, as well as the testing and automated control of silicon
+photonic systems for high-throughput optical interconnects.
+Anthony Rizzo received his B.S. in Physics from Haverford College,
+Haverford, PA in 2017 and his M.S., M.Phil., and Ph.D., all in Electrical
+Engineering, from Columbia University, New York, NY in 2019, 2021, and
+2022, respectively. He completed his doctoral research in the Lightwave
+Research Laboratory at Columbia University under Professor Keren Bergman,
+where he led the first demonstration of data transmission using an integrated
+Kerr frequency comb source and silicon photonic transmitter. He is currently a
+Research Scientist at the Air Force Research Laboratory (AFRL) Information
+Directorate in Rome, NY, with a focus in large-scale silicon photonic systems
+for quantum information processing and artificial intelligence.
+Yuyang Wang received the B.Eng. degree in electronic engineering from
+Tsinghua University, Beijing, China in 2015, and the M.S. and PhD degrees
+in computer engineering from the University of California, Santa Barbara
+(UCSB), CA, USA, in 2018 and 2021 respectively. He is currently a post-
+doctoral researcher in the Lightwave Research Laboratory under Professor
+Keren Bergman. He was a Design Engineering Intern at Cadence Design
+Systems in 2018 and a Visiting Intern at the Hong Kong University of
+Science and Technology in 2019. His research interests include variation-
+aware modeling, design, and optimization of silicon photonic interconnects
+and systems.
+Asher Novick received his M.Eng. and B.S. degrees in Electrical and
+Computer Engineering from Cornell University, Ithaca, NY, USA, in 2016 and
+2015,respectively. Between 2016 and 2019, he was at Panduit’s Fiber Research
+Lab, where he researched and developed new patentable technologies for
+optical fiber-based communication in data center and enterprise applications.
+He is currently working toward his Ph.D. degree in Electrical Engineering
+in the Lightwave Research Laboratory at Columbia University in the City
+of New York. His current research interest is in the modeling, design, and
+testing of silicon photonic systems and devices for scalable and efficient link
+architectures.
+Songli Wang received his B.Eng. in Optoelectronic Information Science and
+Engineering from Harbin Institute of Technology, Harbin, China, in 2019
+and his M.S. in Electrical Engineering from Columbia University, New York,
+NY in 2020. He is currently working towards the Ph.D. degree in Electrical
+Engineering in the Lightwave Research Laboratory at Columbia University.
+His current research interests include modeling, design and testing of silicon
+photonic devices and systems.
+Robert Parsons received the B.S. degree in biomedical engineering from
+George Washington University, Washington, D.C., USA, and the M.S. degree
+in electrical engineering from Columbia University, New York, NY, USA, in
+2020 and 2022, respectively. He is currently working toward the Ph.D. degree
+in electrical engineering with the Lightwave Research Laboratory, Columbia
+University under Professor Keren Bergman. His research interests include the
+modeling, testing, and co-optimization of link architectures and constituent
+silicon photonic devices for high-bandwidth, energy-efficient optical inter-
+connects.
+Kaylx Jang received his B.S. in Electrical Engineering from the University
+of California, Irvine, CA in 2020 and his M.S. in Electrical Engineering from
+Columbia University in 2022. In 2019 and 2020, he interned in the testing
+department at Ayar Labs and in 2021 did a co-op in the Silicon Photonics
+Design team at Nokia (former Elenion). He is working as a Ph.D. student
+in Electrical Engineering in the Lightwave Research Lab under Professor
+Keren Bergman. His research interests include modeling, design, and testing
+of high-performance silicon photonic devices for energy efficient and scalable
+link architectures.
+Maarten Hattink is a graduate student with Columbia University, New
+York, NY, USA. He received the B.S. and M.S. degrees from the Eindhoven
+University of Technology, The Netherlands, in 2015 and 2017, respectively.
+While pursuing these degrees, he worked at Prodrive Technologies B.V. as
+a Software and FPGA Engineer. He is currently working toward the Ph.D.
+degree and his research interest lies in photonic device integration and thermal
+control.
+
+15
+Keren Bergman (S’87–M’93–SM’07–F’09) received the B.S. degree from
+Bucknell University, Lewisburg, PA, in 1988, and the M.S. and Ph.D. degrees
+from the Massachusetts Institute of Technology, Cambridge, in 1991 and
+1994, respectively, all in electrical engineering. Dr. Bergman is currently a
+Charles Batchelor Professor at Columbia University, New York, NY, where she
+also directs the Lightwave Research Laboratory. She leads multiple research
+programs on optical interconnection networks for advanced computing sys-
+tems, data centers, optical packet switched routers, and chip multiprocessor
+nanophotonic networks-on-chip. Dr. Bergman is a Fellow of the IEEE and
+Optica.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf,len=1352
+page_content='1  Process Variation-Aware Compact Model of Strip  Waveguides for Photonic Circuit Simulation  Aneek James, Anthony Rizzo, Yuyang Wang, Asher Novick, Songli Wang, Robert Parsons, Kaylx Jang,  Maarten Hattink, and Keren Bergman  Abstract—We report a novel process variation-aware compact  model of strip waveguides that is suitable for circuit-level sim-  ulation of waveguide-based process design kit (PDK) elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The model is shown to describe both loss and—using a novel  expression for the thermo-optic effect in high index contrast  materials—the thermo-optic behavior of strip waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' A  novel group extraction method enables modeling the effective  index’s (neff) sensitivity to local process variations without the  presumption of variation source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Use of Euler-bend Mach-  Zehnder interferometers (MZIs) fabricated in a 300 mm wafer  run allow model parameter extraction at widths up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5 µm  (highly multi-mode) with strong suppression of higher-order  mode excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Experimental results prove the reported model  can self-consistently describe waveguide phase, loss, and thermo-  optic behavior across all measured devices over an unprecedented  range of optical bandwidth, waveguide widths, and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Index Terms—Silicon photonics, compact modeling, process  variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' INTRODUCTION  S  ILICON photonics (SiPh) has seen explosive growth in  demand as a technology platform, driven by its adoption  in data centers (DC), high performance computing (HPC) [1]–  [3], quantum computing [4]–[8], and radio-frequency com-  munication systems [9]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' SiPh’s rapid rise and matura-  tion has been enabled by its ability to leverage decades of  research in the complementary metal–oxide–semiconductor  (CMOS) industry, drastically reducing the typical research and  development (R&D) costs associated with new semiconductor  technologies [12]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' SiPh, however, has not yet been able  to mimic CMOS yield prediction tools for evaluating photonic  integrated circuits (PICs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Yield is a ubiquitous metric used  across semiconductor manufacturing, with improvements in  yield being strongly correlated with reductions in the time  and costs associated with PIC design cycles [15]–[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The  need for predictive yield models can be mitigated to some  This work was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Advanced Research Projects  Agency–Energy under ENLITENED Grant DE-AR000843 and in part by  the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Defense Advanced Research Projects Agency under PIPES Grant  HR00111920014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' James, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Novick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Parsons, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Jang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Hattink,  and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Bergman are with the Department of Electrical Engineering, Columbia  University, New York, NY 10027, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' (Corresponding author: Aneek James,  e-mail: aej2149@columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Rizzo is with the Air Force Research Laboratory Information Directorate,  Rome, NY 13441, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  © 2023 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Permission from  IEEE must be obtained for all other uses, in any current or future media,  including reprinting/republishing this material for advertising or promotional  purposes, creating new collective works, for resale or redistribution to servers  or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  TABLE I  FEATURES FOR MODELING STRIP WAVEGUIDE PERFORMANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' THE  MODEL IN THIS WORK DESCRIBES PHASE, LOSS AND THERMAL BEHAVIOR  EFFECTS OVER A BROAD RANGE OF WAVELENGTHS AND WAVEGUIDE  GEOMETRIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Model Features  [25]  [26]  This Work  Wavelength [nm]  1550  1520–1570  1450–1650  Nominal Width Range [nm]  480  480–500  400–2500  Considered Variation Sources  w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='t  w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='t  Arbitrary  Statistical Parameter Variations  \x13  \x13  \x13  Waveguide Scattering Losses  \x17  \x17  \x13  Thermo-optic Effect  \x17  \x17  \x13  w - Waveguide Width Variations  t - Waveguide Thickness Variations  degree by designing variation-robust devices [18] or PICs  such that performance variations can be tolerated or cor-  rected for post fabrication [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In each of these cases,  however, quantitative yield data cannot be determined prior  to fabrication—an obstacle that will be exacerbated as the  number of components per PIC in silicon is projected to  scale well into the millions within the next decade [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Circuit designers also need tools to optimize system-level  performance through device-level design choices [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To meet  rising circuit design complexity, commercial foundries must  develop process design kits (PDKs) that include compact  models that are both parameterized over a wide range of  relevant design and environmental variables and describe all  important device figures of merit [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' It is essential  that strip waveguides in particular—a critical component of  most SiPh circuits—are accurately modeled according to their  expected fabricated performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Broadly speaking, there are three ways to construct compact  models: (i) look up table-based models, obtained directly from  measurements or device simulations, (ii) models based on  empirical fit functions, and (iii) physics-based models [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Most previously reported work falls under the look-up table-  based category [25]–[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' These models can be parameterized  using look-up tables (LUTs), where interpolation is used to  predict the performance of designs not explicitly defined in  the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Ensuring that LUT models are accurate over a wide  range of input parameters, however, requires measuring all  waveguide figures of merit for every combination of input  parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a task that scales exponentially with the number of  modeled independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Prior demonstrations methods  also require the explicit connection of the measured effective  and group index variations to a predefined number of process  variation sources, introducing the possibility of error if any  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='01689v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='optics]  4 Jan 2023  2  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  a, Example electric field profile taken from Lumerical MODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b,  Simulated (scatter) and modeled (dashed) effective index vs wavelength for  several waveguide widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Each waveguide was simulated with a thickness of  220 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  systemic deviations exist between the simulation configuration  and the realities of the fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  In this paper, we report to the best of our knowledge, the first  geometry-parameterized compact model of strip waveguides  that can capture device performance over a wide range of  wavelengths and waveguide geometries (see Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Using a  novel derivation of the thermo-optic effect that is accurate for  high-index contrast waveguides, we demonstrate our model’s  ability to describe both scattering loss and the thermo-optic  effect as a function of both design and statistical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A novel group-extraction-based method allows the characteri-  zation of process variations without presumption of a source or  its associated sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This extraction methodology is used  to construct a model from dozens of geometric variations of  Mach-Zehnder Interferometers (MZIs) fabricated in a 300 mm  commercial foundry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' These use of Euler bends in these MZIs  permits the characterization of wide waveguide performance  with minimal higher-order mode excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Experimental re-  sults validate the model’s accuracy in describing the phase,  loss, and thermo-optic performance across the entire wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The model is also implemented in Verilog-A to demonstrate  compatibility with electronic-photonic co-simulation environ-  ments [30]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This work represents a key step toward  the modeling of waveguide-based PDK components, enabling  true-to-measurement circuit simulation at massive integration  densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' PHYSICS-AWARE MODEL DEVELOPMENT  Because the mode condition of an optical waveguide  is  described  via  a  transcendental  equation,  completely  generalized analytical solutions for the effective index (neff)  are impossible to derive [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We therefore propose, as  discussed in [34], finding a behavioral model that accurately  captures its dependence on all design parameters over the  relevant ranges of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In this section, we develop  dependency models for the design parameters available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a-c, Plot of simulated (scatter) and modeled (dashed) neff parameters  �  ∂2neff/∂λ2, ∂neff/∂λ, neff,0  �  vs waveguide width (respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' These  values were for a waveguide with a thickness of 220 nm at a wavelength  of 1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' d, Comparison of the model (dashed) and simulated (scatter)  neff vs waveguide width for different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Simulated at 1550 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The semi-physical nature of the model is then leveraged  to describe both the scattering loss and the thermo-optic  coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Process variations, whether of a design parameter  or not, will be covered in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Wavelength Dependence  The wavelength dependence of the waveguide neff is first  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The neff of several silicon-on-insulator (SOI)  waveguide geometries were simulated in Lumerical MODE  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' From the results, it is shown that the wavelength  dependence over the S-, C-, and L-bands for all geometries is  well-approximated by a second-order Taylor expansion for a  wide range of waveguide widths sufficiently above the cutoff  condition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 1b):  neff, model(λ) =  2  �  i=0  1  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  ∂ineff  ∂λi  ����  λ=λ0  (λ − λ0)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (1)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Geometric Dependence  As the Taylor expansion only captures the wavelength-  dependence, it is clear that the fitting parameters ∂2neff/∂λ2,  ∂neff/∂λ and ∂0neff/∂λ0 (hereafter referred to as neff,0) are  responsible for capturing the dependence on waveguide geom-  etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' With respect to width, all three fitting parameters were  previously found in [35] to be well described by the following  behavioral model:  ∂ineff  ∂λi (w) = pi0 · w2 + pi1w + pi2  w2 + pi3w + pi4  ,  (2)  for a total of fifteen model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To verify correctness  of the model, all three parameters were fitted to the simu-  lation data with (1)-(2) using ordinary least squares (OLS)  a  Thickness  Width  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='6  neff  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='2  1450  1500  1550  1600  1650  Wavelength [nm]  400 nm  580 nm  760 nm  940 nm215 nm  235 nm  255 nm  275 nm  295 nm3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Comparison between modeled (dashed) and simulated (scatter) neff  for higher order modes and the fundamental TM mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' All waveguides were  simulated with a thickness of 220 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The model was able to match all three parameters  over the entire range of the width sweep (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 2a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The  close matching of the modeled and extracted Taylor parameters  means that our modification of (2) still preserves its ability to  match the behavior of effective index as a function of wave-  length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' By extension, these three Taylor parameters allow for  a robust description of neff as a function of waveguide width  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The data also demonstrates this agreement is not  unique to any particular waveguide thickness, with different  thicknesses producing different sub-parameter fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Finally, it  should be noted that both the numerator and denominator in  (2) are polynomials of equal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Our model consequently  predicts that, for a given wavelength, the effective index will  asymptotically approach a constant value as w approaches  infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The value that the model approaches as w tends  towards infinity can be interpreted as the equivalent neff of  an infinite slab of the same thickness:  lim  w→∞ neff(λ, w) = nslab(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (3)  In this way, our behavioral model can elegantly capture all  significant features of effective index for the design parameters  of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The model’s accuracy holds true for higher order  modes as well, provided that they are sufficiently far away  from their respective waveguide cutoff condition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Scattering Loss  Scattering loss due to sidewall roughness (SWR) can be a  significant source of loss in most reported waveguide designs,  making it critical for designers to accurately model [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In this  section, we demonstrate our model’s ability to capture SWR  loss as a function of waveguide geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' It was first noted  in [37] that the traditional Payne and Lacey model of SWR-  induced loss [38], [39] was found to be identical in behavior to  the derivative of the effective index with respect to waveguide  width:  αSWR(λ, w) = R ∂  ∂w [neff(λ, w)] ,  (4)  where R is a proportionality constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As our model can  describe neff as a function of width, a closed-form repre-  sentation of ∂neff/∂w can be exactly derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This equation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  a, Graphical representation of a waveguide simulated with some  sidewall roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The inset is a magnified view of the waveguide to clarify  the definition of σrms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, Scattering losses estimated from FDTD compared  to the fit using our model based on Lumerical MODE data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  can then be fitted to measured waveguide loss data to extract  the proportionality constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We validate this by fitting (4)  to the scattering loss of a 7 µm long SOI waveguide with  some SWR wall roughness in Lumerical 3D-FDTD (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The roughness Root Mean Square (RMS) and correlation  length were arbitrarily chosen to be σrms  = 5 nm and  Lcorr = 1 µm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' These parameters were then used to  generate a random, anisotropic SWR on the waveguide walls  [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Propagation losses were simulated for waveguide widths  ranging from 450 nm to 850 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The results of the fitting are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 4b, with our model closely matching trend of  the scattering loss behavior extracted from FDTD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Thermo-Optic Effect  Our model can also completely describe the thermo-optic  coefficient of an arbitrary waveguide geometry without the  need for any thermal measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The thermo-optic co-  efficient of a waveguide mode most importantly requires  knowledge of the confinement factor, which is the fraction of  a mode’s power confined within each constituent waveguide  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Kawakami showed in [41] that for a waveguide  made up of N materials, each with with an index nk and a  confinement factor Γk:  N  �  k  Γkn2  k = ngneff  (5a)  �  k  Γk = 1,  (5b)  where (5b) is derived from noting that the sum of all con-  finement factors must equal unity due to power conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  n  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  TEO  TE1  TE2  TMO  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  WG Width [um]20m  Width  SOl Waveguide4  A closed-form of the confinement factor for a two-material  waveguide (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' SOI wires) can then be derived:  Γcore = ngneff − n2  clad  n2  core − n2  clad  (6a)  Γclad = n2  core − ngneff  n2  core − n2  clad  ,  (6b)  where Γcore is the power contained in the waveguide core and  Γclad is the power contained in the cladding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Next, we must obtain an expression that describes the  thermo-optic effect on neff in terms of the confinement factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A common approximation of the thermo-optic coefficient of  neff is  ∂neff  ∂T  ≈ Γ1  ∂n1  ∂T + Γ2  ∂n2  ∂T + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' ,  (7)  where δ represents a small perturbation in the values, Γn is  the confinement of the mode within material n and ∂nn/∂T  is the thermo-optic coefficient of material n [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Though  this equation is widely used [43]–[45] and may be accurate  in certain scenarios, to the authors’ knowledge it has never  been demonstrated to be a generally accurate approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  We therefore start from first principles and consider a general  perturbation of the wave equation [46]:  δ  �  β2  eff  �  = Γcore  ω2  c2 δ  �  n2  core  �  + Γclad  ω2  c2 δ  �  n2  clad  �  ,  (8)  where βeff is the effective wavenumber, Γcore is the con-  finement in the waveguide core, Γclad is the confinement in  the waveguide cladding, and ncore and nclad are the core and  cladding indices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Carrying this operation through  and combining with (1) (see Appendix A for details) yields:  neff(λ, w, T) ≈ neff,T0(λ, w) + ∂neff  ∂T (T − T0)  (9a)  ∂neff  ∂T  = Γcore  ncore  neff, T0  ∂ncore  ∂T  + Γclad  nclad  neff, T0  ∂nclad  ∂T  ,  (9b)  where neff,T0 is the neff at some reference temperature T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The  key addition to (9) compared to prior literature is the scaling  of each thermo-optic term by ratio between the material and  effective indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As the index contrast between the core and  cladding decreases, our model will approach the (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Thus  it is clear that our model will outperform (7) in accuracy  when describing high index contrast materials, such as the  SOI waveguide geometries prevalent in SiPh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  With these expressions, our confinement factor and the  thermo-optic coefficient models can be validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The simu-  lated confinement factor is compared to our model prediction  at 1550 nm in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The optical properties of silicon and  silicon dioxide used in our model were taken directly from  [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' There was a near perfect agreement between the modeled  and simulated confinement factor, showing that the general  behavior of confinement factor is captured by our model  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The modeled thermo-optic coefficient is validated  by simulating how the neff of a SOI waveguide varies with  temperature using Lumerical MODE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Silicon was  assumed to have a thermo-optic coefficient of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='9 × 10−4 K−1  [48] and SiO2 was assumed to have a thermo-optic coefficient  of 1 × 10−5 K−1 [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The model and simulations show  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  a Modeled (dashed) and simulated (scatter) confinement factor vs  waveguide width for different thicknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, Comparison between simulated  (scatter), previously reported model (dotted, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' (7)) and our work (dashed  line, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' (9)) describing neff vs Temperature of a 480 x 220 nm waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  exceptional agreement from 300 - 1200 K, despite the fact that  our model does not require any data from thermal simulations  or measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As predicted, the previously reported model  of the thermo-optic effect (7) significantly under-predicts the  expected change in neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' It should be noted that in real devices,  waveguide geometry itself is a function of T due to thermal  expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This can be accounted for by modeling w as a func-  tion of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Experimental results in Section V-C, however, show  that assuming a constant width geometry provides sufficient  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Having a model of the thermo-optic effect that is accurate  over a wide range of conditions like this one holds a great  deal of potential to enable more robust design exploration,  such as evaluating photonic waveguide heater designs [50],  [51], characterizing self-heating in micro-resonators [52], or  studying the effect of ambient temperature fluctuations in a  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Parameter Extraction  The practical utility of a compact model is greatly deter-  mined by the associated parameter extraction procedure to  connect the model to a given foundry process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This is particu-  larly important when developing statistical models, as accurate  parameter extraction is the only way to guarantee that process  variations are accurately reflected in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' A popular  solution is to leverage the phase-sensitivity of interferometric  optical filters—such as Mach-Zehnder interferometers (MZIs),  microresonators, or arrayed waveguide gratings (AWGs)—to  monitor process variations across a wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Regardless of the  chosen device, a shared difficulty lies in accurately guessing  what particular interference fringe position corresponds to a  particular fringe order [25], [26], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Our method is based on  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='75  210 nm  220 nm  230 nm  215 nm  225 nm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  WG Width [um]  b  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='7  neff  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='6  400  600  800  1000  1200  Temperature [K]  This Work - Previous Model  Simulated5  the curve-fitting method presented in [25] and [54], with some  additional steps described to include waveguide dispersion as  an extracted parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The first step in parameter extraction is to characterize  the group index (ng) of a fabricated interferometer from  a wavelength sweep of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To enable this, (1) is  rearranged into a more suitable form:  neff(λ) = 1  2  ∂2neff  ∂λ2 λ2 + Bλ + C  (10a)  B = ∂neff  ∂λ − ∂2neff  ∂λ2 λ0  (10b)  C = 1  2  ∂2neff  ∂λ2 λ2  0 − ∂neff  ∂λ λ0 + neff,0,  (10c)  where B and C are fitting parameters that aggregate the 1st and  0th order terms from (1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Following the procedure  described in [54], it is first noted that the fringe condition of  an inteferometric device is described by  φ = 2π  λ neff(λ)L = 2πm,  (11)  where φ is the phase difference between the interferometry  arms, L is the path length of the interferometer, λ is a partic-  ular fringe wavelength, and m is an integer corresponding to  the particular fringe order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To extract our model parameters,  a wavelength sweep of the interferometric device is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Once this is performed, a peak finding algorithm can be used  to detect the wavelength of all detected fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' A function that  relates the relative fringe locations to the ng of the waveguide  is now required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This can be done by defining a continuous  function that will yield an integer value at each of the detected  fringe locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Let m0 represent the particular fringe order  corresponding to an arbitrarily chosen reference fringe located  at λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The fringe order m of any other fringe can be defined  relative to this reference as  m = m0 +  � λ  λ0  dm  dλ dλ = m0 + ngL ·  � 1  λ − 1  λ0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (12)  This continuous function now allows us to redefine the mea-  sured fringes into a form suitable for parameter extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A reference fringe variable n is now defined by letting  m = (m0 + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Inserting this back into (12) produces:  n = ngL ·  � 1  λn  − 1  λ0  �  ,  (13)  where each relative fringe n is located at an associated  wavelength λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Using (13), the ng of the measured device  is now directly related to the measured fringe locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This  fitting equation must now be extended to our specific model  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The ng of a waveguide is defined to be  ng = neff − λ∂neff  ∂λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (14)  Combining with (10a) yields an expression for ng in terms of  our compact model:  ng = C − 1  2  ∂2neff  ∂λ2 λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (15)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a, Captured spectrum of simulated MZI used for parameter extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The waveguide mode was simulated in Lumerical MODE, and then exported  to a MZI waveguide simulation block in Lumerical INTERCONNECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b,  Linear Regression of fringe wavelengths to extract the ng performed on the  detected fringes from a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' c, Possible neff solutions (black, dashed), along with  the actual solution (red), determined by the ng extracted in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  By inserting (15) back into (13), we can derive an OLS  regression-compatible expression:  n = CΛC − ∂2neff  ∂λ2 ΛS  (16a)  ΛC = L ·  � 1  λn  − 1  λ0  �  (16b)  ΛS = L  2 ·  �  λn − λ2  n  λ0  �  ,  (16c)  where [ΛC, ΛS] are explanatory variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Performing an OLS  regression between n and [ΛC, ΛS] gives us two of our three  fitting parameters in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Finally, B can be calculated by  combining equations (11) and (10a):  B = m  L − 1  2  ∂2neff  ∂λ2 λm − C  λm  ,  (17)  where the only uncertainty is what fringe order m corresponds  to each measured fringe λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Once B is determined from (17),  (10b) and (10c) can be used to determine the original fitting  parameters in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' It should be noted that each detected fringe  (m, λm) location will yield very small variations in the B value  due to resolution-based uncertainty in the exact value for λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  For a best guess, all values Bm taken from each measured  fringe λm should be averaged together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  To validate this method under ideal conditions, an MZI  constructed using 480 nm x 220 nm waveguides is simulated in  Lumerical INTERCONNECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To ensure accuracy, the wave-  guide’s neff was first simulated in MODE and then exported  to a MODE Waveguide element in INTERCONNECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As the  full-width half-maximum (FWHM) of the MZI does not affect  the extracted neff, the waveguides were arbitrarily assumed to  have a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5 dB/cm loss and the coupling coefficient was chosen  to ensure critical coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The spectrum of the simulated MZI  a  Power [dBm]  20  b  10  C  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  neff  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='3  1500  1550  1600  1650  Wavelength [nm]6  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Fringe locations were extracted using  a peak finding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The fringe located closest to the  center of the sweep was arbitrarily chosen as n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Using  (16), OLS regression found ∂2neff/∂λ2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='136 µm−2 and  C = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='9215 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' From here, the family of solutions for  neff is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 6c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Each particular solution corresponds  to a different guess on the fringe orders detected, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' m0 = 52  vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' m0 = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The separation between each neff solution plotted  in 6c is determined by the free-spectral range (FSR) of the  interferometer, with a larger FSR corresponding more widely  separated solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  To determine the correct fringe order of the reference we  use the fact that, from the simulations performed in Section  II, we know the waveguide geometry has an neff of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='411 at  the reference fringe location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In Section III we explain how  to increase the accuracy of this estimation to avoid errors  introduced by this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' From this, the reference fringe  order is found to be m0 ≈ 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Since fringe orders must  be integer numbers, the result is rounded to the nearest integer  114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' By combining (10a)-(10c), the original fitting coefficients  are found to be ∂neff/∂λ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='078 µm−1 and neff,0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  To evaluate accuracy of our extraction, we define the relative  error between the extracted and simulated neff’s σerror by:  σerror =  ��  (neff, model − neff, sim)2 dλ  �  n2  eff, simdλ  ,  (18)  where neff, sim is the effective index from the MODE simula-  tion, used as a reference to quantify our method’s accuracy,  and neff, model is the result from applying our extraction method  to the simulated MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Upon evaluation, the total relative error  was found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='017%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Since the order of the reference  fringe is correct, the remaining model error is attributed to  inaccuracies in the initial regression fit using (16a)-(16c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' MORE ROBUST neff EXTRACTION UNDER PROCESS  VARIABILITY  The reliability of the extraction is highly sensitive to the  guessed value of the reference fringe order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' For the example  in Section II-E, we used a priori knowledge of the neff at the  reference fringe to estimate its order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Therefore, any deviation  between the assumed and actual waveguide dimensions risks  introducing error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' By noting that the initial order estimate  rounded to the nearest integer, we can use (11) to define  a boundary beyond which our fringe order guess will be  incorrect [25]:  |∆m| = |neff, actual − neff, guess| ≤ λm0  2L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (19)  We can see that, to raise confidence in the guessed fringe order,  either the accuracy of our neff guess must be increased or the  interferometric path length must be decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As explained in  Section II-E, our extraction method begins by directly extract-  ing the ng of a given interferometer via optical sweep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Process  variations will therefore appear as variations in the extracted  values for ∂2neff/∂λ2 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' By measuring several devices  of the same drawn width across the all measured dies, wafers,  and lots, the influence of the random width and thickness  variations can be eliminated by averaging their extracted fitting  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a, Plot of the ng error function for one sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The error function  shows a minimum at roughly 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5 nm, which closely agrees with the actual  waveguide width of 490 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, Convergence of the etch bias estimate for  different numbers of samples averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As the sample size becomes sufficiently large—  with the necessary sample size being a function of the severity  of the process variations—any remaining deviation between  the nominal and averaged parameters will be the result of a  systemic etch biases on the waveguide width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We therefore  propose estimating this etch bias by creating a preliminary neff  model based on the results of a photonic mode solver, such as  Lumerical MODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Using this model, an equivalent waveguide  width can be found by minimizing the error function  min  w  ��  [ng, model(w, λ) − ng, meas(λ)]2 dλ  �  n2g, meas(λ)dλ  ,  (20)  where ng, meas is the extracted model of ng using the averaged  extracted parameters and ng, model is the simulation-based,  width-dependent a priori model of neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The neff of our  equivalent waveguide width can then be plugged into the a pri-  ori model to provide a more accurate fringe order estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  In this way, we can increase the accuracy of our guessed  effective index, regardless of whether the modeled waveguide  composition is accurate to the virtual device composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  We now discuss the robustness of this optimization routine  in the presence of other systemic non-idealities and its ability  to perform etch bias correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To do this, we need a ’ground  truth’ value for neff, which we obtain by simulating all the  non-idealities in Lumerical MODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Subsequently we perform  the parameter extraction using Lumerical INTERCONNECT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  By comparing the extracted neff to the known simulated  value for neff, we can directly evaluate the robustness of our  methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Statistical Geometric Variation  To test the extraction procedure’s accuracy under process  variations, a simulation of 100 random variations on the  a  2  Actual  Guess  Error Function  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  460  480  500  520  540  Guessed Widths [nm]  b  4 nm  26 nm  Etch Bias [nm]  Estimated  11 nm  33 nm  15  18 nm  40 nm  10  20  40  60  80  100  Number of Samples7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Plot of mean error in neff over the simulation bandwidth per simulated  device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Each FSR was simulated with 100 random deviations from the target  waveguide geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Both width and thickness were assumed to have a  3σ = 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  waveguide geometry was run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The nominal waveguide di-  mensions were assumed to be 480 x 220 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To simulate  systemic variations, each waveguide was arbitrarily assumed  to have an etch bias of +10 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Random fluctuations were  simulated by subjecting each device to a normally distributed  variation of 3σ = 5 nm on both the waveguide width and  thickness, as this value is consistent with the worst-case  reported values for geometric variations [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Each mode  profile was then exported to INTERCONNECT and simulated  with interferometer FSRs ranging from 4 - 40 nm to investigate  the effect this had on the extraction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The resulting error  function for one of these samples, with a ground truth width of  490nm, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We see the convergence behavior  of the etch bias estimate evolves as a function of device sample  size increases for several FSR designs in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' It can be seen  that all FSR designs can yield at least an estimated etch bias  within 2 nm of the actual value, indicating the utility of our  etch bias correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 8 shows the relationship between the average, per sam-  ple error and the interferometer FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The error is measured  in three scenarios: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=') a ‘na¨ıve’ case, where the fringe order is  estimated assuming no etch bias;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=') where the fringe order is  estimated through our etch bias prediction methodology, based  on 30 measured samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' and iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=') where the exact neff from  simulations is used to determine the actual fringe orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The  last scenario, that produced an average per sample error of  roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='017% represents an error floor for the first two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  This error floor is completely determined by errors in the  initial ng regression, as well as any fundamental limitations  in our chosen behavioral model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As the FSR is increased,  the average per sample error in both cases improves steadily  until it reaches the aforementioned floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This is consistent  with (19), indicating that a larger FSR corresponds to a wider  margin of error for the fringe order estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' For both the na¨ıve  and bias compensation methods, there is a critical FSR value  beyond which the fringe order is correctly estimated for all  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' It is clear, however, that estimating the presence of  any etch biases drastically improves the fringe order accuracy,  reaching the error floor for a much smaller FSR than when  using the na¨ıve method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  a, ng relative error vs simulated sidewall angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, Comparison  between the simulated (scatter) and estimated (dashed) neff for different  sidewall angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Sidewall Angle  We now consider how the parameter extraction behaves  when used for waveguides with some sidewall angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Up  to this point, our simulations assumed the waveguides to  have no sidewall angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Real waveguides, however, typically  deviate from this ideal [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To study how our bias correction  behaves under these conditions, a SOI waveguide with the  same nominal (480 x 220 nm) design as before was simulated  with a series of sidewall angles from 85 to 90 degrees as  this is a range typical of foundries [25], [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As only the  aggregate behavior is being studied, width and thickness  variations were not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 9a, the minimum  of the error function optimized in the etch bias estimation step  remain roughly constant for all considered sidewall angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  This results in very accurate predictions of the effective index  from our model, even though the fundamental geometry is  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We interpret this as our optimization routine is  picking an ‘equivalent’ waveguide width that matches the  extracted ng profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This equivalent width always seems to  result in a waveguide design with a similar confinement factor  and effective index—and therefore behavior—as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  9b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Material Variation  This method for increasing the accuracy of the guessed neff  relies on the assumption that the material properties of the  fabricated waveguides generally match the assumed material  properties used in the simulation data used to construct the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In practice, however, there can be a great deal of  deviation between the assumed and actual optical properties of  the waveguide materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As a workaround, the authors suggest  extracting and building a model based around the dispersion  of the waveguide ∂2neff/∂λ2, as this waveguide parameter  Naive  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='75  Bias Correction  neff Error [%]  Exact  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='00  10  20  30  40  Target FSR [nm]Naive  Bias Correction  0 =90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  0 =87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  0 =85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='08  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a, Illustration of measured reticles on a custom 300 mm wafer, with a blown-up microscopic image of a die with 135 MZIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, Nominal neff and  ng model extracted from device measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' c, Width-based model extraction for each die tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' d, Total model parameter µm variance σ explained vs  number of principal components included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='e, Plot of the width-independent subparameters for neff,0, ∂neff/∂λ0,and ∂2neff/∂λ2  0 vs V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  can be extracted exactly from measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The nominal  model of ∂2neff/∂λ2 can then replace ng in (20) to estimate  the width of the measured device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This width can then be  used in conjunction with simulation data to assign it an neff  guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Though the limits of such a technique are unclear to the  authors, experimental results in Section V demonstrate to be  effective enough for describing the neff, loss, and thermo-optic  effect for all measured device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' EXTRACTING LOCAL PARAMETER VARIATIONS  Process variations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' thickness variation, cladding and  core index variations) will appear in our model as varia-  tions in the fifteen model parameters that comprise Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Capturing these variations requires the ability to extract their  value locally, which cannot be done just by looking at the  performance of any individual device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' It is commonly assumed  in prior literature that most process parameters slowly vary  across the entire wafer [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This assumption implies that  the values of the parameters comprising our model also  vary slowly across the wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The authors therefore propose  analyzing the performance of several waveguide width designs  in close proximity to each other to locally extract all of the  fifteen model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Each local extraction serves as the  observations of each model parameter that are tracked across  the entire wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The simplest way to create a statistical model is to treat each  of fifteen sub-parameters as independent statistical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  This is not ideal, however, as each additional variable drasti-  cally increases the number of required iterations for accurate  Monte Carlo simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To minimize model complexity, we  would like to represent each sub-parameter as a linear function  of an ensemble of variables:  pni = pni,avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' + ⃗s · ⃗V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (21)  ⃗V is the vector of variables that represent the process varia-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Minimizing model complexity would be the equivalent  of minimizing the size of ⃗V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' ⃗s describes the corresponding  sensitivities of a given parameter to each element in ⃗V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' To  minimize the size of ⃗V , we leverage the fact that each extracted  model parameter will be strongly correlated to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  This is because the variations in each model parameter share  common origins such as wafer thickness, annealing time,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We therefore propose using principal component analysis  (PCA), a technique for transforming a number of possibly  correlated variables into a smaller number of uncorrelated  variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='e principal components) [57], to minimize model  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The chosen principal components are then the  variables that make up ⃗V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The chosen principal components  are then the variables that make up ⃗V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The number of  components in ⃗V is flexible (see Appendix B for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Since our waveguide geometry is primarily a function of two  process variables—waveguide width and thickness—we use  only the first principal component to preserve its physical  interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The result is a model of effective index as a  function of width and our process variations–∆w, representing  width variations and an additional variable we will call V ,  representing an aggregate of other process variations, including  thickness variation:  neff,model(w + ∆w, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (22)  This full model of neff is then used in the local optimization  and re-extraction of each measured device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The cost function  b  d  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='75  %1  100  2  neff  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='50  Explained [  15  7  17  73  24  26  75  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='25  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='00  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='75  357  9111315  1000  2000  Number of Components  Width [um]  c  e  17  24  26  2  5  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='25  re/Heue  aneff/a2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='75  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='25  neff,0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='25  2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  WG Width [um]  V9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a, Measured vs modeled optical spectrum of a 480 nm waveguide MZI with ∆L = 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, (i) Histogram of extracted V data along with its  associated Gaussian distribution (red, dashed) overlaid on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' (ii) Spatial map of the average value for V per measured die across the wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' c, Mean and  standard deviation of ∆w, neff, and ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  is defined as the sum of the relative neff and ng errors to match  both the measured fringe locations and FSR respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Thus, we can employ a two-stage direct statistical com-  pact model extraction procedure [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In the first stage, we  use group extraction to obtain the complete set of fifteen  parameters for a uniform device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In a second step, a subset  of model parameters are re-extracted for each member of a  large ensemble of devices measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This approach will  be the most accurate representation of how device performance  varies across the wafer without any presumption of variation  source, statistical distribution, correlation, and the resulting  model sensitivity to the variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' An inherent strength of  this approach over others is that it is potentially useful for  modeling other waveguide geometries as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This potential  is due to the model designer having the option of picking  the number of principal components based on a physical  assumption on the key process variables or optimize the  percentage of explained parameter variance (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  While further investigation would be required to confirm this,  the methodology’s flexibility holds a great deal of promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' EXPERIMENTAL DEMONSTRATION  We measured 7 reticles, each with 135 MZIs consisting  of 27 different waveguide widths (w) from 400 nm to 2500  nm and 5 different arm length delays (∆L) from 100 nm to  500 nm, fabricated on a custom 300 mm full wafer through  AIM Photonics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' All 135 MZI were measured on  reticle 2 while a smaller subset of 30 MZIs were measured  on each of the remaining reticles, totaling 315 measured  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Devices with the same waveguide width are placed  adjacently to minimize the impact of local process variations  on device performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' All MZIs have a nominal waveguide  height of 220 nm, and grating couplers designed for quasi-TE  polarization are utilized for optical I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The two arms of each  MZIs consist of symmetric waveguide bends to mitigate the  impact of bending on the ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' For devices with waveguides  beyond the single-mode cutoff width, Euler bends are used to  maintain single mode operation and high mode isolation [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A tunable laser was swept from 1450–1610 nm at a 10 pm  resolution to characterize the transmission spectrum of each  MZI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Nominal Extraction  A nominal model neff is created by averaging the extracted  parameters for all measured devices as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We  apply the extraction method described in Section II-E to every  collected transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' A preliminary model is built  using the simulation data described in Section II to estimate  the expected device FSR for each waveguide width variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  This estimated FSR is then fed into a peak finding algorithm  to extract the ng parameters, and then estimate fringe orders—  and, therefore the neff—of each measured device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As the  measured ng deviated a great deal from the simulated values,  the technique described in Section III-C is employed where a  preliminary model based on ∂2neff/∂λ2 is created and used  to estimate waveguide geometry to estimate the fringe order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  All three Taylor-expansion parameters are then derived using  (10a)-(10c), and then averaged across for each width variation  across the entire wafer to create a nominal experimental model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The extraction is then repeated locally for devices that are  close in proximity to one another to extract local values for  the model’s sub-parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10d shows that using (31) this first principal component  can explain 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='7% of all variance in the sub-parameter values  across the wafer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The authors determined that due to the clear  connection between the V and the three model parameters  a  0  100  (i)  (ii)  Power [dBm]  100  μAw  MVo  C  50  50  100  Measurement  Model  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='1  (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=')  (iv)  1500  1550  1600  yauo  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='50  Wavelength [nm]  b  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='25  (i)  (ii)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='38  (v)  (vi)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='07  Ong  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='005  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='47  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='75  2  1  1  2  2  0  Width [um]  Width [um]  V10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a, Measured thermo-optic response of a measured MZI device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b,  Extracted value of ∂Tchip/∂Theater vs waveguide width using (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  that determine device behavior as w → ∞, this principal  component was likely capturing width-independent sources of  variance such as thickness variations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The authors  will now show that this provides a model robust enough for  capturing statistical behavior while preserving the goal for  clear physical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Statistical Extraction  The sum of the relative neff and ng errors is optimized using  the Nelder-Mead algorithm [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The drawn waveguide width  and the extracted V from the local extraction performed in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10c are used as the initial guesses for w and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Despite  the limited sample size of collected data, we can already see  several notable preliminary statistical trends in ∆w as a result  of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The model of neff extracted from each local  optimization was found to result in a close agreement between  the measured and modeled MZI performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The extracted  values for V exhibits the intended physical behavior of process  parameter that varies slowly across the wafer (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 11b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Local  optimization yielded a total, average intra-die, and average  local device standard deviation σV of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='603, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='386, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='167  respectively, showing a correlation between device proximity  and their extracted V values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The mean values of V for die  both (i) in close proximity to each other and (ii) equidistant  from the center of the wafer tend to be similar in value, as  shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Decoupling the process variations of V from the width  variations ∆w enables extraction of width-dependent systemic  effects, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 11c(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Our method estimates that  waveguide widths with smaller mean errors also tend to have  smaller σ∆w (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 11c(ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This carries over as an explanation  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a, Microscopic image of a die with waveguide spiral test structures  for measuring width-dependent loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Inset: magnified image showing three of  the test structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, Propagation loss measurement and fit data for a 440 nm  waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  for why for w = 2µm, neff varies more than for w = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='2µm,  allowing insight on what waveguide geometry best minimizes  both σneff and σng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This sort of process insight for circuit  designers is only possible due to the group benchmarking of  all device performance within a localized area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Thermo-optic Effect Model Validation  To validate the thermo-optic effect model developed in  Section II-D, we re-characterized the MZI transmission spectra  from a single die of the chip shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The thermal  characterization was performed by adhering a Thorlabs TLK-  H polyimide heater to the side of the chip stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The heater  was controlled by a Thorlabs TC200 Temperature Controller to  set the heater temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Thermal paste was applied between  the chip and the chip stage to minimize thermal resistance  between the chip and the heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The thermal response of one  of the tested MZI is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The fringe closest to  1550 nm is tracked at each temperature step and plotted against  temperature to extract ∂λ/∂T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This value is then compared to  our predicted value for ∂λ/∂T gained by taking the derivative  of λ in (11) with respect to temperature  ∂λ  ∂Tchip  =  ∆L  m  ∂neff  ∂Tchip  ∂Tchip  Theater  1 − ∆L  m  ∂neff,model  ∂λ  ,  (23)  where m is the order of the tracked fringe, ∆L is the path  length difference between the two arms, and ∂Tchip/∂Theater  represents the heat transfer efficiency from the heater to the  chip itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This last term is included as the authors only know  0  a  Power [dBm]  10  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='8 °C  20  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='8 °C  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='7°C  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='073 nm/K  30  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='9 °C  1527  1528  1529  1530  1531  1532  Wavelength [nm]  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='80  500  1000  1500  2000  Width [nm]a  COLUMBIA UNIVERSITY  DARPA  IN THE CITY OF NEW YORI  ch Lightwave Research Laborato  1CM  Power [dBm]  10  15  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='916 dB/cm  2  4  Spiral Length [cm]  400 um11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  a, Plot of measured (scatter) and modeled (line) propagation loss  vs ∂neff/∂w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Slope of fit represents R in (4), while the intercept represents  non-SWR loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' b, Plot of measured (scatter) and modeled (line) propagation  loss vs waveguide width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  the temperature of the resistive heater rather than the chip  temperature itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We know ∂Tchip/∂Theater ≤ 1 as heater  cannot raise the temperature of the chip to a value higher than  its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The extracted parameters for ∆w and V are used in  calculating (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  On average, the measured thermo-optic effect was found to  be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='91× our model’s (9) prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This value was found to  be independent of waveguide geometry with the exception of  400 nm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This error for 400 nm is assumed to be be-  cause this width is close enough to the cutoff condition for our  model to lose accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In contrast, the measured thermo-optic  effect was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='21× the previously reported model’s prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  This implies that either the chip’s change in temperature is  greater than the heater’s or the previously reported model is  incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The change in temperature of the PIC can only ever  be smaller than the heater’s temperature delta, making the old  model’s prediction clearly nonphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Scattering Loss Model Validation  To validate the scattering loss model, we measured a die  with 25 spiral loss structures consisting of 5 different wave-  guide widths (w) from 400 nm to 500 nm and 5 different spiral  lengths (∆L) from 1 cm to 5 cm, fabricated on a custom 300  mm full wafer through AIM Photonics (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Again, all  spiral structures have a nominal waveguide height of 220 nm,  and grating couplers designed for quasi-TE polarization are  utilized for optical I/O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The losses of each spiral length were  recorded, and then fit to a linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The slope of the  this fit was taken to be the propagation loss associated with  each waveguide width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The results of our model fit are shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The model built in Section V-B was used to build  a model of ∂neff/∂w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Fitting our modeled ∂neff/∂w to the  measured propagation loss yields proportionality constant of  R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='206 × 10−8 cm and (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 14a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The intercept of the loss  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' a, Cadence Virtuoso schematic of the MZI test circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' All circuit  models were written in Verilog-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The optical stimulus is provided by  a continuous-wave (CW) Laser and detected with a photodetector (PD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  b, Comparison of measured and simulated performance for an MZI with  w = 2 µm and ∆L = 100 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  fit is interpreted as the aggregate non-SWR loss, with a value  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='901 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 14b shows the excellent agreement between  our model and the data, predicting the similar propagation  losses of both the 440 nm and 460 nm waveguides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As  mentioned in Section V-B, both 400 and 420 nm waveguide  widths are likely near the cutoff condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Since (4) is only  valid sufficiently far away from this condition, those data  points are not included in the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Verilog-A Implementation  To demonstrate its compatibility with electronic-photonic  co-simulation, the circuit model was implemented in Verilog-  A within Cadence Virtuoso (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 15a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' As Verilog-A does not  inherently support optical signals, some compatibility code as  well as a small library of photonic device models were built  based upon on previously reported demonstrations [32], [59],  [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' CONCLUSION  In summary, we have demonstrated a novel compact model  that can greatly expand the accuracy of circuit-level simulation  capabilities of silicon PICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In contrast to prior work that  focused on providing metrology information that could be use  to fabrication engineers [26], [61], [62], we present this PDK  model as a tool suitable for true-to-measurement circuit simu-  lation and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' By leveraging this underlying physical  behavior and locally extracting process variations by perform-  ing group extraction, we have demonstrated a framework for  building a model of neff that is entirely driven by measurement  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' This model was shown to accurately describing the phase,  a  Loss [dB/cm]  R = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='206e-08 cm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='9  αnon-SwWR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='901 dB  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='7  neff/Ow [um-1]  b  Loss [dB/cm]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='7  440  450  460  470  480  490  500  Width [nm]a  ght<0:3  50:50 Coupler  50:50 Coupler  Waveguides  eftLig1<0:3>  rightLig1<0:3>  leftLig1<0:3>  rightLig1<:3>  leftLig2<0:3>  coupler  coupler  rightLig2<0:3>  eftLig2<0:3>  rightLig2<0:3>  inLight<0:3tatic_waveguide_R2_d1outLight<0:3>  CW Laser  lastightout<0:3  PD  b  Transmission [dB]  20  Measured  VerilogA  1520  1530  1540  1550  1560  Wavelength [nm]12  loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' and thermo-optic behavior of the measured integrated  waveguides over 4× the optical bandwidth and over 80×  the range of waveguides widths reported in prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' We  envision that the advancement over prior demonstrations this  work represents can support the development of waveguide-  based PDK components and enable the robust optimization of  next generation PICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' ACKNOWLEDGEMENTS  This work was supported in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Advanced  Research Projects Agency–Energy under ENLITENED Grant  DE-AR000843 and in part by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Defense Advanced Re-  search Projects Agency under PIPES Grant HR00111920014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The authors thank AIM Photonics for chip fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Derivation of Thermo-Optic Model  Starting from (8) from [46], the effect of a thermal pertur-  bation on the effective index is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Carrying out this  perturbation and following the chain rule yields:  2β ∂β  ∂T = 2Γcore  ω2  c2 ncore  ∂ncore  ∂T  + 2Γclad  ω2  c2 nclad  ∂nclad  ∂T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (24)  Noting that β = ωneff/c and inserting above, the relationship  simplifies to (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Combining with (1) yields:  neff = neff, T0 + ∂neff  ∂T (T − T0)  (25a)  ∂neff  ∂T  = Γcore  ncore  neff  ∂ncore  ∂T  + Γclad  nclad  neff  ∂nclad  ∂T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (25b)  It is noted that neff appears on both sides of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Multiplying both sides by effective index yields a quadratic  equation whose solution is:  neff = neff, T0  2  + 1  2  �  n2  eff, T0 + 4n  ′(T − T0)  (26a)  n  ′ = Γcorencore  ∂ncore  ∂T  + Γcladnclad  ∂nclad  ∂T  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (26b)  The expression can be simplified by noting that n2  eff, T0 ≫ 4n  ′  for typical values for the thermo-optic coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Under-  standing this, it is clear that the behavior of the square root  term is approximately linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The 1st order Taylor expansion  of the square root term is:  neff, T0 + 1  2  4n  ′  �  n2  eff, T0 + 4n  ′(T − T0)  (T − T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (27)  Noting again that n2  eff, T0 ≫ 4n  ′, (27) simplifies to:  neff, T0 +  2n  ′  neff, T0  (T − T0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (28)  Replacing the square root term in (26) with this expression  and simplifying will then yield (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Principal Component Analysis  To start, we form a matrix X our of our list of local sub-  parameter extractions, where each column represents a model  parameter and each row is an observation of said parameter:  X =  �  ������  ∂0neff  ∂λ0  0,1  ∂0neff  ∂λ0  1,1  · ·  ∂2neff  ∂λ2  4,1  ∂0neff  ∂λ0  0,2  ∂0neff  ∂λ0  1,2  · ·  ∂2neff  ∂λ2  4,2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  ∂0neff  ∂λ0  0,n  ∂0neff  ∂λ0  1,n  · ·  ∂2neff  ∂λ2  4,n  �  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  (29)  A covariance matrix S is then created from X and find its  eigenvectors:  S =  �  ����  ⃗v0  ⃗v1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  ⃗vn  �  ����  �  ����  λ0  0  · ·  0  0  λ1  · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  0  0  · ·  λn  �  ����  �  ����  ⃗v0  ⃗v1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  ⃗vn  �  ����  −1  ,  (30)  where  [v0, v1, · · · , vn]  lists  the  eigenvectors  and  [λ0, λ1, · · · , λn]  are  their  associated  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  The  eigenvectors of the correlation matrix represent the directions  of the axes where there is the most variance (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' the most  information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Each eigenvalue λi is proportional to how much  variance is captured by its associated principal component  vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Picking the eigenvectors with the largest eigenvalues  allows us to reduce data dimensionality at the expense of  some accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' The percentage of variability explained by a  principal component is calculated as  �M  i=0 λi  �N  i=0 λi  ,  (31)  where λi is the eigenvalue for each eigenvector, M is the  number of principal components the designer has chosen  to include, and N is the maximum number of principal  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  REFERENCES  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Rizzo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Daudlin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Novick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' James, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Gopal, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Murthy,  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Cheng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
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+page_content=' van Niekerk, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Deena-  dayalan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
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+page_content=' Rumley, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
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+page_content=' Wade, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Anderson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Ardalan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Bhargava, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
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+page_content=' Bogaerts,  “Compact silicon photonics circuit to extract multiple parameters  for process control monitoring,” OSA Continuum, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  379–390, Feb 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Available: https://opg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='optica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='org/osac/  abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='URI=osac-3-2-379  [62] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Boning, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' El-Henawy, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Zhang, “Variation-aware methods  and models for silicon photonic design-for-manufacturability,” Journal  of Lightwave Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' 1776–1783, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Aneek James received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' in Electrical and Electronics Engineering from  the University of Georgia, Athens, GA in 2017 and his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=', in  Electrical Engineering from Columbia University, New York, NY in 2019  and 2021, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He is working as a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' candidate in Electrical  Engineering in the Lightwave Research Laboratory under Professor Keren  Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' His research interests include modeling fabrication variations in  silicon photonic devices, as well as the testing and automated control of silicon  photonic systems for high-throughput optical interconnects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Anthony Rizzo received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' in Physics from Haverford College,  Haverford, PA in 2017 and his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=', and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=', all in Electrical  Engineering, from Columbia University, New York, NY in 2019, 2021, and  2022, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He completed his doctoral research in the Lightwave  Research Laboratory at Columbia University under Professor Keren Bergman,  where he led the first demonstration of data transmission using an integrated  Kerr frequency comb source and silicon photonic transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He is currently a  Research Scientist at the Air Force Research Laboratory (AFRL) Information  Directorate in Rome, NY, with a focus in large-scale silicon photonic systems  for quantum information processing and artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Yuyang Wang received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degree in electronic engineering from  Tsinghua University, Beijing, China in 2015, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' and PhD degrees  in computer engineering from the University of California, Santa Barbara  (UCSB), CA, USA, in 2018 and 2021 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He is currently a post-  doctoral researcher in the Lightwave Research Laboratory under Professor  Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He was a Design Engineering Intern at Cadence Design  Systems in 2018 and a Visiting Intern at the Hong Kong University of  Science and Technology in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' His research interests include variation-  aware modeling, design, and optimization of silicon photonic interconnects  and systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Asher Novick received his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degrees in Electrical and  Computer Engineering from Cornell University, Ithaca, NY, USA, in 2016 and  2015,respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Between 2016 and 2019, he was at Panduit’s Fiber Research  Lab, where he researched and developed new patentable technologies for  optical fiber-based communication in data center and enterprise applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  He is currently working toward his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degree in Electrical Engineering  in the Lightwave Research Laboratory at Columbia University in the City  of New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' His current research interest is in the modeling, design, and  testing of silicon photonic systems and devices for scalable and efficient link  architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Songli Wang received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' in Optoelectronic Information Science and  Engineering from Harbin Institute of Technology, Harbin, China, in 2019  and his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' in Electrical Engineering from Columbia University, New York,  NY in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He is currently working towards the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degree in Electrical  Engineering in the Lightwave Research Laboratory at Columbia University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  His current research interests include modeling, design and testing of silicon  photonic devices and systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Robert Parsons received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degree in biomedical engineering from  George Washington University, Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=', USA, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degree  in electrical engineering from Columbia University, New York, NY, USA, in  2020 and 2022, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He is currently working toward the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degree  in electrical engineering with the Lightwave Research Laboratory, Columbia  University under Professor Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' His research interests include the  modeling, testing, and co-optimization of link architectures and constituent  silicon photonic devices for high-bandwidth, energy-efficient optical inter-  connects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Kaylx Jang received his B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' in Electrical Engineering from the University  of California, Irvine, CA in 2020 and his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' in Electrical Engineering from  Columbia University in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' In 2019 and 2020, he interned in the testing  department at Ayar Labs and in 2021 did a co-op in the Silicon Photonics  Design team at Nokia (former Elenion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He is working as a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' student  in Electrical Engineering in the Lightwave Research Lab under Professor  Keren Bergman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' His research interests include modeling, design, and testing  of high-performance silicon photonic devices for energy efficient and scalable  link architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  Maarten Hattink is a graduate student with Columbia University, New  York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degrees from the Eindhoven  University of Technology, The Netherlands, in 2015 and 2017, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  While pursuing these degrees, he worked at Prodrive Technologies B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' as  a Software and FPGA Engineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' He is currently working toward the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  degree and his research interest lies in photonic device integration and thermal  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='  15  Keren Bergman (S’87–M’93–SM’07–F’09) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degree from  Bucknell University, Lewisburg, PA, in 1988, and the M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' degrees  from the Massachusetts Institute of Technology, Cambridge, in 1991 and  1994, respectively, all in electrical engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Bergman is currently a  Charles Batchelor Professor at Columbia University, New York, NY, where she  also directs the Lightwave Research Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' She leads multiple research  programs on optical interconnection networks for advanced computing sys-  tems, data centers, optical packet switched routers, and chip multiprocessor  nanophotonic networks-on-chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
+page_content=' Bergman is a Fellow of the IEEE and  Optica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQfuP3I/content/2301.01689v1.pdf'}
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+Received:
+Added at production
+Revised:
+Added at production
+Accepted:
+Added at production
+DOI: xxx/xxxx
+RESEARCH ARTICLE
+Data-driven modelling of turbine wake interactions and flow
+resistance in large wind farms
+Andrew Kirby*1 | François-Xavier Briol2 | Thomas D. Dunstan3 | Takafumi Nishino1
+1Department of Engineering Science,
+University of Oxford, Oxford, UK
+2Department of Statistical Science,
+University College London, London, UK
+3Informatics Lab, UK MetOffice, Exeter,
+UK
+Correspondence
+*Andrew Kirby, Department of
+Engineering Science, University of
+Oxford, Oxford, OX1 3PJ, UK. Email:
+andrew.kirby@trinity.ox.ac.uk
+Abstract
+Turbine wake and local blockage effects are known to alter wind farm power production in
+two different ways: (1) by changing the wind speed locally in front of each turbine; and (2)
+by changing the overall flow resistance in the farm and thus the so-called farm blockage
+effect. To better predict these effects with low computational costs, we develop data-driven
+emulators of the ‘local’ or ‘internal’ turbine thrust coefficient C∗
+T as a function of turbine
+layout. We train the model using a multi-fidelity Gaussian Process (GP) regression with a
+combination of low (engineering wake model) and high-fidelity (Large-Eddy Simulations)
+simulations of farms with different layouts and wind directions. A large set of low-fidelity
+data speeds up the learning process and the high-fidelity data ensures a high accuracy. The
+trained multi-fidelity GP model is shown to give more accurate predictions of C∗
+T compared
+to a standard (single-fidelity) GP regression applied only to a limited set of high-fidelity
+data. We also use the multi-fidelity GP model of C∗
+T with the two-scale momentum theory
+(Nishino & Dunstan 2020, J. Fluid Mech. 894, A2) to demonstrate that the model can be
+used to give fast and accurate predictions of large wind farm performance under various
+mesoscale atmospheric conditions. This new approach could be beneficial for improving
+annual energy production (AEP) calculations and farm optimisation in the future.
+KEYWORDS:
+Class file; LATEX 2ε; Wiley NJD
+1
+INTRODUCTION
+The installed capacity of wind energy is projected to increase rapidly in the next decades. A major challenge in the optimisation
+of wind farm design is the accurate prediction of wind farm performance 1. Existing wind farm models struggle to make accurate
+predictions of wind farm power production. This is partly because the ‘global blockage effect’ reduces the velocity upstream of large
+farms and hence the energy yield 2. It remains unclear how global blockage should be modelled and this is the subject of a large-scale
+field campaign 3.
+Wind farms are typically modelled using engineering ‘wake’ models. These models predict the velocity deficit in the wakes behind
+turbines 4 5. To account for interactions between multiple turbines, the wake velocity deficits are superposed 6,7. Simple wake models
+can give predictions of wind farm performance with very low computational cost ( 10−3 CPU hours per simulation 1). However, wake
+arXiv:2301.01699v1  [physics.flu-dyn]  4 Jan 2023
+
+2
+KIRBY et al
+models do not account for the response of the atmospheric boundary layer (ABL) to the wind farm which is likely to be important
+for large wind farms 8. It has been found that wake models compare poorly to Large-Eddy Simulations (LES) of large wind farms 9.
+Wind farms are also modelled in numerical weather prediction (NWP) models using farm parameterisation schemes. In these pa-
+rameterisations, farms are often modelled as a momentum sink and a source of turbulent kinetic energy 10. Turbine-wake interactions
+cannot be adequately predicted using these schemes. A new scheme was proposed 11 which uses a correction factor to model turbine
+interactions. More recently, data-driven approaches have been proposed 12 to model these effects in wind farm parameterisations.
+Data-driven modelling of wind farm flows is a promising new approach 13. Data from high-fidelity simulations with complex flow
+physics can be used to make predictions with low computational cost. Recent studies have applied machine learning techniques
+to data from a single turbine or from an existing wind farm. The data for these studies are from measurements 14,15,16,17, LES 18 or
+Reynolds-Averaged Navier-Stokes (RANS) simulations 19,20,21. A limitation of these approaches is that they are not generalisable to
+different turbine layouts unless they rely on wake superposition techniques to model farm flows. Another approach is modelling
+the effect of turbine layout using geometric parameters 17 or using the layout as a graph input to a neural network 22,23. However,
+these alternative approaches may struggle to fully capture the complex two-way interaction with the ABL as it seems impractical to
+prepare a data set that covers the entire range of scales involved in wind farm flows 1.
+The problem of modelling wind farm flows can be split into ‘internal’ turbine-scale and ‘external’ farm-scale problems 24. The
+‘internal’ problem is to determine a ‘local’ or ‘internal’ turbine thrust coefficient, C∗
+T , which represents the flow resistance inside a
+wind farm, i.e., how the turbine thrust changes with wind speed within the farm. Nishino 25 proposed an analytical model for an upper
+limit of C∗
+T by using an analogy to the classic Betz analysis. This analytical model is a function of turbine-scale induction factor but
+is independent of turbine layout and wind direction. Previous studies 24 25 8 showed that C∗
+T is usually lower than the limit predicted
+by Nishino’s model and can vary significantly with turbine layout due to wake and turbine blockage effects.
+The aim of this study is to develop statistical emulators of C∗
+T as a function of turbine layout and wind direction. The novelty of
+this approach is that we are modelling the effect of turbine-wake interactions on C∗
+T rather than turbine power. Both turbine-scale
+flows (e.g., wake effects) and farm-scale flows (e.g. farm blockage and mesoscale atmospheric response) affect turbine power within
+a farm. Therefore to create an emulator of turbine power, either (1) a very large set of expensive data such as finite-size wind farm
+LES is needed which covers a range of large-scale atmospheric conditions or (2) the model would not be generalisable to different
+mesoscale atmospheric responses. An emulator of C∗
+T is however applicable to different atmospheric responses modelled separately,
+following the concept of the two-scale momentum theory 24 8.
+In section 2 we give the definitions of key wind farm parameters in the two-scale momentum theory 24. Section 3 summarises
+the methodology of the LES and wake model simulations, followed by the machine learning approaches to develop the emulators
+in section 4. In section 5 we present the results from the trained emulators. These results are discussed in section 6 and concluding
+remarks are given in section 7.
+2
+TWO-SCALE MOMENTUM THEORY
+By considering the conservation of momentum for a control volume with and without a large wind farm over the land or sea surface,
+the following non-dimensional farm momentum (NDFM) equation can be derived 24,
+C∗
+T
+λ
+Cf0
+β2 + βγ = M
+(1)
+where β is the farm wind-speed reduction factor defined as β ≡ UF /UF 0 (with UF defined as the average wind speed in the nominal
+wind farm-layer of height HF , and UF 0 is the farm-layer-averaged speed without the wind farm present); λ is the array density
+defined as λ ≡ nA/SF (where n is the number of turbines in the farm, A is the rotor swept area and SF is the farm footprint area);
+
+KIRBY et al
+3
+C∗
+T is the internal turbine thrust coefficient defined as C∗
+T ≡ �n
+i=1 Ti/ 1
+2 ρU2
+F nA (where Ti is thrust of turbine i in the farm and ρ
+is the air density); Cf0 is the natural friction coefficient of the surface defined as Cf0 ≡ ⟨τw0⟩/ 1
+2 ρU2
+F 0 (where τw0 is the bottom
+shear stress without the farm present); γ is the bottom friction exponent defined as γ ≡ logβ(⟨τw⟩/τw0) (where ⟨τw⟩ is the bottom
+shear stress averaged across the farm); M is the momentum availability factor defined as,
+M =
+Momentum supplied by the atmosphere to the farm site with turbines
+Momentum supplied by the atmosphere to the farm site without turbines.
+(2)
+noting that this includes pressure gradient forcing, Coriolis force, net injection of streamwise momentum through top and side
+boundaries and time-dependent changes in streamwise velocity 24. The height of the farm-layer, HF , is used to define the reference
+velocities UF and UF 0. Equation 1 is valid so long as the same of HF is used for both the internal and external problem. HF is typically
+between 2Hhub and 3Hhub 8 (where Hhub is the turbine hub-height) and in this study we use a fixed definition of HF = 2.5Hhub.
+Patel 26 used an NWP model to demonstrate that, for most cases, M varied almost linearly with β (for a realistic range of β
+between 0.8 and 1). Therefore, M can be approximated by
+M = 1 + ζ(1 − β)
+(3)
+where ζ is the ‘momentum response’ factor or ‘wind extractability’ factor. Patel 26 found ζ to be time-dependent and vary between
+5 and 25 for a typical offshore site (note that ζ = 0 corresponds to the case where momentum available to the farm site is assumed
+to be fixed, i.e., M = 1).
+Nishino 25 proposed an analytical model for C∗
+T given by,
+C∗
+T = 4α(1 − α) =
+16C′
+T
+(4 + C′
+T )2
+(4)
+where α is the turbine-scale wind speed reduction factor defined as α ≡ UT /UF (UT is the streamwise velocity averaged over the
+rotor swept area) and C′
+T ≡ T/ 1
+2 ρU2
+T A is a turbine resistance coefficient describing the turbine operating conditions.
+For a given farm configuration at a farm site (i.e., for given set of C∗
+T , λ, Cf0, γ and ζ) the farm wind-speed reduction factor β
+can be calculated using equation 1. The (farm-averaged) power coefficient Cp is defined as Cp ≡ �n
+i=1 Pi/ 1
+2 ρU3
+F 0nA (Pi is power
+of turbine i in the farm). Using the calculated value of β, Cp can be calculated by using the expression,
+Cp = β3C∗
+p
+(5)
+where C∗
+p is the (farm-averaged) ‘local’ or ‘internal’ turbine power coefficient defined as C∗
+p ≡ �n
+i=1 Pi/ 1
+2 ρU3
+F nA.
+3
+WIND FARM SIMULATIONS
+In this study we model wind farms as arrays of actuator discs (or aerodynamically ideal turbines operating below the rated wind
+speed). This is because, in real wind farms, the effects of turbine wake interactions on the farm performance are most significant
+when they operate below the rated wind speed. The ‘internal’ thrust coefficient C∗
+T is an important wind farm parameter which
+includes the effect of turbine interactions (including both wake and local blockage effects). In this study we will be modelling the
+effect of turbine layout on C∗
+T for aligned turbine layouts with various wind directions and a fixed turbine resistance of C′
+T = 1.33.
+We chose C′
+T = 1.33 because it leads to a turbine induction factor of 1/4 which is close to a typical value for modern large wind
+turbines. As such we will be considering
+C∗
+T = f(Sx, Sy, θ)
+(6)
+
+4
+KIRBY et al
+Figure 1 Design of numerical experiments: a) input parameters, b) maximin design of LES.
+where Sx is the turbine spacing in the x direction, Sy is the turbine spacing in the y direction and θ is the wind direction relative
+to the x direction (see figure 1a). However the true function C∗
+T cannot be easily evaluated so we will instead investigate C∗
+T using
+computer codes. One computer code we will use is LES (see section 3.1) to estimate C∗
+T
+C∗
+T,LES = fLES(Sx, Sy, θ).
+(7)
+We assume that the function fLES is close to the true function f because of the accuracy of LES to model wind farm flows. We will
+also use a wake model (see section 3.2) to provide cheap approximations of C∗
+T according to
+C∗
+T,wake = fwake(Sx, Sy, θ).
+(8)
+Engineering problems are often investigated using complex computer models. Evaluating the output of such computer models
+for a given input can be very computationally expensive. Therefore a common objective is to create a cheap statistical model of
+the expensive computer model; this is commonly known as emulation of computer models 27 28. In this study we aim to develop a
+statistical emulator which can cheaply emulate fLES.
+The emulators will only be valid for aligned layouts of wind turbines and for a given turbine resistance (here we use C′
+T = 1.33).
+We consider the input parameters for a realistic range of turbine spacings 1: Sx ∈ [5D, 10D], Sy ∈ [5D, 10D] and θ ∈ [0o, 45o]
+where D is the diameter of the turbine rotor swept area. In this study D is set as 100m and the turbine hub height is also 100m. We
+only need to consider wind directions of θ ∈ [0o, 45o] because of symmetry in the aligned turbine layouts. If θ is negative than the
+turbine layout given by (Sx, Sy, θ) is exactly the same as (Sx, Sy, −θ). When θ > 45o, then (Sx, Sy, θ) and (Sy, Sx, 90o − θ) give
+identical layouts.
+In this study we build several emulators to predict fLES. The models are trained using data from low-fidelity (wake model)
+and high fidelity (LES) wind farm simulations. One evaluation of C∗
+T,wake takes approximately 130 seconds on a single CPU and
+C∗
+T,LES requires around 400 CPU hours on a supercomputer. We use a space filling maximin design 29 30 to select training points
+in the parameter space. The maximin algorithm selects points which maximises the minimum distance to other points and to the
+boundaries. This provides a good coverage of the domain which ensures that the emulators can give good predictions across the
+whole of the domain 31. Figure 1b shows the LES training points in the parameter space.
+
+b)
+a)
+10
+40
+9
+Wind
+30
+200
+10
+5
+0
+6
+8
+10
+S/DKIRBY et al
+5
+Figure 2 LES a) instantaneous and b) time-averaged flow fields over a periodic turbine array (Sx/D = 7.59, Sy/D = 5.47 and
+θ = 37.6o).
+3.1
+Large-Eddy Simulations
+This study uses the data from 50 high-fidelity (LES) simulations of wind farms published in a previous study 8. Here we give a brief
+summary of the LES methodology. The LES models a neutrally stratified atmospheric boundary layer over a periodic array of actuator
+discs, which face the wind direction θ and exert uniform thrust. The resolution is 24.5m in the horizontal directions (4 points across
+the rotor diameter) and 7.87m in the vertical. This is a coarse horizontal resolution; however using a correction factor for the turbine
+thrust 32 makes the C∗
+T,LES values insensitive to horizontal resolution 8. For all simulations the vertical domain size was fixed at
+1km and the horizontal extent varied with turbine layout but was at least 3.14km. The horizontal boundary conditions were periodic
+(essentially an infinitely-large wind farm). The bottom boundary used a no-slip condition with the value of eddy viscosity specified
+following the Monin-Obukhov similarity theory for a surface roughness length of z0 = 1 × 10−4m. The top boundary had a slip
+condition with zero vertical velocity. The flow was driven by a pressure gradient forcing which was constant and in the direction θ
+throughout the domain. Figure 2 shows the instantaneous and time-averaged hub height velocities from one wind farm LES. See the
+original paper 8 for further details of the LES.
+3.2
+Wake model simulations
+Wake models are a cheap low-fidelity approach to modelling wind farm aerodynamics compared to expensive high-fidelity LES
+simulations 1. We use the wake model proposed by Niayafar and Porté-Agel 33 to evaluate C∗
+T,wake as a cheap approximation of C∗
+T .
+We use the Python package PyWake 34 to implement the wake model. The turbine thrust coefficient CT is needed as an input for
+the wake model. We use the value of C∗
+T predicted by equation 4 as the value of CT . For the turbine operating conditions used in
+this study (C′
+T = 1.33) the wake model has CT equal to 0.75 for all turbines. To model actuator discs, we consider a hypothetical
+turbine which has a constant CT for all wind speeds. We calculate C∗
+T,wake for a single turbine at the back of a large farm (marked X
+in figure 3). The farm simulated using the wake model is 10km long in the streamwise direction and 4km long in the cross-streamwise
+direction. The farm size was chosen so that C∗
+T no longer varied with increasing farm size. The wake growth parameter is calculated
+using k∗ = 0.38I +0.004 where I is the local streamwise turbulence velocity. The local streamwise turbulence intensity is estimated
+using the model proposed by Crespo and Hernández 35. The background turbulence intensity (TI) is set as a typical value of 10%.
+The velocity incident to the turbine is calculated by averaging the velocity across the disc area. We use a 4×3 cartesian grid with
+Gaussian quadrature coordinates and weights on the disc to average the velocity. The disc-averaged velocity, UT is then calculated
+by multiplying the averaged incident velocity by (1 − a) where a is the turbine induction factor set by the value of C′
+T (using the
+expression a = C′
+T /(4 + C′
+T )). To calculate the farm-average velocity, UF , we average the velocity across a volume around the
+
+b)
+a)
+0.4
+30
+30
+0.3
+20
+20
+D
+D
+n/n
+9
+9
+0.2
+10
+10
+0.1
+0
+0
+20
+0
+20
+c/ D
+c/ D6
+KIRBY et al
+Figure 3 Example of wind farm layout for wake model simulations.
+single turbine. The volume has dimensions of Sy in the y direction, Sx in the x direction and 250m in the z direction (the height of
+the nominal farm layer used in the previous LES study 8). To calculate the average velocity, we discretise the volume into 200 points
+in the horizontal directions and 20 points in the vertical. This was sufficient for the calculation of C∗
+T,wake to not vary with further
+discretisation. Figure 3 shows an example of the farm layout for the wake model simulations.
+4
+MACHINE LEARNING METHODOLOGY
+4.1
+Gaussian Process regression
+We will use Gaussian process (GP) regression 36 to build statistical emulators of fLES. A Gaussian process is a stochastic process
+g ∼ GP(m, k) described by a mean function m(v) = E[g(v)] and a covariance function k(v, v′) = E[(g(v) − m(v))(g(v′) − m(v′)].
+In our case v = (Sx, Sy, θ). We will use such a stochastic process as a model of fLES, the true mapping from v to C∗
+T,LES. Each
+realisation from this process will therefore be a function which could plausibly represent this mapping. The mean function represents
+the expected output value at an input v = (Sx, Sy, θ). The covariance function gives the covariance between output values at v and
+v′. Examples of covariance functions include squared exponential, rational quadratic and periodic functions 36. Different covariance
+functions will give differently shaped GPs. For example the squared exponential covariance function will give very smooth GPs
+whereas the periodic function will give GPs with a periodic structure. Other types of structure, for example symmetry, can also be
+encoded in the covariance function. Therefore the expected shape (for example smoothness) of the expected relationship and any
+properties (for example discontinuities or symmetries) need to be considered when choosing a covariance function for GP regression.
+Let V = (v1, ..., vn)T be a collection of design points then mV = (m(v1), ..., m(vn))T is the mean vector and kV V = (k(vi, vj))
+is the covariance matrix. We will start by positing a GP model with mean m and covariance k (called the ‘prior GP’), then condition
+this GP on LES observations; the outcome is a new GP (called the ‘posterior GP’). This gives the posterior distribution g|V, C∗
+T,LES ∼
+GP(mσ2, kσ2). mσ2 is the posterior mean function given by mσ2(v) = m(v) + kvV (kV V + σ2In×n)−1(C∗
+T,LES − mV ) where
+kvV = (k(v, v1), ..., k(v, vn)) and In×n is the identity matrix of size n. The posterior mean function mσ2 is used to make predictions
+at v = (Sx, Sy, θ). The posterior covariance function kσ2 quantifies the uncertainty in our prediction at v = (Sx, Sy, θ). The
+posterior covariance function is given by kσ2(v, v′) = k(v, v′) − kvV (kV V + σ2In×n)−1kV v′.
+Often in GP regression a zero prior mean is used. However, using an informative prior mean can improve the accuracy of the
+trained model. By using a prior mean, many of the trends in fLES can be incorporated into our model prior to making expensive
+
+Volume for
+UF calculation
+10 km
+X
+WindKIRBY et al
+7
+Figure 4 Demonstration of basic GP regression: a) shows the prior mean and covariance function prior to fitting with 3 GPs drawn
+from the distribution shown in colour; b) shows the effect of decreasing the lengthscale hyperparameter; c) the effect of variance
+hyperparameter; and d) the posterior mean and covariance functions.
+evaluations of C∗
+T,LES. Therefore, after training our model will likely better describe the true relationship between Sx, Sy, θ and
+fLES. In this study, we will use both C∗
+T,wake and the analytical model of C∗
+T as the prior mean for the standard GP regression. For
+the wake model prior mean we also vary the specified ambient TI input parameter.
+We expect fLES to be a smooth function of input variables Sx, Sy and θ, and to vary more rapidly with θ than Sx or Sy. Therefore
+we will use an anisotropic squared-exponential covariance function,
+k(v, v′) = σ2
+f exp
+�
+− (Sx − S′
+x)2
+2l2
+1
+�
+exp
+�
+−
+(Sy − S′
+y)2
+2l2
+2
+�
+exp
+�
+− (θ − θ′)2
+2l2
+3
+�
+(9)
+where σ2
+f > 0 is the signal variance hyperparameter and li > 0 is the lengthscale hyperparameter for each dimension. This is also
+called an ARD (automatic relevance detection) kernel. If we consider v = v′ then we can see that σ2
+f determines the variance of g(v).
+Therefore σ2
+f determines the prior uncertainty the model has about the value of g(v). As the lengthscale hyperparameter li gets
+smaller then k(v, v′) decreases (for v ̸= v′). Equally if li increases then k(v, v′) will also increase. A GP with a small li will therefore
+vary more rapidly across the parameter space in the ith dimension.
+Due to numerical issues associated with the matrix inversion/linear system solve operations in the formulae for the posterior GP,
+it is common to add a nugget σ2 > 0 to the kernel matrix. The hyperparameters σ2
+f and li are selected automatically during the
+fitting process by maximising the log marginal likelihood 36. This approach selects the model which maximises the fit to the data.
+Figure 4 shows the impact of the hyperparameters in an example GP regression setting (using the squared exponential covariance
+function). The mean function and 95% credible interval (+/-1.96 times the standard deviation) prior to fitting are shown in figure
+4a with 3 GPs drawn from the distribution (coloured lines). The effect of decreasing the lengthscale hyperparameter li is shown in
+figure 4b. The prior mean and 95% credible interval are unchanged however the example GPs drawn vary more rapidly because of
+the shorter lengthscale. Figure 4c shows the same setup as figure 4a but with a smaller value of σ2
+f. The example GPs still vary slowly
+but the magnitude of the variations is now smaller. Figure 4d shows the GPs conditioned on observations with hyperparameters
+selected by maximising the log marginal likelihood.
+
+a) α = 1.0, l = 1.0
+b) α² = 1.0, l = 0.5
+2
+2
+9
+0
+9
+0
+-2
+-2
+0
+2
+4
+6
+0
+2
+4
+6
+c) ² = 0.5, l = 1.0
+d)o
+= 1.41. = 1.57
+2
+2
+9
+0
+-2
+-2
+0
+6
+0
+2
+2
+4
+4
+6
+Observations
+Mean function
+95% credible interval8
+KIRBY et al
+Figure 5 Demonstration of a) basic GP regression and b) multi-fidelity GP regression. In this example f(x) = 1 + sin(6x) for the
+high-fidelity data and f(x) = −0.5 + 0.5sin(6x) for the low-fidelity data.
+4.2
+Non-linear multi-fidelity Gaussian Process regression
+In many applications there are several computational models available. These models can have varying accuracies and computational
+costs. The models which are more computationally expensive typically give more accurate predictions. The GP regression frame-
+work can be extended to combine information from low and high-fidelity models 37. This type of modelling uses the low-fidelity
+observations to speed up the learning process and the high-fidelity observations to ensure accuracy. In our scenario we will com-
+bine evaluations of from a low-fidelity (C∗
+T,wake) and a high-fidelity (C∗
+T,LES) model. Note that for the multi-fidelity models in this
+study we set the ambient TI to 10% for the wake model and use a zero prior mean. We will keep the number of high-fidelity training
+points fixed at 50 and we will vary the number of low-fidelity training points used.
+We combine information from our high and low-fidelity models using a nonlinear information fusion algorithm 38. The framework
+is based on the autoregressive multi-fidelity scheme given by:
+ghigh(v) = ρ(glow(v)) + δ(v)
+(10)
+where glow(v) is a model with a GP denoted fwake and ghigh(v) is a model with a GP denoted fLES. ρ is a model with a GP
+which maps the low-fidelity output to the high-fidelity output and δ(v) is a model with a GP which is a bias term. The non-linear
+multi-fidelity framework can learn non-linear space-dependent correlations between models of different accuracies. To reduce the
+computational cost and complexity of implementation the autoregressive scheme given by equation 10 is simplified. Firstly, the GP
+prior glow(v) is replaced by the GP posterior glow,∗(v) and secondly the GPs ρ and δ are assumed to be independent. Equation 10
+can then be summarised as
+ghigh(v) = hhigh(v, glow,∗(v))
+(11)
+where hhigh is a model with a GP which has both v and glow,∗(v) as inputs. More details of hhigh and the implementation of the
+multi-fidelity framework are given in Perdikaris et. al. 38.
+Figure 5 shows an example of how a multi-fidelity GP can outperform a standard GP regression. We implement the non-linear
+multi-fidelity framework using the ‘emukit’ package 39. We first maximise the log marginal likelihood whilst keeping the Gaussian
+noise variance fixed at a low value of 1 × 10−6. The fitting process is then repeated whilst allowing the Gaussian noise variance to
+be optimised too. This is to prevent a high noise local optima from being selected.
+
+a)
+b)
+3
+3
+High fidelity
+High fidelity
+2
+2
+1
+1
+9
+9
+0
+0
+-1
+-1
+Low fidelity
+2
+2
+-0.5
+0.0
+0.5
+-1.0
+1.0
+-1.0
+-0.5
+0.0
+0.5
+1.0
+True function
+Observations
+Posterior mean function
+95% credible intervalKIRBY et al
+9
+5
+RESULTS
+In this study, we build various statistical emulators of fLES using different techniques and compare the performance. A summary
+of the techniques is shown in the list below:
+1 Standard Gaussian Process regression (see section 4.1)
+a GP-analytical-prior: Gaussian Process using analytical model (equation 4) prior mean
+b GP-wake-TI10-prior: Gaussian Process using wake model (section 3.2) with ambient TI=10% prior mean
+c GP-wake-TI1-prior: Gaussian Process using wake model with ambient TI=1% prior mean
+d GP-wake-TI5-prior: Gaussian Process using wake model with ambient TI=5% prior mean
+e GP-wake-TI15-prior: Gaussian Process using wake model with ambient TI=15% prior mean
+2 Non-linear multi-fidelity Gaussian Process regression (see section 4.2)
+a MF-GP-nlow500: multi-fidelity Gaussian Process using 500 low-fidelity training points
+b MF-GP-nlow250: multi-fidelity Gaussian Process using 250 low-fidelity training points
+c MF-GP-nlow1000: multi-fidelity Gaussian Process using 1000 low-fidelity training points
+The code used to produce the results in this section is available open-access at the following GitHub repository: https://github.
+com/AndrewKirby2/ctstar_statistical_model.
+5.1
+Performance of standard GP regression
+We first assessed the accuracy of the standard GP models (section 4.1) by performing leave-one-out cross-validation (LOOCV). This
+is a method of estimating the accuracy of a statistical model when making predictions on data not used to train the model. We trained
+our model on 49 of the 50 training points and then calculated the prediction accuracy for the single high-fidelity data point which is
+excluded from the training set. This is then repeated for all data points in turn, and we took the average accuracy as an estimate of
+the model test accuracy. The standard GP models were implemented using the ‘GPy’ package 40.
+The standard GP gave accurate predictions of fLES with average errors of less than 2%. Table 1 shows the accuracy of the stan-
+dard GP models compared to the analytical and wake models. We calculated the errors by using the expression |mσ2 −C∗
+T,LES|/0.75
+where mσ2 is the posterior mean function of the emulator. The reference value for C∗
+T of 0.75 was chosen because this is the pre-
+diction from the analytical model. Both GP models give similar maximum errors of approximately 6%. Using the wake model as a prior
+mean gave a lower mean absolute error of 1.26%. The GP models reduced the average prediction error and significantly reduced
+the maximum error compared to the wake model and analytical model of C∗
+T .
+Table 1 Accuracy of models for C∗
+T prediction.
+Model
+MAE (%)
+Maximum error (%)
+GP-analytical-prior
+1.87
+6.09
+GP-wake-TI10-prior
+1.26
+6.11
+Analytical model
+5.26
+22.0
+Wake model (TI=10%)
+4.60
+9.28
+
+10
+KIRBY et al
+Figure 6 Posterior variance function of GP-wake-TI10-prior model.
+Figure 7 Sensitivity of fitted GP models to the ambient TI chosen for wake model prior means.
+The model GP-wake-TI10-prior has a high degree of confidence when making predictions in regions of the parameter space.
+Figure 6 shows the square root of the posterior covariance function kσ2, which quantities the uncertainty of the emulator. The
+uncertainty is uniform throughout the parameter space with regions of slightly higher uncertainty at θ = 0o and 45o.
+We also assessed the sensitivity of the model accuracy to the ambient TI used in the wake model prior mean. Figure 7 shows
+the impact of ambient TI on the wake model prior mean and the fitted GP model. Increasing the ambient TI increased the value of
+C∗
+T,wake. This is because of the enhanced wake recovery behind wind turbines. Increasing the ambient TI in the wake model results
+in C∗
+T,wake overpredicting C∗
+T,LES. The MAE from the LOOCV procedure for each fitted GP is shown in the bottom right corner.
+
+b)=5°
+c)=100
+d)  =15°
+a)=00
+e)=200
+10
+10
+10
+10
+10
+Dβ
+8
+8
+8
+8
+6
+6
+6
+6 
+6
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+f) =250
+g)  =300
+h)=35°
+i)=40°
+j)=45°
+10
+10
+10
+10
+10
+D
+8
+8
+8
+8
+6
+6
+6
+6
+6
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+Sα/D
+Sα/D
+Sα/D
+Sα/D
+Sαc/D
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Vk.2a) Ambient TI=1%
+b) Ambient TI=5%
+0.8
+0.8
+0.7
+0.7
+*
+山秋
+0.6
+0.6
+GP-wake-TI1-prior
+.·
+GP-wake-TI5-prior
+0.5
+0.5
+MAE=3.02%
+MAE=2.16%
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+CT,LES
+C*,LES
+c) Ambient TI=10%
+d) Ambient TI=15%
+0.8
+0.8
+0.7
+0.7
+0.6
+0.6
+GP-wake-TI10-prior
+GP-wake-TI15-prior
+0.5
+0.5
+MAE=1.25%
+MAE=1.04%
+0.60
+0.65
+0.55
+0.70
+0.75
+0.80
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Wake model prior mean
+Standard GP modelKIRBY et al
+11
+The fitted GPs became more accurate when the wake model ambient TI was increased. Increasing the ambient TI for the wake
+model causes the wakes to recover faster. The wakes become shorter in the streamwise direction and wider in the spanwise direction.
+As such, C∗
+T,wake becomes less sensitive to the turbine layout. When an ambient TI of 1% and 5% is used for the wake model,
+C∗
+T,wake is more sensitive to turbine layout than C∗
+T,LES (figures 7a and 7b). When the ambient TI is increased to 10% and above,
+the relationship between C∗
+T,wake and C∗
+T,LES becomes simpler (figures 7c and 7d). This seems to explain why the fitted GPs
+become more accurate.
+5.2
+Performance of non-linear multi-fidelity GP regression
+We then assessed the accuracy of the multi-fidelity GP models (section 4.2). All models used the 50 high-fidelity (C∗
+T,LES) training
+points and a varying number of low-fidelity (C∗
+T,wake) training points (using an ambient TI of 10% for C∗
+T,wake). The results from
+LOOCV are shown in table 2. For the LOOCV we train our model on 49 out of the 50 high-fidelity data points and all low-fidelity
+data points. Then we average the error in predicting the high-fidelity data point left of the training set and repeat this in turn for
+data points. Increasing the number of low-fidelity training points from 250 to 500 reduced the mean and maximum error. However,
+increasing this to 1000 low-fidelity training points did not increase accuracy and increased the fitting and prediction time. This is
+because the number of high-fidelity training points is fixed. There is a threshold where the model of the relationship between fLES
+and fwake, denoted ρ, limits the final accuracy of the emulator of fLES.
+The posterior mean mσ2 of glow(v) is an emulator of fwake and ghigh(v) is an emulator of fLES. Figure 8 gives the predictions
+from the posterior mean of ghigh(v) (for MF-GP-nlow500). The lowest mσ2 values were for a wind direction of θ = 0o. mσ2
+Table 2 Performance of the multi-fidelity Gaussian Process models.
+Model
+MAE (%)
+Maximum error (%)
+Training time (s)
+Prediction time (s)
+MF-GP-nlow250
+1.46
+7.12
+6.15
+0.00157
+MF-GP-nlow500
+0.828
+3.75
+9.73
+0.00167
+MF-GP-nlow1000
+0.866
+3.55
+26.8
+0.00236
+Figure 8 Posterior mean function for ghigh(v) of MF-GP-nlow500.
+
+b)=50
+d)  =15°
+a)=00
+c)=100
+e)=200
+10
+10
+10
+10
+10
+Dβ
+8
+8
+8
+8
+6
+6
+6
+6 
+6
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+f) =250
+g)  =300
+h) =35°
+i)=40°
+j）)θ =45°
+10
+10
+10
+10
+10
+D
+8
+8
+8
+8
+6
+6
+6
+6
+6
+5
+10
+10
+5
+10
+5
+10
+5
+5
+10
+Sα/D
+Sα/D
+Sα/D
+Sα/D
+Sαc/D
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+mg212
+KIRBY et al
+Figure 9 Posterior variance function for ghigh(v) of MF-GP-nlow500.
+increased rapidly with θ reaching a maximum of slightly over 0.75 at θ = 10o. For large values of θ (above θ = 25o) there were
+local minima in mσ2 which appear in figure 8 as diagonal strips of low mσ2 values. The main diagonal strip occurs along the line
+of Sy = Sx tan(θ). There are two smaller strips either side of with positions given by Sy = 2 tan(θ) and Sy = 0.5 tan(θ) (this is
+discussed further in section 6).
+The uncertainty the model MF-GP-nlow500 has in predicting fLES is shown in figure 9. The model uncertainty is uni-
+form throughout the parameter space with slightly higher values at θ = 0o and 45o. Compared to the posterior variance of
+GP-wake-TI10-prior (shown in figure 6) the uncertainty is lower. By incorporating information from C∗
+T,wake, the multi-fidelity GP
+model has more confidence about predicting fLES.
+The prediction errors from the LOOCV (for MF-GP-nlow500) are shown in figure 10. The box plot of prediction errors in figure
+10a shows that this model had no significant bias whereas both the wake and analytical models systemically overestimated C∗
+T,LES.
+Figures 10b-d show that for the statistical model there appears to be no part of the parameter space which had larger errors.
+The multi-fidelity approach used in this study builds a statistical model of both the low-fidelity (fwake) and high-fidelity (fLES)
+model. We can use the posterior means of glow(v) and ghigh(v) to see the differences between the wake model and LES. The
+posterior mean for both models are shown in figure 11. For the wake model the change in mσ2 with θ is greater than for the LES
+(especially between θ = 0o and 10o). For larger values of θ, there is a larger difference in mσ2 between waked and unwaked layouts
+for the low-fidelity model compared to the high-fidelity one. This suggests than the wake model is more sensitive to changes in wind
+directions than the LES.
+
+a) =00
+b) =5°
+c)=100
+d)θ=15°
+e)=20°
+10
+10
+10
+10
+10
+D∞
+8
+8
+8 
+8
+6
+6
+6
+6
+6
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+f）θ =25°
+g) =300
+h) =350
+i)=40°
+j)θ=45°
+10
+10
+10
+10
+10
+Dβ
+8
+8
+8
+8
+6
+6
+6
+6 
+6
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+Sα/D
+Sα/D
+Sα/D
+Sα/D
+S/D
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+Vkg?KIRBY et al
+13
+Figure 10 Comparison of LOOCV prediction errors (%) for different models a) and LOOCV prediction error (%) of MF-GP-nlow500
+against input parameters b) Sx/D, c) Sy/D and d) θ(o). Note that for the box plot in a) the orange line is the median LOOCV error
+and the box is the interquartile range of LOOCV error.
+Figure 11 Posterior mean function of MF-GP-nlow500 for different values of θ for a) to e) ghigh(v) and f) to j) glow(v).
+5.3
+Prediction of wind farm performance
+We use the predicted values of C∗
+T,LES from the emulators to predict the power output of wind farms under various mesoscale
+atmospheric conditions, following the concept of the two-scale momentum theory. We predict the (farm-averaged) turbine power
+coefficient Cp using C∗
+T,LES predictions from MF-GP-nlow500. We call this prediction of farm performance Cp,model. Firstly, we
+use the C∗
+T,LES prediction from the LOOCV procedure as C∗
+T in equation 1 to calculate β for a given value of wind extractability ζ.
+We substitute this value of β into the expression Cp = β3C∗
+T
+3
+2 C′
+T
+− 1
+2 (which is only valid for actuator discs) to calculate Cp,model.
+We compare the value of Cp,model with the turbine power coefficient recorded in the LES, Cp,LES. The effect of the coarse LES
+
+a)
+b)
+5.0
+20
+(%)
+Overprediction
+Overprediction
+Prediction errors (
+l errors
+2.5
+10
+0.0
+0
+2.5
+Underprediction
+Underprediction
+10
+5.0
+MF-GP-nlow500
+Wake
+Analytical
+5
+6
+7
+8
+9
+10
+model
+model
+Sα/D
+d)
+c)
+5.0
+5.0
+Prediction errors
+2.5
+errors
+2.5
+0.0
+0.0
+Prediction
+2.5
+2.5
+5.0
+1
+5.0
+5
+6
+7
+8
+9
+10
+10
+20
+30
+40
+0
+Sy/D
+ (°)b)=100
+d)  =300
+a)=00
+c)=200
+e)=400
+10
+10
+10
+10
+10
+Dβ
+8
+8
+8
+8
+6
+6
+6
+6 
+6
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+f)=0°
+g)=100
+h)=20°
+i)=300
+j)=40°
+10
+10
+10
+10
+10
+D
+8
+8
+8
+8
+6
+6
+6
+6
+6
+5
+10
+5
+10
+5
+10
+5
+10
+5
+10
+Sα/D
+Sα/D
+Sα/D
+Sα/D
+Sαc/D
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+mg214
+KIRBY et al
+Figure 12 Comparison of Cp predictions with LES results for a realistic range of ζ values.
+resolution on turbine thrust (and hence also ABL response and Cp) has already been corrected 8. The LES was performed with
+periodic horizontal boundary conditions and a fixed momentum supply, i.e., ζ = 0. However, the Cp,LES has also been adjusted for
+a given ζ by scaling the velocity fields assuming Reynolds number independence 8.
+Similarly, the analytical model of C∗
+T can be used to give a theoretical prediction of wind farm performance called Cp,Nishino 8,
+which is given by
+Cp,Nishino =
+64C′
+T
+(4 + C′
+T )3
+�
+���
+−ζ +
+�
+ζ2 + 4
+�
+16C′
+T
+(4+C′
+T )2
+λ
+Cf0 + 1
+�
+(1 + ζ)
+2
+�
+16C′
+T
+(4+C′
+T )2
+λ
+Cf0 + 1
+�
+�
+���
+3
+.
+(12)
+We will compare the accuracy of both Cp,model and Cp,Nishino in predicting Cp,LES.
+Both Cp,model and Cp,LES are shown in figure 12 for a realistic range of wind extractability factors, along with the results from
+Cp,Nishino (equation 12). Cp,Nishino provides an approximate upper limit of farm-averaged Cp as it predicts very well the effects
+of array density and large-scale atmospheric response. The statistical model accurately predicts the effect of turbine layout on farm
+performance which becomes more important with larger ζ values. As ζ increases, there is a larger difference between Cp,LES and
+Cp,Nishino. Also, Cp,model becomes slightly less accurate when ζ increases.
+Table 3 shows the average prediction errors of Cp,model and Cp,Nishino. We quantified the mean absolute error using two
+different reference powers. Using Cp,LES as the reference power, Cp,Nishino had an error of around 5% and the error increases
+
+a)（=0
+b)（=5
+X
+0.04
+A
+0.15
+= 0.03
+P
+C
+C
+4
+0.02
+0.10
+0.01
+5
+10
+15
+20
+5
+10
+15
+20
+>/Cf0
+入/Cf0
+= 10
+d)（
+= 15
+）（
+0.25
+0.30
+△
+K
+△
+0.25
+0.20
+A
+C
+A
+0.20
+0.15
+0.15
+A
+A
+0.10
+1
+5
+10
+15
+20
+5
+10
+15
+20
+>/Cf0
+入/Cfo
+e)（= 20
+f)（
+= 25
+0.35
+必
+0.35
+X
+X
+0.30
+△
+AA
+△
+≥ 0.30
+公
+0.25
+X
+0.25
+又
+0.20
+0.20
+5
+10
+15
+20
+5
+10
+15
+20
+>/Cf0
+入/Cf0
+Cp,LES
+C
+X
+△
+p,NishinoKIRBY et al
+15
+Table 3 Comparison of models for Cp prediction.
+1
+50
+�50
+i=1 |Cp,i − Cp,LES|/Cp,LES
+1
+50
+�50
+i=1 |Cp,i − Cp,LES|/Cp,Betz
+ζ
+Cp,Nishino
+Cp,model
+ζ
+Cp,Nishino
+Cp,model
+0
+2.82%
+2.15%
+0
+0.142%
+0.108%
+5
+4.38%
+1.48%
+5
+0.954%
+0.338%
+10
+5.16%
+1.35%
+10
+1.67%
+0.459%
+15
+5.66%
+1.30%
+15
+2.24%
+0.542%
+20
+6.02%
+1.26%
+20
+2.72%
+0.601%
+25
+6.30%
+1.24%
+25
+3.11%
+0.648%
+with ζ. The mean absolute error of Cp,model was typically less than 1.5% and this decreased slightly as ζ increases (due to the
+reference power Cp,LES increasing). We also use the power of an isolated ideal turbine, Cp,Betz, as a reference power. Cp,Betz is
+calculated using the actuator disc theory with the expression Cp,Betz = 64C′
+T /(4 + C′
+T )3 (note that in this study C′
+T = 1.33 and
+hence Cp,Betz = 0.563). In this case the mean absolute error increased with ζ for both Cp,model and Cp,Nishino. However, the
+average prediction error of Cp,model remained below 0.65%.
+6
+DISCUSSION
+Data-driven modelling of the internal turbine thrust coefficient C∗
+T is a novel approach to modelling turbine-wake interactions. Data-
+driven models of wind farm performance typically focus on predicting the power output, which, however, depends on flow physics
+across a wide range of scales. Current data-driven approaches are either not generalisable to different atmospheric responses, or
+would require a very large set of expensive training data, such as finite-size wind farm LES data. Data-driven models of C∗
+T captures
+the effects of turbine-wake interactions, whilst also being applicable to different atmospheric responses (following the concept of
+the two-scale momentum theory).
+The statistical emulator of C∗
+T developed in this study was able to predict the farm power Cp of Kirby et. al. 8 with an average error
+of less than 0.65%. The high accuracy and very low computational cost of this approach shows the potential of this approach for
+modelling turbine-wake interactions. It has several advantages over traditional approaches using the superposition of wake models.
+Information from turbulence-resolving LES is included which ensures a high accuracy. It will also be more advantageous as wind
+farms become larger because wake models struggle to capture the complex multi-scale flows physics which are important for large
+farms. The statistical model of C∗
+T may therefore allow fast and accurate predictions of wind farm performance.
+All emulators developed in this study gave substantially better predictions of C∗
+T,LES compared to the analytical and wake
+models. Both the mean and maximum prediction errors were reduced by the emulators. The standard GP regression approach had
+a mean prediction error of 1.26% and maximum error of approximately 6%. The accuracy depends on the size of the LES data set
+and could be further decreased with a larger training set. The multi-fidelity GP approach gave more accurate predictions of C∗
+T,LES
+compared to the standard GP regression. This is because non-linear information fusion algorithm has incorporated information from
+many low-fidelity data points to improve the emulator of the high-fidelity (LES) model. This approach has the advantage that, unlike
+the standard GP regression approach, it is not necessary to evaluate the prior mean before making a prediction. Therefore, to predict
+C∗
+T it is only necessary to evaluate the posterior mean of the high-fidelity emulator for a specific turbine layout.
+The shape of the posterior mean in figure 8 gives insights into the physics of turbine-wake interactions. This is because C∗
+T,LES
+is low when a layout has a high degree of turbine-wake interactions. For the turbine operating conditions used, C∗
+T,LES is close to
+0.75 when a layout has a small degree of wake interactions. Figure 8a shows C∗
+T,LES when the wind direction is perfectly aligned
+
+16
+KIRBY et al
+Figure 13 Alignment of turbines for different combinations of Sx, Sy and θ.
+with the rows of turbines (θ = 0). This gives wind farms with a high degree of wake interactions which results in low C∗
+T,LES values.
+For θ = 0o, increasing Sx/D increases C∗
+T because there is a larger streamwise distance between turbines for the wakes to recover.
+When the cross-streamwise spacing (Sy/D) is increased the degree of wake interactions increases, i.e., C∗
+T,LES decreases. This is
+because there is a lower array density which results in a lower turbulence intensity within the farm and hence slower wake recovery.
+Yang 41 found that increasing the cross-streamwise spacing in infinitely-large wind farms increased the power of individual turbines
+and concluded that this was due to reduced wake interactions. However, the increase in turbine power found by Yang 41 may be also
+explained by to a faster farm-averaged wind speed caused by a reduced array density rather than reduced wake interactions.
+When the wind direction θ increases, C∗
+T,LES increases to a maximum of just over 0.75 at θ = 10o (figure 8c). This result agrees
+qualitatively with another study 42 in which it was found that the maximum farm power was produced by an intermediate wind
+direction. When θ increases above 20o regions of low C∗
+T,LES appear diagonally (see figures 8f-j). The regions of low C∗
+T,LES are
+centred on the surfaces given by Sy = 2Sx tan(θ), Sy = Sx tan(θ) and Sy = 0.5Sx tan(θ). These regions correspond to turbines
+being aligned along different axes throughout the farm (see figure 13). There are longer streamwise distance between turbines for
+these arrangements (compared to θ = 0o) and so the C∗
+T,LES values are higher than for θ = 0o.
+The accuracy of the statistical emulators could be further improved in future studies. Both the standard and multi-fidelity GP
+models can be improved by adding more evaluations of C∗
+T,LES. From table 2, the accuracy of the multi-fidelity GP models did not
+improve once we used more than 500 C∗
+T,wake evaluations. This shows that the error in predicting C∗
+T,LES for MF-GP-nlow500 is
+not due to the model of fwake. Instead the error arises from the learnt relationship between fwake and fLES.
+The statistical emulators developed are not applicable to all wind farms because of the limited nature of our data set. A limitation
+of the developed model is that it is only applicable to farms with perfectly aligned layouts. It should also be noted that our model
+was trained on data from simulations of a neutrally stratified boundary layer. Therefore a larger LES data set with an extended
+parameter space would be required to account for the effect of atmospheric stability on wake interactions and the resulting C∗
+T .
+Another limitation of our model is that it assumes all turbines have the same resistance coefficient C′
+T . It is likely that this condition
+can be strictly satisfied only in the fully developed region of a large farm where the wind speed does not change in the streamwise
+or cross-streamwise directions.
+Although we considered only actuator discs in this study for demonstration, the proposed approach using a data-driven model
+of CT ∗ can be applied to power prediction of real turbines as well in future studies. In this study, we calculate Cp,model using the
+expression Cp,model = β3C∗
+T
+3
+2 C′
+T
+− 1
+2 . This assumes that the relationship between C∗
+p and C′
+T is given by C∗
+p = C∗
+T
+3
+2 C′
+T
+− 1
+2 , which
+is only valid for actuator discs. For real turbines, the relationship between C∗
+p and C′
+T can be calculated using BEM theory 43 according
+to the turbine design and operating conditions (noting that the turbine induction factor can still be estimated as a = C′
+T /(4 + C′
+T )).
+Cp,model can then be calculated using equation 5 with β found using equation 1. However, for a data-driven model of C∗
+T to be
+applicable to real turbines, it will be necessary to model the impact of a variable C′
+T rather than assuming a fixed C′
+T value as in this
+study.
+
+b) Su = Sαtan(0)
+= 2Stan(0KIRBY et al
+17
+7
+CONCLUSIONS
+In this study we proposed a new data-driven approach to modelling turbine wake interactions and resulting flow resistance in large
+wind farms. We developed statistical emulators of the farm-internal turbine thrust coefficient C∗
+T,LES as a function of turbine layout
+and wind direction. C∗
+T represents the flow resistance within a wind farm and reflects the characteristics of the turbine-scale flows
+including wake and turbine blockage effects. We developed several emulators using both standard GP regression and multi-fidelity
+GP regression. The standard GP was trained using data from 50 infinitely-large wind farm LES (and using a low-fidelity wake model
+as a prior mean). The multi-fidelity GP was trained using data from both LES and wake model simulations. We estimated the test
+accuracy of the model by performing leave-one-out cross-validation and assessed the error in predicting C∗
+T,LES. All emulators had
+a mean test error of less than 2% for predicting C∗
+T,LES. The multi-fidelity GP gave the best performance with a mean prediction
+error of 0.849% and maximum prediction error of 3.78% with no bias for under or over-prediction. This is low compared to the mean
+error of the wake model (4.60%) and analytical C∗
+T model (5.26%) which both had a bias for overpredicting C∗
+T,LES.
+We used an emulator of C∗
+T,LES to make predictions of wind farm performance under various mesoscale atmospheric conditions
+(characterised by the wind extractability factor ζ) using the two-scale momentum theory 24. Our predictions of farm power produc-
+tion had an average error of less than 1.5% under realistic wind extractability scenarios compared to the LES. When the error in
+power prediction is expressed relative to the power of an isolated ideal turbine the average prediction error is less than 0.7%. We
+also used a previously proposed analytical model of C∗
+T
+25 to predict farm power output with an average error of less than 3.5% (with
+the power of an isolated turbine as the reference power). The analytical model correctly predicts the trends in farm performance
+with array density under different scenarios of large-scale atmospheric response, although it tends to overpredict the power where
+turbine-wake interactions are important. Using statistical emulators of C∗
+T is a new approach to modelling turbine-wake interactions
+and flow resistance within large wind farms. The approach can be extended in future studies by increasing the size of the training
+data set, for example, to account for the effects of C′
+T and atmospheric stability conditions on C∗
+T . The very low computational cost
+and high accuracy of the model could be beneficial for future wind farm optimisation.
+ACKNOWLEDGMENTS
+The first author (AK) acknowledges the NERC-Oxford Doctoral Training Partnership in Environmental Research (NE/S007474/1) for
+funding and training.
+Author contributions
+T.N. derived the theory. A.K. and T.D.D. performed the simulations. F-X.B. provided assistance and guidance for the machine learning
+methodology. A.K. wrote the paper with corrections from T.N., F-X.B and T.D.D.
+Financial disclosure
+None reported.
+Conflict of interest
+The authors report no conflict of interest.
+
+18
+KIRBY et al
+Data availability statement
+The data and code that support the findings of this study are openly available at https://github.com/AndrewKirby2/ctstar_statistical_
+model. This includes the results from the wind farm LES and wake model simulations. The repository also includes the code for the
+results presented in sections 5.1, 5.2 and 5.3.
+Author ORCID
+A. Kirby, https://orcid.org/0000-0001-8389-1619; F-X. Briol https://orcid.org/0000-0002-0181-2559; T. Nishino, https://orcid.
+org/0000-0001-6306-7702.
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+interactions in large wind farms, Wind Energy, xxxx.
+
diff --git a/5NAzT4oBgHgl3EQfu_1f/content/tmp_files/load_file.txt b/5NAzT4oBgHgl3EQfu_1f/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,976 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf,len=975
+page_content='Received:  Added at production  Revised:  Added at production  Accepted:  Added at production  DOI: xxx/xxxx  RESEARCH ARTICLE  Data-driven modelling of turbine wake interactions and flow  resistance in large wind farms  Andrew Kirby*1 | François-Xavier Briol2 | Thomas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Dunstan3 | Takafumi Nishino1  1Department of Engineering Science,  University of Oxford, Oxford, UK  2Department of Statistical Science,  University College London, London, UK  3Informatics Lab, UK MetOffice, Exeter,  UK  Correspondence  Andrew Kirby, Department of  Engineering Science, University of  Oxford, Oxford, OX1 3PJ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Email:  andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='kirby@trinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='uk  Abstract  Turbine wake and local blockage effects are known to alter wind farm power production in  two different ways: (1) by changing the wind speed locally in front of each turbine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' and (2)  by changing the overall flow resistance in the farm and thus the so-called farm blockage  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' To better predict these effects with low computational costs, we develop data-driven  emulators of the ‘local’ or ‘internal’ turbine thrust coefficient C∗  T as a function of turbine  layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We train the model using a multi-fidelity Gaussian Process (GP) regression with a  combination of low (engineering wake model) and high-fidelity (Large-Eddy Simulations)  simulations of farms with different layouts and wind directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A large set of low-fidelity  data speeds up the learning process and the high-fidelity data ensures a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The  trained multi-fidelity GP model is shown to give more accurate predictions of C∗  T compared  to a standard (single-fidelity) GP regression applied only to a limited set of high-fidelity  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We also use the multi-fidelity GP model of C∗  T with the two-scale momentum theory  (Nishino & Dunstan 2020, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' 894, A2) to demonstrate that the model can be  used to give fast and accurate predictions of large wind farm performance under various  mesoscale atmospheric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This new approach could be beneficial for improving  annual energy production (AEP) calculations and farm optimisation in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  KEYWORDS:  Class file;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' LATEX 2ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Wiley NJD  1  INTRODUCTION  The installed capacity of wind energy is projected to increase rapidly in the next decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A major challenge in the optimisation  of wind farm design is the accurate prediction of wind farm performance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Existing wind farm models struggle to make accurate  predictions of wind farm power production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is partly because the ‘global blockage effect’ reduces the velocity upstream of large  farms and hence the energy yield 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' It remains unclear how global blockage should be modelled and this is the subject of a large-scale  field campaign 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Wind farms are typically modelled using engineering ‘wake’ models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' These models predict the velocity deficit in the wakes behind  turbines 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' To account for interactions between multiple turbines, the wake velocity deficits are superposed 6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Simple wake models  can give predictions of wind farm performance with very low computational cost ( 10−3 CPU hours per simulation 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However, wake  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='01699v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='flu-dyn]  4 Jan 2023  2  KIRBY et al  models do not account for the response of the atmospheric boundary layer (ABL) to the wind farm which is likely to be important  for large wind farms 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' It has been found that wake models compare poorly to Large-Eddy Simulations (LES) of large wind farms 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Wind farms are also modelled in numerical weather prediction (NWP) models using farm parameterisation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In these pa-  rameterisations, farms are often modelled as a momentum sink and a source of turbulent kinetic energy 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Turbine-wake interactions  cannot be adequately predicted using these schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A new scheme was proposed 11 which uses a correction factor to model turbine  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' More recently, data-driven approaches have been proposed 12 to model these effects in wind farm parameterisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Data-driven modelling of wind farm flows is a promising new approach 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Data from high-fidelity simulations with complex flow  physics can be used to make predictions with low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Recent studies have applied machine learning techniques  to data from a single turbine or from an existing wind farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The data for these studies are from measurements 14,15,16,17, LES 18 or  Reynolds-Averaged Navier-Stokes (RANS) simulations 19,20,21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A limitation of these approaches is that they are not generalisable to  different turbine layouts unless they rely on wake superposition techniques to model farm flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Another approach is modelling  the effect of turbine layout using geometric parameters 17 or using the layout as a graph input to a neural network 22,23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However,  these alternative approaches may struggle to fully capture the complex two-way interaction with the ABL as it seems impractical to  prepare a data set that covers the entire range of scales involved in wind farm flows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The problem of modelling wind farm flows can be split into ‘internal’ turbine-scale and ‘external’ farm-scale problems 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The  ‘internal’ problem is to determine a ‘local’ or ‘internal’ turbine thrust coefficient, C∗  T , which represents the flow resistance inside a  wind farm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', how the turbine thrust changes with wind speed within the farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Nishino 25 proposed an analytical model for an upper  limit of C∗  T by using an analogy to the classic Betz analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This analytical model is a function of turbine-scale induction factor but  is independent of turbine layout and wind direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Previous studies 24 25 8 showed that C∗  T is usually lower than the limit predicted  by Nishino’s model and can vary significantly with turbine layout due to wake and turbine blockage effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The aim of this study is to develop statistical emulators of C∗  T as a function of turbine layout and wind direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The novelty of  this approach is that we are modelling the effect of turbine-wake interactions on C∗  T rather than turbine power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Both turbine-scale  flows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', wake effects) and farm-scale flows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' farm blockage and mesoscale atmospheric response) affect turbine power within  a farm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore to create an emulator of turbine power, either (1) a very large set of expensive data such as finite-size wind farm  LES is needed which covers a range of large-scale atmospheric conditions or (2) the model would not be generalisable to different  mesoscale atmospheric responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' An emulator of C∗  T is however applicable to different atmospheric responses modelled separately,  following the concept of the two-scale momentum theory 24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  In section 2 we give the definitions of key wind farm parameters in the two-scale momentum theory 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Section 3 summarises  the methodology of the LES and wake model simulations, followed by the machine learning approaches to develop the emulators  in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In section 5 we present the results from the trained emulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' These results are discussed in section 6 and concluding  remarks are given in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  2  TWO-SCALE MOMENTUM THEORY  By considering the conservation of momentum for a control volume with and without a large wind farm over the land or sea surface,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  the following non-dimensional farm momentum (NDFM) equation can be derived 24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  C∗  T  λ  Cf0  β2 + βγ = M  (1)  where β is the farm wind-speed reduction factor defined as β ≡ UF /UF 0 (with UF defined as the average wind speed in the nominal  wind farm-layer of height HF ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' and UF 0 is the farm-layer-averaged speed without the wind farm present);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' λ is the array density  defined as λ ≡ nA/SF (where n is the number of turbines in the farm, A is the rotor swept area and SF is the farm footprint area);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  KIRBY et al  3  C∗  T is the internal turbine thrust coefficient defined as C∗  T ≡ �n  i=1 Ti/ 1  2 ρU2  F nA (where Ti is thrust of turbine i in the farm and ρ  is the air density);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Cf0 is the natural friction coefficient of the surface defined as Cf0 ≡ ⟨τw0⟩/ 1  2 ρU2  F 0 (where τw0 is the bottom  shear stress without the farm present);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' γ is the bottom friction exponent defined as γ ≡ logβ(⟨τw⟩/τw0) (where ⟨τw⟩ is the bottom  shear stress averaged across the farm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' M is the momentum availability factor defined as,  M =  Momentum supplied by the atmosphere to the farm site with turbines  Momentum supplied by the atmosphere to the farm site without turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  (2)  noting that this includes pressure gradient forcing, Coriolis force, net injection of streamwise momentum through top and side  boundaries and time-dependent changes in streamwise velocity 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The height of the farm-layer, HF , is used to define the reference  velocities UF and UF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Equation 1 is valid so long as the same of HF is used for both the internal and external problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' HF is typically  between 2Hhub and 3Hhub 8 (where Hhub is the turbine hub-height) and in this study we use a fixed definition of HF = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5Hhub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Patel 26 used an NWP model to demonstrate that, for most cases, M varied almost linearly with β (for a realistic range of β  between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='8 and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore, M can be approximated by  M = 1 + ζ(1 − β)  (3)  where ζ is the ‘momentum response’ factor or ‘wind extractability’ factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Patel 26 found ζ to be time-dependent and vary between  5 and 25 for a typical offshore site (note that ζ = 0 corresponds to the case where momentum available to the farm site is assumed  to be fixed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', M = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Nishino 25 proposed an analytical model for C∗  T given by,  C∗  T = 4α(1 − α) =  16C′  T  (4 + C′  T )2  (4)  where α is the turbine-scale wind speed reduction factor defined as α ≡ UT /UF (UT is the streamwise velocity averaged over the  rotor swept area) and C′  T ≡ T/ 1  2 ρU2  T A is a turbine resistance coefficient describing the turbine operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  For a given farm configuration at a farm site (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', for given set of C∗  T , λ, Cf0, γ and ζ) the farm wind-speed reduction factor β  can be calculated using equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The (farm-averaged) power coefficient Cp is defined as Cp ≡ �n  i=1 Pi/ 1  2 ρU3  F 0nA (Pi is power  of turbine i in the farm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Using the calculated value of β, Cp can be calculated by using the expression,  Cp = β3C∗  p  (5)  where C∗  p is the (farm-averaged) ‘local’ or ‘internal’ turbine power coefficient defined as C∗  p ≡ �n  i=1 Pi/ 1  2 ρU3  F nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  3  WIND FARM SIMULATIONS  In this study we model wind farms as arrays of actuator discs (or aerodynamically ideal turbines operating below the rated wind  speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is because, in real wind farms, the effects of turbine wake interactions on the farm performance are most significant  when they operate below the rated wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The ‘internal’ thrust coefficient C∗  T is an important wind farm parameter which  includes the effect of turbine interactions (including both wake and local blockage effects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In this study we will be modelling the  effect of turbine layout on C∗  T for aligned turbine layouts with various wind directions and a fixed turbine resistance of C′  T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We chose C′  T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='33 because it leads to a turbine induction factor of 1/4 which is close to a typical value for modern large wind  turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' As such we will be considering  C∗  T = f(Sx, Sy, θ)  (6)  4  KIRBY et al  Figure 1 Design of numerical experiments: a) input parameters, b) maximin design of LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  where Sx is the turbine spacing in the x direction, Sy is the turbine spacing in the y direction and θ is the wind direction relative  to the x direction (see figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However the true function C∗  T cannot be easily evaluated so we will instead investigate C∗  T using  computer codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' One computer code we will use is LES (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='1) to estimate C∗  T  C∗  T,LES = fLES(Sx, Sy, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  (7)  We assume that the function fLES is close to the true function f because of the accuracy of LES to model wind farm flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We will  also use a wake model (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2) to provide cheap approximations of C∗  T according to  C∗  T,wake = fwake(Sx, Sy, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  (8)  Engineering problems are often investigated using complex computer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Evaluating the output of such computer models  for a given input can be very computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore a common objective is to create a cheap statistical model of  the expensive computer model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' this is commonly known as emulation of computer models 27 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In this study we aim to develop a  statistical emulator which can cheaply emulate fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The emulators will only be valid for aligned layouts of wind turbines and for a given turbine resistance (here we use C′  T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We consider the input parameters for a realistic range of turbine spacings 1: Sx ∈ [5D, 10D], Sy ∈ [5D, 10D] and θ ∈ [0o, 45o]  where D is the diameter of the turbine rotor swept area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In this study D is set as 100m and the turbine hub height is also 100m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We  only need to consider wind directions of θ ∈ [0o, 45o] because of symmetry in the aligned turbine layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' If θ is negative than the  turbine layout given by (Sx, Sy, θ) is exactly the same as (Sx, Sy, −θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' When θ > 45o, then (Sx, Sy, θ) and (Sy, Sx, 90o − θ) give  identical layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  In this study we build several emulators to predict fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The models are trained using data from low-fidelity (wake model)  and high fidelity (LES) wind farm simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' One evaluation of C∗  T,wake takes approximately 130 seconds on a single CPU and  C∗  T,LES requires around 400 CPU hours on a supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We use a space filling maximin design 29 30 to select training points  in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The maximin algorithm selects points which maximises the minimum distance to other points and to the  boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This provides a good coverage of the domain which ensures that the emulators can give good predictions across the  whole of the domain 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 1b shows the LES training points in the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  b)  a)  10  40  9  Wind  30  200  10  5  0  6  8  10  S/DKIRBY et al  5  Figure 2 LES a) instantaneous and b) time-averaged flow fields over a periodic turbine array (Sx/D = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='59, Sy/D = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='47 and  θ = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='6o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='1  Large-Eddy Simulations  This study uses the data from 50 high-fidelity (LES) simulations of wind farms published in a previous study 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Here we give a brief  summary of the LES methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The LES models a neutrally stratified atmospheric boundary layer over a periodic array of actuator  discs, which face the wind direction θ and exert uniform thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The resolution is 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5m in the horizontal directions (4 points across  the rotor diameter) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='87m in the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is a coarse horizontal resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' however using a correction factor for the turbine  thrust 32 makes the C∗  T,LES values insensitive to horizontal resolution 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For all simulations the vertical domain size was fixed at  1km and the horizontal extent varied with turbine layout but was at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='14km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The horizontal boundary conditions were periodic  (essentially an infinitely-large wind farm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The bottom boundary used a no-slip condition with the value of eddy viscosity specified  following the Monin-Obukhov similarity theory for a surface roughness length of z0 = 1 × 10−4m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The top boundary had a slip  condition with zero vertical velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The flow was driven by a pressure gradient forcing which was constant and in the direction θ  throughout the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 2 shows the instantaneous and time-averaged hub height velocities from one wind farm LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' See the  original paper 8 for further details of the LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2  Wake model simulations  Wake models are a cheap low-fidelity approach to modelling wind farm aerodynamics compared to expensive high-fidelity LES  simulations 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We use the wake model proposed by Niayafar and Porté-Agel 33 to evaluate C∗  T,wake as a cheap approximation of C∗  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We use the Python package PyWake 34 to implement the wake model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The turbine thrust coefficient CT is needed as an input for  the wake model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We use the value of C∗  T predicted by equation 4 as the value of CT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For the turbine operating conditions used in  this study (C′  T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='33) the wake model has CT equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75 for all turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' To model actuator discs, we consider a hypothetical  turbine which has a constant CT for all wind speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We calculate C∗  T,wake for a single turbine at the back of a large farm (marked X  in figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The farm simulated using the wake model is 10km long in the streamwise direction and 4km long in the cross-streamwise  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The farm size was chosen so that C∗  T no longer varied with increasing farm size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The wake growth parameter is calculated  using k∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='38I +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='004 where I is the local streamwise turbulence velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The local streamwise turbulence intensity is estimated  using the model proposed by Crespo and Hernández 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The background turbulence intensity (TI) is set as a typical value of 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The velocity incident to the turbine is calculated by averaging the velocity across the disc area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We use a 4×3 cartesian grid with  Gaussian quadrature coordinates and weights on the disc to average the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The disc-averaged velocity, UT is then calculated  by multiplying the averaged incident velocity by (1 − a) where a is the turbine induction factor set by the value of C′  T (using the  expression a = C′  T /(4 + C′  T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' To calculate the farm-average velocity, UF , we average the velocity across a volume around the  b)  a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='4  30  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='3  20  20  D  D  n/n  9  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2  10  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='1  0  0  20  0  20  c/ D  c/ D6  KIRBY et al  Figure 3 Example of wind farm layout for wake model simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  single turbine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The volume has dimensions of Sy in the y direction, Sx in the x direction and 250m in the z direction (the height of  the nominal farm layer used in the previous LES study 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' To calculate the average velocity, we discretise the volume into 200 points  in the horizontal directions and 20 points in the vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This was sufficient for the calculation of C∗  T,wake to not vary with further  discretisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 3 shows an example of the farm layout for the wake model simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  4  MACHINE LEARNING METHODOLOGY  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='1  Gaussian Process regression  We will use Gaussian process (GP) regression 36 to build statistical emulators of fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A Gaussian process is a stochastic process  g ∼ GP(m, k) described by a mean function m(v) = E[g(v)] and a covariance function k(v, v′) = E[(g(v) − m(v))(g(v′) − m(v′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  In our case v = (Sx, Sy, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We will use such a stochastic process as a model of fLES, the true mapping from v to C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Each  realisation from this process will therefore be a function which could plausibly represent this mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The mean function represents  the expected output value at an input v = (Sx, Sy, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The covariance function gives the covariance between output values at v and  v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Examples of covariance functions include squared exponential, rational quadratic and periodic functions 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Different covariance  functions will give differently shaped GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For example the squared exponential covariance function will give very smooth GPs  whereas the periodic function will give GPs with a periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Other types of structure, for example symmetry, can also be  encoded in the covariance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore the expected shape (for example smoothness) of the expected relationship and any  properties (for example discontinuities or symmetries) need to be considered when choosing a covariance function for GP regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Let V = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', vn)T be a collection of design points then mV = (m(v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', m(vn))T is the mean vector and kV V = (k(vi, vj))  is the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We will start by positing a GP model with mean m and covariance k (called the ‘prior GP’), then condition  this GP on LES observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' the outcome is a new GP (called the ‘posterior GP’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This gives the posterior distribution g|V, C∗  T,LES ∼  GP(mσ2, kσ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' mσ2 is the posterior mean function given by mσ2(v) = m(v) + kvV (kV V + σ2In×n)−1(C∗  T,LES − mV ) where  kvV = (k(v, v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', k(v, vn)) and In×n is the identity matrix of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The posterior mean function mσ2 is used to make predictions  at v = (Sx, Sy, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The posterior covariance function kσ2 quantifies the uncertainty in our prediction at v = (Sx, Sy, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The  posterior covariance function is given by kσ2(v, v′) = k(v, v′) − kvV (kV V + σ2In×n)−1kV v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Often in GP regression a zero prior mean is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However, using an informative prior mean can improve the accuracy of the  trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' By using a prior mean, many of the trends in fLES can be incorporated into our model prior to making expensive  Volume for  UF calculation  10 km  X  WindKIRBY et al  7  Figure 4 Demonstration of basic GP regression: a) shows the prior mean and covariance function prior to fitting with 3 GPs drawn  from the distribution shown in colour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' b) shows the effect of decreasing the lengthscale hyperparameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' c) the effect of variance  hyperparameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' and d) the posterior mean and covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  evaluations of C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore, after training our model will likely better describe the true relationship between Sx, Sy, θ and  fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In this study, we will use both C∗  T,wake and the analytical model of C∗  T as the prior mean for the standard GP regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For  the wake model prior mean we also vary the specified ambient TI input parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We expect fLES to be a smooth function of input variables Sx, Sy and θ, and to vary more rapidly with θ than Sx or Sy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore  we will use an anisotropic squared-exponential covariance function,  k(v, v′) = σ2  f exp  �  − (Sx − S′  x)2  2l2  1  �  exp  �  −  (Sy − S′  y)2  2l2  2  �  exp  �  − (θ − θ′)2  2l2  3  �  (9)  where σ2  f > 0 is the signal variance hyperparameter and li > 0 is the lengthscale hyperparameter for each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is also  called an ARD (automatic relevance detection) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' If we consider v = v′ then we can see that σ2  f determines the variance of g(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Therefore σ2  f determines the prior uncertainty the model has about the value of g(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' As the lengthscale hyperparameter li gets  smaller then k(v, v′) decreases (for v ̸= v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Equally if li increases then k(v, v′) will also increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A GP with a small li will therefore  vary more rapidly across the parameter space in the ith dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Due to numerical issues associated with the matrix inversion/linear system solve operations in the formulae for the posterior GP,  it is common to add a nugget σ2 > 0 to the kernel matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The hyperparameters σ2  f and li are selected automatically during the  fitting process by maximising the log marginal likelihood 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This approach selects the model which maximises the fit to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Figure 4 shows the impact of the hyperparameters in an example GP regression setting (using the squared exponential covariance  function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The mean function and 95% credible interval (+/-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='96 times the standard deviation) prior to fitting are shown in figure  4a with 3 GPs drawn from the distribution (coloured lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The effect of decreasing the lengthscale hyperparameter li is shown in  figure 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The prior mean and 95% credible interval are unchanged however the example GPs drawn vary more rapidly because of  the shorter lengthscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 4c shows the same setup as figure 4a but with a smaller value of σ2  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The example GPs still vary slowly  but the magnitude of the variations is now smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 4d shows the GPs conditioned on observations with hyperparameters  selected by maximising the log marginal likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  a) α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0, l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  b) α² = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0, l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  2  2  9  0  9  0  2  2  0  2  4  6  0  2  4  6  c) ² = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5, l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  d)o  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='57  2  2  9  0  2  2  0  6  0  2  2  4  4  6  Observations  Mean function  95% credible interval8  KIRBY et al  Figure 5 Demonstration of a) basic GP regression and b) multi-fidelity GP regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In this example f(x) = 1 + sin(6x) for the  high-fidelity data and f(x) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5sin(6x) for the low-fidelity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2  Non-linear multi-fidelity Gaussian Process regression  In many applications there are several computational models available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' These models can have varying accuracies and computational  costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The models which are more computationally expensive typically give more accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The GP regression frame-  work can be extended to combine information from low and high-fidelity models 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This type of modelling uses the low-fidelity  observations to speed up the learning process and the high-fidelity observations to ensure accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In our scenario we will com-  bine evaluations of from a low-fidelity (C∗  T,wake) and a high-fidelity (C∗  T,LES) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Note that for the multi-fidelity models in this  study we set the ambient TI to 10% for the wake model and use a zero prior mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We will keep the number of high-fidelity training  points fixed at 50 and we will vary the number of low-fidelity training points used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We combine information from our high and low-fidelity models using a nonlinear information fusion algorithm 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The framework  is based on the autoregressive multi-fidelity scheme given by:  ghigh(v) = ρ(glow(v)) + δ(v)  (10)  where glow(v) is a model with a GP denoted fwake and ghigh(v) is a model with a GP denoted fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' ρ is a model with a GP  which maps the low-fidelity output to the high-fidelity output and δ(v) is a model with a GP which is a bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The non-linear  multi-fidelity framework can learn non-linear space-dependent correlations between models of different accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' To reduce the  computational cost and complexity of implementation the autoregressive scheme given by equation 10 is simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Firstly, the GP  prior glow(v) is replaced by the GP posterior glow,∗(v) and secondly the GPs ρ and δ are assumed to be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Equation 10  can then be summarised as  ghigh(v) = hhigh(v, glow,∗(v))  (11)  where hhigh is a model with a GP which has both v and glow,∗(v) as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' More details of hhigh and the implementation of the  multi-fidelity framework are given in Perdikaris et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Figure 5 shows an example of how a multi-fidelity GP can outperform a standard GP regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We implement the non-linear  multi-fidelity framework using the ‘emukit’ package 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We first maximise the log marginal likelihood whilst keeping the Gaussian  noise variance fixed at a low value of 1 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The fitting process is then repeated whilst allowing the Gaussian noise variance to  be optimised too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is to prevent a high noise local optima from being selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  a)  b)  3  3  High fidelity  High fidelity  2  2  1  1  9  9  0  0  1  1  Low fidelity  2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  True function  Observations  Posterior mean function  95% credible intervalKIRBY et al  9  5  RESULTS  In this study, we build various statistical emulators of fLES using different techniques and compare the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A summary  of the techniques is shown in the list below:  1 Standard Gaussian Process regression (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='1)  a GP-analytical-prior: Gaussian Process using analytical model (equation 4) prior mean  b GP-wake-TI10-prior: Gaussian Process using wake model (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2) with ambient TI=10% prior mean  c GP-wake-TI1-prior: Gaussian Process using wake model with ambient TI=1% prior mean  d GP-wake-TI5-prior: Gaussian Process using wake model with ambient TI=5% prior mean  e GP-wake-TI15-prior: Gaussian Process using wake model with ambient TI=15% prior mean  2 Non-linear multi-fidelity Gaussian Process regression (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2)  a MF-GP-nlow500: multi-fidelity Gaussian Process using 500 low-fidelity training points  b MF-GP-nlow250: multi-fidelity Gaussian Process using 250 low-fidelity training points  c MF-GP-nlow1000: multi-fidelity Gaussian Process using 1000 low-fidelity training points  The code used to produce the results in this section is available open-access at the following GitHub repository: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  com/AndrewKirby2/ctstar_statistical_model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='1  Performance of standard GP regression  We first assessed the accuracy of the standard GP models (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='1) by performing leave-one-out cross-validation (LOOCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This  is a method of estimating the accuracy of a statistical model when making predictions on data not used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We trained  our model on 49 of the 50 training points and then calculated the prediction accuracy for the single high-fidelity data point which is  excluded from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is then repeated for all data points in turn, and we took the average accuracy as an estimate of  the model test accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The standard GP models were implemented using the ‘GPy’ package 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The standard GP gave accurate predictions of fLES with average errors of less than 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Table 1 shows the accuracy of the stan-  dard GP models compared to the analytical and wake models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We calculated the errors by using the expression |mσ2 −C∗  T,LES|/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75  where mσ2 is the posterior mean function of the emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The reference value for C∗  T of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75 was chosen because this is the pre-  diction from the analytical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Both GP models give similar maximum errors of approximately 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Using the wake model as a prior  mean gave a lower mean absolute error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='26%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The GP models reduced the average prediction error and significantly reduced  the maximum error compared to the wake model and analytical model of C∗  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Table 1 Accuracy of models for C∗  T prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Model  MAE (%)  Maximum error (%)  GP-analytical-prior  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='87  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='09  GP-wake-TI10-prior  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='26  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='11  Analytical model  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='26  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  Wake model (TI=10%)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='60  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='28  10  KIRBY et al  Figure 6 Posterior variance function of GP-wake-TI10-prior model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Figure 7 Sensitivity of fitted GP models to the ambient TI chosen for wake model prior means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The model GP-wake-TI10-prior has a high degree of confidence when making predictions in regions of the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Figure 6 shows the square root of the posterior covariance function kσ2, which quantities the uncertainty of the emulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The  uncertainty is uniform throughout the parameter space with regions of slightly higher uncertainty at θ = 0o and 45o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We also assessed the sensitivity of the model accuracy to the ambient TI used in the wake model prior mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 7 shows  the impact of ambient TI on the wake model prior mean and the fitted GP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Increasing the ambient TI increased the value of  C∗  T,wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is because of the enhanced wake recovery behind wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Increasing the ambient TI in the wake model results  in C∗  T,wake overpredicting C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The MAE from the LOOCV procedure for each fitted GP is shown in the bottom right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  b)=5°  c)=100  d)  =15°  a)=00  e)=200  10  10  10  10  10  Dβ  8  8  8  8  6  6  6  6  6  5  10  5  10  5  10  5  10  5  10  f) =250  g)  =300  h)=35°  i)=40°  j)=45°  10  10  10  10  10  D  8  8  8  8  6  6  6  6  6  5  10  5  10  5  10  5  10  5  10  Sα/D  Sα/D  Sα/D  Sα/D  Sαc/D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='030  Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2a) Ambient TI=1%  b) Ambient TI=5%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='02%  MAE=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='80  CT,LES  C*,LES  c) Ambient TI=10%  d) Ambient TI=15%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='6  GP-wake-TI10-prior  GP-wake-TI15-prior  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='80  Wake model prior mean  Standard GP modelKIRBY et al  11  The fitted GPs became more accurate when the wake model ambient TI was increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Increasing the ambient TI for the wake  model causes the wakes to recover faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The wakes become shorter in the streamwise direction and wider in the spanwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  As such, C∗  T,wake becomes less sensitive to the turbine layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' When an ambient TI of 1% and 5% is used for the wake model,  C∗  T,wake is more sensitive to turbine layout than C∗  T,LES (figures 7a and 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' When the ambient TI is increased to 10% and above,  the relationship between C∗  T,wake and C∗  T,LES becomes simpler (figures 7c and 7d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This seems to explain why the fitted GPs  become more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2  Performance of non-linear multi-fidelity GP regression  We then assessed the accuracy of the multi-fidelity GP models (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' All models used the 50 high-fidelity (C∗  T,LES) training  points and a varying number of low-fidelity (C∗  T,wake) training points (using an ambient TI of 10% for C∗  T,wake).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The results from  LOOCV are shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For the LOOCV we train our model on 49 out of the 50 high-fidelity data points and all low-fidelity  data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Then we average the error in predicting the high-fidelity data point left of the training set and repeat this in turn for  data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Increasing the number of low-fidelity training points from 250 to 500 reduced the mean and maximum error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However,  increasing this to 1000 low-fidelity training points did not increase accuracy and increased the fitting and prediction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is  because the number of high-fidelity training points is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' There is a threshold where the model of the relationship between fLES  and fwake, denoted ρ, limits the final accuracy of the emulator of fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The posterior mean mσ2 of glow(v) is an emulator of fwake and ghigh(v) is an emulator of fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 8 gives the predictions  from the posterior mean of ghigh(v) (for MF-GP-nlow500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The lowest mσ2 values were for a wind direction of θ = 0o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' mσ2  Table 2 Performance of the multi-fidelity Gaussian Process models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Model  MAE (%)  Maximum error (%)  Training time (s)  Prediction time (s)  MF-GP-nlow250  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='46  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='12  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='00157  MF-GP-nlow500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='828  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='00167  MF-GP-nlow1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='866  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='55  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='00236  Figure 8 Posterior mean function for ghigh(v) of MF-GP-nlow500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  b)=50  d)  =15°  a)=00  c)=100  e)=200  10  10  10  10  10  Dβ  8  8  8  8  6  6  6  6  6  5  10  5  10  5  10  5  10  5  10  f) =250  g)  =300  h) =35°  i)=40°  j）)θ =45°  10  10  10  10  10  D  8  8  8  8  6  6  6  6  6  5  10  10  5  10  5  10  5  5  10  Sα/D  Sα/D  Sα/D  Sα/D  Sαc/D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='80  mg212  KIRBY et al  Figure 9 Posterior variance function for ghigh(v) of MF-GP-nlow500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  increased rapidly with θ reaching a maximum of slightly over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75 at θ = 10o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For large values of θ (above θ = 25o) there were  local minima in mσ2 which appear in figure 8 as diagonal strips of low mσ2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The main diagonal strip occurs along the line  of Sy = Sx tan(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' There are two smaller strips either side of with positions given by Sy = 2 tan(θ) and Sy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5 tan(θ) (this is  discussed further in section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The uncertainty the model MF-GP-nlow500 has in predicting fLES is shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The model uncertainty is uni-  form throughout the parameter space with slightly higher values at θ = 0o and 45o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Compared to the posterior variance of  GP-wake-TI10-prior (shown in figure 6) the uncertainty is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' By incorporating information from C∗  T,wake, the multi-fidelity GP  model has more confidence about predicting fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The prediction errors from the LOOCV (for MF-GP-nlow500) are shown in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The box plot of prediction errors in figure  10a shows that this model had no significant bias whereas both the wake and analytical models systemically overestimated C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Figures 10b-d show that for the statistical model there appears to be no part of the parameter space which had larger errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The multi-fidelity approach used in this study builds a statistical model of both the low-fidelity (fwake) and high-fidelity (fLES)  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We can use the posterior means of glow(v) and ghigh(v) to see the differences between the wake model and LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The  posterior mean for both models are shown in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For the wake model the change in mσ2 with θ is greater than for the LES  (especially between θ = 0o and 10o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For larger values of θ, there is a larger difference in mσ2 between waked and unwaked layouts  for the low-fidelity model compared to the high-fidelity one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This suggests than the wake model is more sensitive to changes in wind  directions than the LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  a) =00  b) =5°  c)=100  d)θ=15°  e)=20°  10  10  10  10  10  D∞  8  8  8  8  6  6  6  6  6  5  10  5  10  5  10  5  10  5  10  f）θ =25°  g) =300  h) =350  i)=40°  j)θ=45°  10  10  10  10  10  Dβ  8  8  8  8  6  6  6  6  6  5  10  5  10  5  10  5  10  5  10  Sα/D  Sα/D  Sα/D  Sα/D  S/D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='030  Vkg?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='KIRBY et al  13  Figure 10 Comparison of LOOCV prediction errors (%) for different models a) and LOOCV prediction error (%) of MF-GP-nlow500  against input parameters b) Sx/D, c) Sy/D and d) θ(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Note that for the box plot in a) the orange line is the median LOOCV error  and the box is the interquartile range of LOOCV error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Figure 11 Posterior mean function of MF-GP-nlow500 for different values of θ for a) to e) ghigh(v) and f) to j) glow(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='3  Prediction of wind farm performance  We use the predicted values of C∗  T,LES from the emulators to predict the power output of wind farms under various mesoscale  atmospheric conditions, following the concept of the two-scale momentum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We predict the (farm-averaged) turbine power  coefficient Cp using C∗  T,LES predictions from MF-GP-nlow500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We call this prediction of farm performance Cp,model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Firstly, we  use the C∗  T,LES prediction from the LOOCV procedure as C∗  T in equation 1 to calculate β for a given value of wind extractability ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We substitute this value of β into the expression Cp = β3C∗  T  3  2 C′  T  − 1  2 (which is only valid for actuator discs) to calculate Cp,model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We compare the value of Cp,model with the turbine power coefficient recorded in the LES, Cp,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The effect of the coarse LES  a)  b)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  20  (%)  Overprediction  Overprediction  Prediction errors (  l errors  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  Underprediction  Underprediction  10  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  MF-GP-nlow500  Wake  Analytical  5  6  7  8  9  10  model  model  Sα/D  d)  c)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  Prediction errors  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  errors  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='0  Prediction  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='0  5  6  7  8  9  10  10  20  30  40  0  Sy/D  (°)b)=100  d)  =300  a)=00  c)=200  e)=400  10  10  10  10  10  Dβ  8  8  8  8  6  6  6  6  6  5  10  5  10  5  10  5  10  5  10  f)=0°  g)=100  h)=20°  i)=300  j)=40°  10  10  10  10  10  D  8  8  8  8  6  6  6  6  6  5  10  5  10  5  10  5  10  5  10  Sα/D  Sα/D  Sα/D  Sα/D  Sαc/D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='80  mg214  KIRBY et al  Figure 12 Comparison of Cp predictions with LES results for a realistic range of ζ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  resolution on turbine thrust (and hence also ABL response and Cp) has already been corrected 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The LES was performed with  periodic horizontal boundary conditions and a fixed momentum supply, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However, the Cp,LES has also been adjusted for  a given ζ by scaling the velocity fields assuming Reynolds number independence 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Similarly, the analytical model of C∗  T can be used to give a theoretical prediction of wind farm performance called Cp,Nishino 8,  which is given by  Cp,Nishino =  64C′  T  (4 + C′  T )3  �  ���  −ζ +  �  ζ2 + 4  �  16C′  T  (4+C′  T )2  λ  Cf0 + 1  �  (1 + ζ)  2  �  16C′  T  (4+C′  T )2  λ  Cf0 + 1  �  �  ���  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  (12)  We will compare the accuracy of both Cp,model and Cp,Nishino in predicting Cp,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Both Cp,model and Cp,LES are shown in figure 12 for a realistic range of wind extractability factors, along with the results from  Cp,Nishino (equation 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Cp,Nishino provides an approximate upper limit of farm-averaged Cp as it predicts very well the effects  of array density and large-scale atmospheric response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The statistical model accurately predicts the effect of turbine layout on farm  performance which becomes more important with larger ζ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' As ζ increases, there is a larger difference between Cp,LES and  Cp,Nishino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Also, Cp,model becomes slightly less accurate when ζ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Table 3 shows the average prediction errors of Cp,model and Cp,Nishino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We quantified the mean absolute error using two  different reference powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Using Cp,LES as the reference power, Cp,Nishino had an error of around 5% and the error increases  a)（=0  b)（=5  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='04  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='15  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='03  P  C  C  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='01  5  10  15  20  5  10  15  20  >/Cf0  入/Cf0  = 10  d)（  = 15  ）（  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='30  △  K  △  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='30  △  AA  △  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='30  公  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='20  5  10  15  20  5  10  15  20  >/Cf0  入/Cf0  Cp,LES  C  X  △  p,NishinoKIRBY et al  15  Table 3 Comparison of models for Cp prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  1  50  �50  i=1 |Cp,i − Cp,LES|/Cp,LES  1  50  �50  i=1 |Cp,i − Cp,LES|/Cp,Betz  ζ  Cp,Nishino  Cp,model  ζ  Cp,Nishino  Cp,model  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='30%  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='24%  25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='11%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='648%  with ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The mean absolute error of Cp,model was typically less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5% and this decreased slightly as ζ increases (due to the  reference power Cp,LES increasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We also use the power of an isolated ideal turbine, Cp,Betz, as a reference power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Cp,Betz is  calculated using the actuator disc theory with the expression Cp,Betz = 64C′  T /(4 + C′  T )3 (note that in this study C′  T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='33 and  hence Cp,Betz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='563).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In this case the mean absolute error increased with ζ for both Cp,model and Cp,Nishino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However, the  average prediction error of Cp,model remained below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  6  DISCUSSION  Data-driven modelling of the internal turbine thrust coefficient C∗  T is a novel approach to modelling turbine-wake interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Data-  driven models of wind farm performance typically focus on predicting the power output, which, however, depends on flow physics  across a wide range of scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Current data-driven approaches are either not generalisable to different atmospheric responses, or  would require a very large set of expensive training data, such as finite-size wind farm LES data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Data-driven models of C∗  T captures  the effects of turbine-wake interactions, whilst also being applicable to different atmospheric responses (following the concept of  the two-scale momentum theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The statistical emulator of C∗  T developed in this study was able to predict the farm power Cp of Kirby et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' 8 with an average error  of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='65%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The high accuracy and very low computational cost of this approach shows the potential of this approach for  modelling turbine-wake interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' It has several advantages over traditional approaches using the superposition of wake models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Information from turbulence-resolving LES is included which ensures a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' It will also be more advantageous as wind  farms become larger because wake models struggle to capture the complex multi-scale flows physics which are important for large  farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The statistical model of C∗  T may therefore allow fast and accurate predictions of wind farm performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  All emulators developed in this study gave substantially better predictions of C∗  T,LES compared to the analytical and wake  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Both the mean and maximum prediction errors were reduced by the emulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The standard GP regression approach had  a mean prediction error of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='26% and maximum error of approximately 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The accuracy depends on the size of the LES data set  and could be further decreased with a larger training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The multi-fidelity GP approach gave more accurate predictions of C∗  T,LES  compared to the standard GP regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is because non-linear information fusion algorithm has incorporated information from  many low-fidelity data points to improve the emulator of the high-fidelity (LES) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This approach has the advantage that, unlike  the standard GP regression approach, it is not necessary to evaluate the prior mean before making a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore, to predict  C∗  T it is only necessary to evaluate the posterior mean of the high-fidelity emulator for a specific turbine layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The shape of the posterior mean in figure 8 gives insights into the physics of turbine-wake interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is because C∗  T,LES  is low when a layout has a high degree of turbine-wake interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For the turbine operating conditions used, C∗  T,LES is close to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75 when a layout has a small degree of wake interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Figure 8a shows C∗  T,LES when the wind direction is perfectly aligned  16  KIRBY et al  Figure 13 Alignment of turbines for different combinations of Sx, Sy and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  with the rows of turbines (θ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This gives wind farms with a high degree of wake interactions which results in low C∗  T,LES values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  For θ = 0o, increasing Sx/D increases C∗  T because there is a larger streamwise distance between turbines for the wakes to recover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  When the cross-streamwise spacing (Sy/D) is increased the degree of wake interactions increases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=', C∗  T,LES decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is  because there is a lower array density which results in a lower turbulence intensity within the farm and hence slower wake recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Yang 41 found that increasing the cross-streamwise spacing in infinitely-large wind farms increased the power of individual turbines  and concluded that this was due to reduced wake interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However, the increase in turbine power found by Yang 41 may be also  explained by to a faster farm-averaged wind speed caused by a reduced array density rather than reduced wake interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  When the wind direction θ increases, C∗  T,LES increases to a maximum of just over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='75 at θ = 10o (figure 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This result agrees  qualitatively with another study 42 in which it was found that the maximum farm power was produced by an intermediate wind  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' When θ increases above 20o regions of low C∗  T,LES appear diagonally (see figures 8f-j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The regions of low C∗  T,LES are  centred on the surfaces given by Sy = 2Sx tan(θ), Sy = Sx tan(θ) and Sy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5Sx tan(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' These regions correspond to turbines  being aligned along different axes throughout the farm (see figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' There are longer streamwise distance between turbines for  these arrangements (compared to θ = 0o) and so the C∗  T,LES values are higher than for θ = 0o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The accuracy of the statistical emulators could be further improved in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Both the standard and multi-fidelity GP  models can be improved by adding more evaluations of C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' From table 2, the accuracy of the multi-fidelity GP models did not  improve once we used more than 500 C∗  T,wake evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This shows that the error in predicting C∗  T,LES for MF-GP-nlow500 is  not due to the model of fwake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Instead the error arises from the learnt relationship between fwake and fLES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  The statistical emulators developed are not applicable to all wind farms because of the limited nature of our data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A limitation  of the developed model is that it is only applicable to farms with perfectly aligned layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' It should also be noted that our model  was trained on data from simulations of a neutrally stratified boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Therefore a larger LES data set with an extended  parameter space would be required to account for the effect of atmospheric stability on wake interactions and the resulting C∗  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Another limitation of our model is that it assumes all turbines have the same resistance coefficient C′  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' It is likely that this condition  can be strictly satisfied only in the fully developed region of a large farm where the wind speed does not change in the streamwise  or cross-streamwise directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Although we considered only actuator discs in this study for demonstration, the proposed approach using a data-driven model  of CT ∗ can be applied to power prediction of real turbines as well in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' In this study, we calculate Cp,model using the  expression Cp,model = β3C∗  T  3  2 C′  T  − 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This assumes that the relationship between C∗  p and C′  T is given by C∗  p = C∗  T  3  2 C′  T  − 1  2 , which  is only valid for actuator discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' For real turbines, the relationship between C∗  p and C′  T can be calculated using BEM theory 43 according  to the turbine design and operating conditions (noting that the turbine induction factor can still be estimated as a = C′  T /(4 + C′  T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Cp,model can then be calculated using equation 5 with β found using equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' However, for a data-driven model of C∗  T to be  applicable to real turbines, it will be necessary to model the impact of a variable C′  T rather than assuming a fixed C′  T value as in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  b) Su = Sαtan(0)  = 2Stan(0KIRBY et al  17  7  CONCLUSIONS  In this study we proposed a new data-driven approach to modelling turbine wake interactions and resulting flow resistance in large  wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We developed statistical emulators of the farm-internal turbine thrust coefficient C∗  T,LES as a function of turbine layout  and wind direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' C∗  T represents the flow resistance within a wind farm and reflects the characteristics of the turbine-scale flows  including wake and turbine blockage effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We developed several emulators using both standard GP regression and multi-fidelity  GP regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The standard GP was trained using data from 50 infinitely-large wind farm LES (and using a low-fidelity wake model  as a prior mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The multi-fidelity GP was trained using data from both LES and wake model simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We estimated the test  accuracy of the model by performing leave-one-out cross-validation and assessed the error in predicting C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' All emulators had  a mean test error of less than 2% for predicting C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The multi-fidelity GP gave the best performance with a mean prediction  error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='849% and maximum prediction error of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='78% with no bias for under or over-prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' This is low compared to the mean  error of the wake model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='60%) and analytical C∗  T model (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='26%) which both had a bias for overpredicting C∗  T,LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  We used an emulator of C∗  T,LES to make predictions of wind farm performance under various mesoscale atmospheric conditions  (characterised by the wind extractability factor ζ) using the two-scale momentum theory 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Our predictions of farm power produc-  tion had an average error of less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5% under realistic wind extractability scenarios compared to the LES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' When the error in  power prediction is expressed relative to the power of an isolated ideal turbine the average prediction error is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' We  also used a previously proposed analytical model of C∗  T  25 to predict farm power output with an average error of less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='5% (with  the power of an isolated turbine as the reference power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The analytical model correctly predicts the trends in farm performance  with array density under different scenarios of large-scale atmospheric response, although it tends to overpredict the power where  turbine-wake interactions are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Using statistical emulators of C∗  T is a new approach to modelling turbine-wake interactions  and flow resistance within large wind farms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The approach can be extended in future studies by increasing the size of the training  data set, for example, to account for the effects of C′  T and atmospheric stability conditions on C∗  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The very low computational cost  and high accuracy of the model could be beneficial for future wind farm optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The first author (AK) acknowledges the NERC-Oxford Doctoral Training Partnership in Environmental Research (NE/S007474/1) for  funding and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Author contributions  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' derived the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' performed the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' F-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content=' provided assistance and guidance for the machine learning  methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content=' wrote the paper with corrections from T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Financial disclosure  None reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  Conflict of interest  The authors report no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  18  KIRBY et al  Data availability statement  The data and code that support the findings of this study are openly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content=' This includes the results from the wind farm LES and wake model simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' The repository also includes the code for the  results presented in sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content=' F-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Briol https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content=' Nishino, https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
+page_content='  org/0000-0001-6306-7702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
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+page_content=' (2022), Data-driven modelling of wind turbine wake  interactions in large wind farms, Wind Energy, xxxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NAzT4oBgHgl3EQfu_1f/content/2301.01699v1.pdf'}
diff --git a/5dE2T4oBgHgl3EQfOgbi/content/tmp_files/2301.03750v1.pdf.txt b/5dE2T4oBgHgl3EQfOgbi/content/tmp_files/2301.03750v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7e7551b678700885812ec887d31b050296a054f2
--- /dev/null
+++ b/5dE2T4oBgHgl3EQfOgbi/content/tmp_files/2301.03750v1.pdf.txt
@@ -0,0 +1,5252 @@
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE
+INTEGRALS
+ETHAN SUSSMAN
+Abstract. We discuss the meromorphic continuation of certain hypergeometric integrals modeled
+on the Selberg integral, including the 3-point and 4-point functions of BPZ’s minimal models of
+2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the “Coulomb gas formalism”).
+This is accomplished via a geometric analysis of the singularities of the integrands. In the case
+that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call
+“DF-symmetric,” we show that a number of apparent singularities are removable, as required for the
+construction of the minimal models via these methods.
+Contents
+1.
+Introduction
+1
+2.
+Associahedra
+10
+3.
+Meromorphic continuation
+26
+4.
+Removing singularities
+40
+Appendix A.
+The N = 2 case
+48
+Appendix B.
+Explicit coordinates on [0, 1)N
+tb
+52
+References
+55
+1. Introduction
+Let
+△N = {(x1, . . . , xN) ∈ [0, 1]N : x1 ≤ · · · ≤ xN}
+(1)
+denote the standard N-simplex, which we consider as a subset of CN. We study in this note
+Selberg-like integrals, by which we mean definite integrals of the form
+SN[F](α, β, γ) =
+�
+△N
+� N
+�
+j=1
+xαj
+j (1 − xj)βj
+��
+�
+1≤j<k≤N
+(xk − xj)2γj,k
+�
+F(x1, . . . , xN) dx1 · · · dxN, (2)
+for N ∈ N+, F ∈ C∞(△N), and α = {αj}N
+j=1, β = {βj}N
+j=1, γ = {γj,k = γk,j}1≤j<k≤N ⊂ C such
+that the integrand above is absolutely integrable on △N. Integrals of this form are relevant to an
+array of topics in mathematical physics [FW08]. Here, we discuss a particular application to the
+Coulomb gas formalism (a.k.a. “free field realization,” “Feigin–Fuchs representation,” etcetera) of 2D
+CFT [DF84][DF85a][DF85b][FS89][PM97, Chp. 9][FW08]. This approach of Dotsenko–Fateev to the
+construction of the “minimal models” of Belavin–Polyakov–Zamolodchikov (BPZ) [BPZ84] has been
+the subject of substantial interest, but it appears that it has not yet been placed on entirely rigorous
+mathematical footing. The construction in [FS89][FS92] of the 3-point coefficients of the (1, s)- and
+(r, 1)-primary fields and their descendants in the minimal models is satisfactorily rigorous, but it
+has remained an open problem to handle the rest of the primary fields at a similarly satisfactory
+degree of rigor. From our perspective, the issue is an insufficient treatment of the meromorphic
+Date: January 8th, 2023 (Last update).
+2020 Mathematics Subject Classification. Primary 32A20; Secondary 33C60, 33C90, 81T40.
+1
+arXiv:2301.03750v1  [math-ph]  10 Jan 2023
+
+2
+ETHAN SUSSMAN
+continuation of Selberg-like integrals, which are instead treated somewhat formally in the original
+works.
+The issue is that Dotsenko & Fateev (DF) take some of the γ’s to be −1 — see e.g. [DF85a,
+Appendix A][FS92, p. 27][FW08, §2] — and then the integrand above is, say for F = 1, no longer
+integrable over the integral’s domain. As a consequence, the integrals in [DF85a, Appendix A] are
+formal. One way of making sense of them is via meromorphic continuation in the exponents of the
+integrand. Dotsenko and Fateev suggest this, but they do not prove that a suitable meromorphic
+continuation exists, nor do they discuss the singularities of the extension in sufficient detail to
+justify their manipulations. A construction of Kanie–Tsuchiya [KT86a][KT86b] rediscovered by
+Mimachi-Yoshida [MY02][MY03][Yos03] yields the existence of some meromorphic continuation
+defined for almost all values of the exponents. But, this extension is not quite sufficient for our
+purposes: it has removable singularities that, while removable, are nontrivial to actually prove
+removable. In particular, the Kanie–Tsuchiya construction has an apparent (but isolated, in a
+suitable sense) singularity at γ = −1 (see [KT86b, §5, above Thm. 5]), along with at a few other
+problematic affine hyperplanes in the space of possible parameters.
+Most of the rigorous work on the analysis of integrals of Dotsenko–Fateev type — see e.g.
+[FK15a][FK15b][FK15c][FK15d][LV19] for some recent work — focuses on screened multipoint
+correlation functions with at most one screening charge screening per insertion point. Such integrals
+are related to the N = 1 case of SN(α, β, γ). Not much has been done about the N > 1 case.
+Moreover, while a fair amount of work has gone into the study of general hypergeometric integrals
+associated to hyperplane arrangements — the literature on this topic is large, so we just cite
+[Var95][AK11] — it does not seem possible to deduce the specific, concrete results below from results
+in the current literature.
+We will identify indexed collections of complex numbers (and tuples thereof) with column vectors.
+For example, we identify γ with an element of CN(N−1)/2 and
+(α, β, γ) ∈ CN × CN × CN(N−1)/2
+(3)
+with an element of C2N+N(N−1)/2. Similar identifications will be made throughout the rest of the
+paper without further comment. Let
+ΩN =
+�
+(α, β, γ) ∈ C2N+N(N−1)/2 :
+N
+�
+j=1
+xαj
+j (1 − xj)βj
+�
+1≤j<k≤N
+(xk − xj)2γj,k ∈ L1(△N)
+�
+(4)
+denote the (open, nonempty) subset of C2N+N(N−1)/2 consisting of the (α, β, γ) ∈ C2N+N(N−1)/2
+for which the integrand in eq. (2) is absolutely integrable on △N. We begin with SN[F] defined as
+a function SN[F] : ΩN → C. It can be checked – see §2 – that, letting
+αj,∗(α, β, γ) =
+j
+�
+j0=1
+αj0 + 2
+�
+j0,k∈{1,...,N}
+1≤j0<k≤j
+γj0,k,
+βj,∗(α, β, γ) =
+N
+�
+j0=N−j+1
+βj0 + 2
+�
+j0,k∈{1,...,N}
+N−j+1≤j0<k≤N
+γj0,k
+(5)
+for each j ∈ {1, . . . , N}, and letting
+γj,k,∗(α, β, γ) = 2
+�
+j0,k0∈{1,...,N}
+j≤j0<k0≤k
+γj0,k0
+(6)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+3
+for each pair of j, k ∈ {1, . . . , N} with j < k,
+ΩN =
+� N
+�
+j=1
+{ℜαj,∗ > −j}
+�
+∩
+� N
+�
+j=1
+{ℜβj,∗ > −j}
+�
+∩
+�
+�
+1≤j<k≤N
+{ℜγj,k,∗ > −(k − j)}
+�
+.
+(7)
+So, ΩN is nonempty, open, and convex (in particular, connected) and contains all (α, β, γ) ∈
+C2N+N(N−1)/2 such that the real parts of the components of α, β, γ are sufficiently large.
+To simplify the formula above, let γ0,k,∗ = αk,∗ and γN+1−j,N+1,∗ = βj,∗. Then
+ΩN =
+�
+0≤j<k≤N+1
+{(α, β, γ) ∈ C2N+N(N−1)/2 : ℜγj,k,∗ > −(k − j)}.
+(8)
+Our first goal is to prove that SN[F] can be analytically continued to a subset
+˙ΩN ⊆ C2N+N(N−1)/2
+(9)
+having full measure in C2N+N(N−1)/2.
+In order to describe precisely the structure of the singularity at C2N+N(N−1)/2\ ˙ΩN, we introduce
+some terminology. Let T(N) denote the collection of maximal families I of consecutive subsets
+I ⊊ {0, . . . , N + 1} such that
+• 2 ≤ |I| ≤ N + 1 for all I ∈ I and
+• if I, I′ ∈ I satisfy I ∩ I′ ̸= ∅, then either I ⊆ I′ or I′ ⊆ I.
+“T” stands either for “tree” in “full binary trees” or “Tamari” in Tamari lattice [Tam62][Gey94], and
+the elements of T(N) can be thought of as specifying the valid ways of adding a maximal number of
+nonredundant parentheses to a string of N + 2 identical characters. There are #T(N) = CN+1 such
+ways, where CN+1 is the (N + 1)st Catalan number. To each I ∈ I, we associate the facet
+fI = {(x1, . . . , xN) ∈ △N : xj = xk for all j, k ∈ I}
+(10)
+of △N, where x0 = 0 and xN+1 = 1. Let oI ∈ N denote the order of vanishing of F at fI. (So,
+oI = 0 unless F is vanishing identically at fI.)
+Theorem 1.1. There exist entire functions SN;reg,I[F] : C2N+N(N−1)/2 → C associated to the
+I ∈ T(N) such that
+SN[F](α, β, γ) =
+�
+I∈T(N)
+� �
+I∈I
+Γ(oI + |I| − 1 + γmin I,max I,∗)
+�
+SN;reg,I[F](α, β, γ)
+(11)
+for all (α, β, γ) ∈ ΩN.
+■
+Here, Γ : C\{−n : n ∈ N} → C is the Gamma function. As a consequence of the theorem, there
+exists an entire function SN;reg[F] : C2N+N(N−1)/2 → C such that
+SN[F](α, β, γ) =
+�
+�
+0≤j<k≤N+1
+Γ(k − j + γj,k,∗)
+�
+SN;reg[F](α, β, γ)
+(12)
+for all (α, β, γ) ∈ ΩN.
+Corollary 1.1.1. The function SN[F] : ΩN → C admits an analytic continuation ˙SN[F] : ˙ΩN → C
+to the domain
+˙ΩN = C2N+N(N−1)/2
+α,β,γ
+��� N
+�
+j=1
+{αj,∗ ∈ Z≤−j−δj}
+�
+∪
+� N
+�
+j=1
+{βj,∗ ∈ Z≤−j−
+δ
+j}
+�
+∪
+�
+�
+1≤j<k≤N
+{γj,k,∗ ∈ Z≤−(k−j)−dj,k}
+��
+,
+(13)
+where Z≤n = {m ∈ Z : m ≤ n} and δj = δj[F] = o{0,...,j},
+δ
+j =
+δ
+j[F] = o{N−j+1,...,N+1}, and
+dj,k = dj,k[F] = o{j,...,k}.
+■□
+
+4
+ETHAN SUSSMAN
+The set ˙ΩN contains all elements of C2N+N(N−1)/2 lying outside of a locally finite arrangement
+of affine hyperplanes.
+Consider F ∈ C[x1, . . . , xN]. Letting [F]d1,...,dN denote the coefficient of xd1
+1 · · · xdN
+N
+in F, and
+letting refl : (x1, . . . , xN) �→ (1 − x1, . . . , 1 − xN), we have
+δj[F] = min{d1 + · · · + dj : [F]d1,...,dj,dj+1,...,dN ̸= 0 for some dj+1, . . . , dN ∈ N},
+(14)
+δ
+j[F] = min{dN + · · · + dN+1−j : [F ◦ refl]d1,...,dN−j,dN+1−j,...,dN ̸= 0 for some d1, . . . , dN−j ∈ N}.
+(15)
+Example. The simplest case is when N = 1 and F = 1 identically, when the Selberg integral is given
+by
+S1(α, β, γ) = B(α, β) =
+� 1
+0
+xα(1 − x)β dx = Γ(1 + α)Γ(1 + β)
+Γ(2 + α + β)
+,
+(16)
+defined initially for ℜα, ℜβ > −1 via the definite integral and then extended meromorphically via
+the formula on the right-hand side above (or via another method). This is Euler’s β-function. One
+method of meromorphic continuation involves the “Pochhammer contour”
+b−1a−1ba ∈ π1(C\{0, 1}),
+(17)
+where a, b are the generators of π1(C\{0, 1}) corresponding to one (say, counterclockwise) circuit
+around each of 0, 1 respectively.
+0
+1
+Figure 1. The Pochhammer contour in C\{0, 1}, up to homotopy.
+Then, b−1a−1ba can be lifted to a closed contour p in the cover M of C\{0, 1} corresponding to
+the commutator subgroup of π1(C\{0, 1}). Then, choosing the basepoint of p appropriately,
+B(α, β) =
+1
+1 − e−2πiα
+1
+1 − e−2πiβ
+�
+p
+xα(1 − x)β dx,
+(18)
+where we are now considering xα(1 − x)β as an analytic function on M. The theorem above tells us
+that there exist entire S1;reg,(••)•, S1;reg,•(••) such that
+B(α, β) = Γ(1 + α)S1;reg,(••)•(α, β) + Γ(1 + β)S1;reg,•(••)(α, β).
+(19)
+This splitting is not so obvious from the formula B(α, β) = Γ(1 + α)Γ(1 + β)/Γ(2 + α + β).
+■
+Example. Now consider the case when N = 2 and F = 1. It can be computed that the Selberg-like
+integral is then
+S2(α, β, γ) = Γ(1 + α1)Γ(1 + β2)Γ(2 + 2γ1,2 + α1 + α2)Γ(1 + 2γ1,2)
+Γ(2 + α1 + 2γ1,2)Γ(3 + α1 + α2 + β2 + 2γ1,2)
+3F2(a, b; 1),
+(20)
+where a = (a1, a2, a3) = (1 + α1, −β1, 2 + 2γ1,2 + α1 + α2) and b = (b1, b2) = (2 + α1 + 2γ1,2, 3 +
+α1 + α2 + β2 + 2γ1,2), where pFq denotes the generalized hypergeometric function. For N = 2, the
+theorem above reads
+S2(α, β, γ) = Γ(1 + α1)Γ(2 + α1 + α2 + 2γ1,2)S2;reg,((••)•)•(α, β, γ)+
+Γ(1 + α1)Γ(1 + β2)S2;reg,(••)(••)(α, β, γ) + Γ(1 + β2)Γ(2 + β1 + β2 + 2γ1,2)S2;reg,•(•(••))(α, β, γ)
++ Γ(1 + 2γ1,2)Γ(2 + β1 + β2 + 2γ1,2)S2;reg,•((••)•)(α, β, γ)
++ Γ(1 + 2γ1,2)Γ(2 + α1 + α2 + 2γ1,2)S2;reg,(•(••))•(α, β, γ),
+(21)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+5
+but once again this splitting is not so obvious from the exact formula eq. (20). This example is
+explored more in the appendix.
+■
+The proof below is lower-brow than the twisted homological constructions of [KT86a, §5][KT86b]
+and Aomoto [Aom87], as it is based on the method described in [Var95, Chp. 10]. This involves
+the geometric analysis of the singularities of the Selberg(-like) integrand. The key observation is
+that if the N-simplex is blown up to the N-dimensional associahedron [Sta63][MSS02, §1.6][Pos09]
+(see Figure 2, Figure 6), then the Selberg integrand – which is not polyhomogeneous on △N –
+becomes one-step polyhomogeneous (a.k.a “classical”) on the resolution. See §2 for details. This
+observation appears, in an essentially equivalent form (albeit with different terminology), already in
+[KT86a][KT86b][MY03], though the term “associahedron” does not appear there. Closely related
+observations have appeared in the physics literature [Miz17][CKW18][CMT19][Miz20].
+The application of polyhomogeneity to the proof of the theorem above is given in §3. In a nutshell,
+the classicality of the lift of the Selberg integrand on the associahedron allows us to reduce the
+problem to what is essentially a product of one-dimensional cases. The faces of the associahedron
+are in bijective correspondence with the quantities defined in eq. (5), eq. (6). The correspondence is
+depicted in Figure 2 in the case N = 3. The quantities αj,∗, βj,∗, γj,k,∗ are the orders of the Selberg
+α1 + α2 + α3 + 2γ1,2 + 2γ1,3 + 2γ2,3
+2γ2,3
+α1 + α2 + 2γ1,2
+2γ1,2 + 2γ1,3 + 2γ2,3
+α1
+2γ1,2
+β1 + β2 + β3 + 2γ1,2 + 2γ1,3 + 2γ2,3
+β3
+β2 + β3 + 2γ2,3
+Figure 2. The 3-dimensional associahedron, with its faces labeled by the associated
+functions in eq. (5), eq. (6). The C4 = 14 vertices are in correspondence with the 14
+elements T(3).
+integrand at the corresponding faces. Each I ∈ T(N) is associated with a minimal facet of the
+associahedron, and the I ∈ I are associated with the faces containing that facet. Thus, we have a
+geometric interpretation of each of the terms in eq. (11).
+Note that ˙ΩN does not contain (α, β, γ) with γj,k = −1 for |j − k| = 1, so the theorem above is
+insufficient for the construction of the minimal models. Moreover, the theorem cannot be sharpened
+while maintaining generality. Indeed, the proof of the theorem shows that if F > 0 everywhere in
+△N (including the boundary), then
+SN;reg[F](α, β, γ) ̸= 0
+(22)
+for any (α, β, γ) ∈ R2N+N(N−1)/2 for which both of
+• γj,k,∗ ∈ Z≤−(k−j) for precisely one pair of j, k ∈ {0, . . . , N + 1} with j < k,
+• γj,k,∗ > −(k − j) for all other j, k
+hold, as for such (α, β, γ) the quantity SN;reg[F](α, β, γ) is proportional to a convergent integral of a
+positive integrand over the corresponding face of the associahedron. Consequently, SN[F] : ΩN → C
+cannot be analytically continued to the complement of any strictly smaller collection of hyperplanes
+than that in eq. (13).
+
+6
+ETHAN SUSSMAN
+However, for the desired application, we do not need full generality. Of special importance is the
+case when α, β, γ are each “constant,” meaning that, for some α, β, γ ∈ C,
+• αi = α and βi = β for all i ∈ {1, . . . , N}, and
+• γj,k = γ for all j, k ∈ {1, . . . , N} with j < k.
+In this case, we simply write
+SN[F](α, β, γ) =
+�
+△N
+� N
+�
+j=1
+xα
+j (1 − xj)β��
+�
+1≤j<k≤N
+(xk − xj)2γ�
+F(x1, . . . , xN) dx1 · · · dxN.
+(23)
+We now consider F ∈ C[x1, . . . , xN]SN , i.e. symmetric polynomial F. This case includes, of course,
+Selberg’s original example, in which F = 1, as well as the 3-point coefficients of the (1, s)- and
+(r, 1)-primary fields and their descendants in the BPZ minimal models. The computation of such
+integrals is listed as an open problem in [KT86a]. Below, we will introduce a more general notion
+of “DF-symmetric” Selberg-like integrals, this including the other 3-point coefficients. But, for the
+purposes of an introductory discussion we focus on the – already interesting – symmetric case.
+The integral eq. (23) is defined initially on the subset UN[F] ⊂ C3
+α,β,γ given by
+UN[F] =
+�
+(α, β, γ) ∈ C3 : ℜj(α + (j − 1)γ) > −1 − δj[F] and
+ℜj(β + (j − 1)γ) > −1 −
+δ
+j[F] for all j ∈ {1, . . . , N}, and ℜγ > −
+1
+N − 1
+�
+,
+(24)
+which contains
+UN = UN[1] =
+�
+(α, β, γ) ∈ C3 : min{ℜα, ℜβ}+min{0, (N −1)γ} > −1 and ℜγ > −
+1
+N − 1
+�
+. (25)
+An immediate corollary of the theorem above is that the function SN[F] : UN[F] → C defined by
+eq. (23) admits an analytic continuation ˙SN[F](α, β, γ) : ˙UN[F] → C to the domain ˙UN[F] ⊋ UN[F]
+given by
+˙UN[F] = C3
+α,β,γ
+��� N
+�
+j=1
+{j(α + (j − 1)γ) ∈ Z≤−j−δj}
+�
+∪
+� N
+�
+j=1
+{j(β + (j − 1)γ) ∈ Z≤−j−
+δ
+j}
+�
+∪
+� N−1
+�
+j=1
+{j(j + 1)γ ∈ Z−j}
+��
+.
+(26)
+Example. Consider F = 1, i.e. the Selberg integral. In this case, Selberg proved in [Sel44] that
+SN(α, β, γ) = SN[1](α, β, γ) is given by
+SN(α, β, γ) = 1
+N!
+� N
+�
+j=1
+Γ(1 + α + (j − 1)γ)Γ(1 + β + (j − 1)γ)Γ(1 + jγ)
+Γ(2 + α + β + (N + j − 2)γ)Γ(1 + γ)
+�
+.
+(27)
+See [FW08] for a review of the history of this result.
+■
+The example of the Selberg integral suggests that, in the symmetric case, eq. (26) is not the
+maximal domain of analyticity. Set
+degj[F] = max{d1 + · · · + dj : [F]d1,...,dj,dj+1,...,dN ̸= 0 for some dj+1, . . . , dN ∈ N}.
+(28)
+(Since F is symmetric, degj[F] = degj[F ◦ refl].) Then:
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+7
+Theorem 1.2. For any F ∈ C[x1, . . . , xN]SN , there exists an entire function SN;Reg[F] : C3
+α,β,γ → C
+such that
+SN[F](α, β, γ) =
+� N
+�
+j=1
+Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯
+δ
+j + β + (j − 1)γ)Γ(1 + jγ)
+Γ(2 + ¯dj + α + β + (N + j − 2)γ)Γ(1 + γ)
+�
+× SN;Reg[F](α, β, γ)
+(29)
+for all (α, β, γ) ∈ UN, where ¯δj = ⌈j−1δj[F]⌉, ¯
+δ
+j = ⌈j−1
+δ
+j[F]⌉, and ¯dj = ⌊(N − j + 1)−1 degj[F]⌋
+for each j ∈ {1, . . . , N}.
+■
+Thus, SN[F](α, β, γ) admits an analytic continuation ˚SN[F](α, β, γ) : ˚UN[F] → C to the domain
+˚UN[F] ⊋ ˙UN[F] defined by
+˚UN[F] = C3
+α,β,γ
+��� N
+�
+j=1
+{α + ¯δj + (j − 1)γ ∈ Z≤−1}
+�
+∪
+� N
+�
+j=1
+{β + ¯
+δ
+j + (j − 1)γ ∈ Z≤−1}
+�
+∪
+� N−1
+�
+j=1
+{(j + 1)γ ∈ Z≤−1, γ /∈ Z}
+��
+.
+(30)
+Observe that eq. (30) allows γ = −1.
+The Γ(2+ ¯dj+α+β+(N+j−2)γ) term in the denominator of eq. (29) implies that ˚SN[F](α, β, γ) =
+0 for all
+(α, β, γ) ∈ ˚UN[F] ∩ {α + β + (N + j − 2)γ ∈ Z≤−2− ¯dj for some j ∈ {1, . . . , N}}.
+(31)
+When constructing the 3-point coefficients of the BPZ minimal models, this is one mechanism
+preventing the fusion of (0, s)-primary fields (which are not included in the model) with the primary
+fields that are included. In BPZ’s terminology, this is the truncation of the operator algebras, as
+originally derived via the constraint of OPE associativity — see [BPZ84, §6][PM97, Chp. 7.3.2].
+In the case of the original Selberg integral, Theorem 1.2 describes precisely the singularities
+and zeroes of the meromorphic continuation of the original integral, and SN;Reg = SN;Reg[1] is just
+constant. The functions S2[F] and S2;Reg[F] are explored in §A.
+The proof of the theorem above consists of three steps:
+(1) The first step is the removal of the fictitious singularities of ˙SN[F](α, β, γ) only in γ (as
+required e.g. in the Coulomb gas formalism with both kinds of screening charges).
+The basic idea is to employ the relation – which can be found in a heuristic form in
+[DF85a, Ap. A] – between the symmetrization of SN[F](α, β, γ) and the “DF-like” integral
+IN[F](α, β, γ) =
+�
+□N
+� N
+�
+j=1
+xαj
+j (1 − xj)βj
+�
+×
+�
+�
+1≤j<k≤N
+(xk − xj + i0)2γj,k
+�
+F dx1 · · · dxN,
+(32)
+where □N = [0, 1]N. We can analytically continue IN[F] via an argument similar to that
+used to prove Theorem 1.1. Unlike that of SN[F](α, β, γ), this extension has no singularities
+associated with hyperplanes of constant γ.
+The true singularities of the extension of
+SN[F](α, β, γ) associated with hyperplanes of constant γ show up in the relation with the
+extension of IN[F](α, β, γ).
+(2) The second step removes the other unwanted singularities away from the loci of two or
+more unwanted singularities, via some identities proven via Aomoto [Aom87] in the F = 1
+case (and [DF85a, Ap. A], at a physicist’s level of rigor). The use of these identities for
+computing the original Selberg integral is sketched in [FW08]. It seems there cannot be a
+
+8
+ETHAN SUSSMAN
+similar computation of SN[F] in the deg F > 1 case, so a statement about the singularities
+is the best we can do.
+The simplex △N ⊂ RN can be thought of as a subset of
+(C\{0, 1})N = (CP 1\{0, 1, ∞})N
+(33)
+via the embedding R �→ C �→ CP 1, and the rough idea of this step of the proof is to relate
+the integrals above to the result of replacing △N with L⊠N△N for L one of the six linear
+fractional transformations preserving CP 1\{0, 1, ∞}. Only three of these are essentially
+different, and one of these three is just the identity and therefore uninteresting. The other
+two integrals each have meromorphic extensions with different manifest singularities. Using
+Proposition 4.2, these functions can be related to each other, and this can be used to
+remove most of the apparent singularities that are not present in all three functions. Some
+singularities are present in the relations between the integrals, and these cannot be removed.
+(3) The third step is the application of Hartog’s theorem to remove the remaining removable
+singularities, which now lie on a codimension two subvariety of C3.
+This argument is carried out in §4.1. The version more relevant to [DF85a] (with the additional
+steps needed) is in §4.2.
+We call IN[F] a “DF-like” integral because similar integrals appear, albeit at a somewhat formal
+level, in [DF85a]. A similar construction appears in [Fel89]. Let ΣT(N) denote the collection of
+maximal collections I of pairs (x0, S) of x0 ∈ {0, 1} and nonempty subsets S ⊆ {1, . . . , N} such
+that, given (x0, S), (x0, Q) ∈ I, either S ⊆ Q or Q ⊆ S.
+Theorem 1.3. There exist entire functions IN;reg,I[F] : C2N+N(N−1)/2
+α,β,γ
+→ C associated to the
+I ∈ ΣT(N) such that
+IN[F](α, β, γ) =
+�
+I∈ΣT(N)
+�
+�
+(1,S)∈I
+Γ
+�
+|S| +
+�
+j∈S
+βj + 2
+�
+j,k∈S,j<k
+γj,k
+��
+×
+�
+�
+(0,S)∈I
+Γ
+�
+|S| +
+�
+j∈S
+αj + 2
+�
+j,k∈S,j<k
+γj,k
+��
+IN;reg,I[F](α, β, γ)
+(34)
+for all (α, β, γ) for which the left-hand side is a well-defined integral.
+■
+In particular, IN[F](α, β, γ) admits an analytic extension ˙IN[F](α, β, γ) to an open, dense set
+˙VN = C2N+N(N−1)/2
+α,β,γ
+���
+�
+S⊆{1,...,N}
+{αS,∗ ∈ Z≤−|S|}
+�
+∪
+�
+�
+S⊆{1,...,N}
+{βS,∗ ∈ Z≤−|S|}
+��
+.
+(35)
+We end this section by discussing the removal of removable singularities for the integrals in
+[DF84][DF85a][DF85b]. Given λ ∈ C and S ⊆ {1, . . . , N}, let DFSym(N, S, λ) denote the set of
+F ∈ C[x1, x−1
+1 , . . . , xN, x−1
+N ] such that:
+• given any σ ∈ SN such that σ : S → S, F = F ◦ σ (where we are identifying σ with the map
+CN ∋ {xi}N
+i=1 → {xσ(i)}N
+i=1 ∈ CN), and
+• for any j ∈ S and k ∈ {1, . . . , N}\S,
+λ
+� ∂
+∂xj
+F
+����
+xj=xk
+=
+∂
+∂xk
+�
+F
+���
+xj=xk
+�
+∈ C[x1, x−1
+1
+. . . , ˆxj, ˆx−1
+j
+. . . , xN].
+(36)
+In particular, DFSym(N, {1, . . . , N}, λ) = C[x1, x−1
+1 , . . . , xN, x−1
+N ]SN , so in this sense DF-symmetry
+is a generalization of symmetry. (Our disallowal of Laurent polynomials in the symmetric case was
+without loss of generality, as we could shuffle factors of x1 · · · xN between the polynomial and the
+rest of the Selberg integrand. However, it is useful here to allow general Laurent polynomials.)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+9
+For each λ and S, DFSym(N, S, λ) is a (unital) C-subalgebra of C[x1, x−1
+1
+. . . , xN, x−1
+N ], and
+λ−
+�
+j∈S
+xj + λ+
+�
+k∈{1,...,N}\S
+xk ∈ DFSym(N, S, λ),
+λ−
+�
+j∈S
+1
+xj
++ λ+
+�
+k∈{1,...,N}\S
+1
+xk
+∈ DFSym(N, S, λ)
+(37)
+whenever λ−1
+− (λ− + λ+) = λ, so DFSym(N, S, λ) contains polynomials of all degrees. The key
+method of constructing DF-symmetric Laurent polynomials is the following:
+Example. For any M ∈ N+ and matrix-valued polynomials ϕ, ψ ∈ xCM×M[x] such that
+• if Aj is the coefficient of xj in ϕ, then Aj is strictly upper-triangular,
+• if Bj is the coefficient of xj in ψ, then Bj is strictly lower-triangular.
+Suppose that the A’s all commute with each other and that the B’s all commute with each other.
+(We do not assume that the A’s commute with the B’s.) Then, the matrix elements of
+exp
+�
+λ−
+�
+j∈S
+ψ(x−1
+j ) + λ+
+�
+k∈{1,...,N}\S
+ψ(x−1
+k )
+�
+exp
+�
+λ−
+�
+j∈S
+ϕ(xj) + λ+
+�
+k∈{1,...,N}\S
+ϕ(xk)
+�
+(38)
+lie in DFSym(N, S, λ), where λ = λ−1
+− (λ−+λ+). The vertex operators which Dotsenko and Fateev in-
+tegrate to define the minimal model 3-point coefficients have this form up to some scalar factors which
+are part of the Selberg integrand. In this example, the coefficients of ϕ are annihilation operations
+on some Fock space, and the coefficients of ψ are creation operators (with all operators truncated
+to some finite dimensional subspace of the Fock space). See [DF84][KT86b][KT86a][Fel89][PM97,
+Chp. 9].
+■
+For a set S ⊆ {1, . . . , N} and α±, β±, γ±, γ0 ∈ C, let αDF0,S, βDF0,S ∈ CN be given by
+αj =
+�
+α+
+(j ∈ S),
+α−
+(j /∈ S),
+(39)
+βj =
+�
+β+
+(j ∈ S),
+β−
+(j /∈ S),
+(40)
+and let γDF0,S ∈ CN(N−1)/2 be given by
+γj,k =
+�
+�
+�
+�
+�
+�
+�
+γ+
+(j, k ∈ S),
+γ0
+(j ∈ S, k /∈ S or vice versa),
+γ−
+(j, k /∈ S).
+(41)
+Let
+˙W DF0,S
+N
+[F] = {(α−, α+, β−, β+, γ−, γ0, γ+) ∈ C7 : (αDF0,S, βDF0,S, γDF0,S) ∈ ˙VN[F]}.
+(42)
+Define, for (α−, α+, β−, β+, γ−, γ0, γ+) ∈ ˙W DF0,S
+N
+[F],
+˙IDF0,S
+N
+[F](α−, α+, β−, β+, γ−, γ0, γ+) = ˙IN[F](αDF0,S, βDF0,S, γDF0,S).
+(43)
+Now let ˙W DF,S
+N
+[F] denote the set of (α+, β+, γ+) ∈ C3 such that γ+ ̸= 0 and, setting
+γ− = γ−1
++ ,
+α− = −γ−α+,
+β− = −γ−β+
+(44)
+— cf. [DF85a, eq. A.2] — it is the case that (α−, α+, β−, β+, γ−, −1, γ+) ∈ ˙W DF0,S
+N
+[F]. This is an
+open and dense subset of C3. Let
+˙IDF,S
+N
+[F](α+, β+, γ+) = ˙IDF0,S
+N
+[F](−γ−1
++ α+, α+, −γ−1
++ β+, β+, γ−1
++ , −1, γ+).
+(45)
+for (α+, β+, γ+) ∈ ˙W DF,S
+N
+[F]. Set N+ = |S| and N− = N − N+.
+
+10
+ETHAN SUSSMAN
+Theorem 1.4. Fix γ+ ∈ C\{0, 1} and S ⊆ {1, . . . , N}. Suppose that
+F ∈ DFSym(N, S, γ−1
++ (1 − γ+)).
+(46)
+There exist an entire function IDF,S
+N;Reg[F](α+, β+, γ+) : C2
+α+,β+ → C such that
+˙IDF,S
+N
+[F](α+, β+, γ+) =
+� �
+±
+N±
+�
+j=1
+sin(π(α± + β± + (N± + j − 2)γ±))
+sin(π(α± + (j − 1)γ±)) sin(π(β± + (j − 1)γ±))
+�
+× IDF,S
+N;Reg[F](α+, β+, γ+)
+(47)
+when α−, β−, γ− are related to α+, β+, γ+ by eq. (44) and the left-hand side is well-defined.
+■
+If desired, it is possible to replace the sines with Γ-functions with appropriate integral shifts.
+Example. When F = 1, Dotsenko and Fateev claim in [DF85a, Eqs. A.8, A.35]1 that the integral
+above is given by
+˙IDF,S
+N
+[1](α+, β+, γ+) ∝ γ2N−N+
+∓
+� N±
+�
+j=1
+e−iπ(j−1)γ± Γ(jγ±) sin(πjγ±)
+Γ(γ±) sin(πγ±)
+�
+×
+� N∓
+�
+j=1
+e−iπ(j−1)γ∓ Γ(jγ∓ − N±) sin(πjγ∓)
+Γ(γ∓) sin(πγ∓)
+�� N±
+�
+j=1
+Γ(1 + α± + (j − 1)γ±)Γ(1 + β± + (j − 1)γ±)
+Γ(2 − 2N∓ + α± + β± + (N± − 2 + j)γ±)
+�
+×
+� N∓
+�
+j=1
+Γ(1 + α∓ + (j − 1)γ∓ − N±)Γ(1 + β∓ + (j − 1)γ∓ − N±)
+Γ(2 − N± + α∓ + β∓ + (N∓ − 2 + j)γ∓)
+�
+(48)
+for each choice of sign.
+■
+2. Associahedra
+We use the term ‘mwc’ to mean manifold-with-corners in the sense of Melrose – e.g. [Mel][HMM97],
+these possessing C∞-structure. In this section, we define the mwcs that will be used to resolve the
+singularities of Selberg- and Dotsenko-Fateev-like integrands:
+• in §2.1, we define the associahedra Kℓ,m,n, used to meromorphically continue the Selberg-like
+integrals, and
+• in §2.2 we define the mwcs Aℓ,m,n, used to meromorphically continue the DF-like integrals.
+Since K0,N,0 is the usual N-dimensional associahedra, we refer to the mwcs defined below as
+associahedra as well, hence the title of this section. If M is a mwc, we use F(M) to denote the set
+of faces of M, where by faces we mean only the boundary hypersurfaces. We use “facet” to refer to
+the higher codimension boundary components.
+It is worth comparing Melrose’s notion of mwc to that of polyhedron.
+A mwc is locally a
+polyhedron, but the converse is not true, as the basic requirement of M being locally diffeomorphic
+to a relatively open neighborhood of [0, ∞)N means that every facet f ⊊ M is the intersection of
+at most N faces. While the (closed) ball, tetrahedron, cube, and dodecahedron are all mwcs, the
+octahedron and icosahedron are not. It is critical for the argument in §3 that the associahedra
+Aℓ,m,n and Kℓ,m,n are not just polyhedra, but rather mwcs. Thus, the notion of “mwc” used here
+plays a similar role to that of “polyhedra in general position” in [Var95, §10.7], but they are not
+equivalent. For the purposes of this paper, we find it more natural (and technically simpler, as it
+avoids the need for polyhedral realizations) to use the language of mwcs.
+1There seem to be a couple typos in [DF85a, Eq. A.35], which we have fixed above. The first few cases of eq. (48)
+have been numerically checked.
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+11
+We keep track of the full C∞-structure of these mwcs below. Were it required, we could keep
+track of Cω- (i.e. real analytic) structure, but since this would require going somewhat beyond the
+existent literature on mwcs, and since this level of precision is not needed for the rest of the paper,
+we will restrict ourselves to the smooth category.
+If f is a facet of M, then the blowup [M; f] is a mwc, and the blowdown map
+bd : [M; f] → M
+(49)
+is smooth. For convenience, we can identify the interior [M; f]◦ with M◦. (If F is a codimension ≤ 1
+facet of M, then we can identify [M; F] with M itself.) Naturally, if f has codimension ≥ 2, then
+F([M; f]) = {[F; f ∩ F] : F ∈ F(M)} ∪ {ff},
+(50)
+where ff = bd−1(f) is the front face of the blowup. Then, given bdfs xF ∈ C∞(M; R+) of the faces
+F ∈ F(M), we can choose bdfs x[F;f∩F], xff of the faces of [M; f] such that, for each F ∈ F(M),
+xF ◦ bd =
+�
+x[F;f∩F]xff
+(f ⊆ F),
+x[F;f∩F]
+(otherwise).
+(51)
+(We identify polyhomogeneous – in particular, smooth – functions on [M; f] with their restrictions
+to the interior, so, going forwards, we can drop the “◦ bd.”) Specifically, in addition to defining
+x[F;f∩F] = xF if f ̸⊆ F, we can take
+xff =
+�
+F∈F(M),f⊆F
+xF,
+(52)
+and, if f ⊆ F, then
+x[F;f∩F] = xF
+�
+�
+F∈F(M),f⊆F
+xF
+�−1
+.
+(53)
+This follows from the analogous result for blowing up a facet of [0, ∞)N. Note that because M is a
+mwc and not just a polyhedron, if F1, . . . , Fd ∈ F(M) are distinct faces with ∩δFδ ̸= ∅, then the
+connected components of ∩δFδ are codimension d facets of M. (The 2D lens is an example of a
+mwc with two faces whose intersection is disconnected.)
+If U is an open subset of a mwc, then U can be considered as a mwc in its own right. We will say
+that some function x ∈ C∞(U; [0, ∞)) is a bdf in U of F ∈ F(M) if it is a bdf of the face F ∩ U of U,
+assuming that F ∩ U ̸= ∅, in which case it is automatically a face of U. Let Rt = Rt ∪ {−∞, +∞}
+denote the “radial” compactification of R. This is a (C∞-)manifold-with-boundary, with 1/t serving
+as a bdf for {∞} in {t > 0} and −1/t serving as a bdf for {−∞} in {t < 0}.
+2.1. The Associahedra Kℓ,m,n. We now define the mwc Kℓ,m,n for ℓ, m, n ∈ N not all zero. The
+blowup procedure below is a generalization of that in [KT86a]. We begin with the set
+△ℓ,m,n = {(x1, . . . , xN) ∈ RN : x1 ≤ · · · ≤ xℓ ≤ 0
+≤ xℓ+1 ≤ · · · ≤ xℓ+m ≤ 1 ≤ xℓ+m+1 ≤ · · · ≤ xN},
+(54)
+where N = ℓ + m + n. This is a compact sub-mwc of RN. Naturally,
+△ℓ,m,n ∼= △ℓ,0,0 × △0,m,0 × △0,0,n.
+(55)
+Also, △ℓ,0,0 ∼= △ℓ, △0,m,0 ∼= △m, and △0,0,n ∼= △n.
+For example, in the case N = 2, we have six cases. These are △2,0,0, △0,2,0, △0,0,2, each of which
+is diffeomorphic to the triangle △2, and △1,1,0, △1,0,1, △0,1,1, each of which is diffeomorphic to the
+square □2.
+If ℓ, n = 0, in which case m = N, then △ℓ,m,n is just the standard N-simplex △N.
+
+12
+ETHAN SUSSMAN
+We call a subset I ⊆ Z/(N + 3)Z consecutive if it is of the form {k mod (N + 3), · · · , k +
+κ mod (N + 3)Z} for some k ∈ Z/(N + 3)Z and κ ∈ N. (Thus, the empty set will not be considered
+consecutive.)
+We label the facets (of any codimension, possibly zero) of △ℓ,m,n using (unordered) partitions I
+of Z/(N + 3)Z into consecutive subsets I, with no two of 0, ℓ + 1, ℓ + m + 2 ∈ Z/(N + 3)Z appearing
+together in any element I ∈ I. Specifically,
+f0,I =
+�
+(x1, . . . , xN) ∈ △ℓ,m,n :
+�
+I ∈ I ⇒
+�
+j ∈ I ⇒ tj = ±∞
+(0 ∈ I)
+j, k ∈ I ⇒ tj = tk
+(0 /∈ I)
+��
+,
+(56)
+where
+• tj = xj for j = 1, . . . , ℓ,
+• tℓ+1 = 0,
+• tℓ+1+j = xℓ+j for j = 1, . . . , m,
+• tℓ+m+2 = 1, and
+• tℓ+m+2+j = xℓ+m+j for j = 1, . . . , n.
+The dimension of f0,I is given by
+dim f0,I = |I| − 3.
+(57)
+For notational simplicity, if I0 ⊆ I is I with the singletons removed, then we define fI0 = f0,I. Thus,
+f∅ denotes the “bulk” of △ℓ,m,n, and the faces of △ℓ,m,n are of the form f{I} for I a consecutive
+pair. Rephrasing eq. (57),
+codim fI =
+�
+I∈I
+(|I| − 1).
+(58)
+As a bdf of f{I} for I = {k mod Z/(N + 3)Z, k + 1 mod Z/(N + 3)} when k ∈ {1, . . . , N + 1}, we
+can take
+xf{I} = tk+1 − tk.
+(59)
+For the remaining two cases of F{0,1} (which only exists if ℓ ≥ 1) and f{N+2,N+3} (which only exists
+if n ≥ 1), we can take xf{0,1} = −1/x1 and xf{N+2,N+3} = 1/xN.
+Let Fℓ,m,n = Fℓ,m,n(△) denote the family of facets fI of △ℓ,m,n such that I = {I} for some
+consecutive subset I ⊂ Z/(N + 3)Z of size |I| ≥ 2 not containing any two of 0, ℓ + 1, ℓ + m + 2.
+In other words, Fℓ,m,n is the set of facets fI for I defining a partition of Z/(N + 3)Z into a single
+interval of length at least two (not containing any two of 0, ℓ + 1, ℓ + m + 2) and a number of
+singletons which are being omitted from the notation.
+For each d ∈ {0, . . . , N}, let Fℓ,m,n;d denote the set of elements of Fℓ,m,n of dimension d. Then,
+the mwc Kℓ,m,n is defined by the iterated blowup
+Kℓ,m,n = [△ℓ,m,n; Fℓ,m,n,0; · · · ; Fℓ,m,n,N] = [· · · [△ℓ,m,n; Fℓ,m,n;0] · · · ; Fℓ,m,n;N].
+(60)
+I.e., we first blow up the elements of the collection Fℓ,m,n;0 (which may be empty, e.g. if ℓ, m, n are
+all nonzero), and then, proceeding from higher to lower codimension, iteratively blow up the lifts of
+the facets in Fℓ,m,n;d (meaning the closures of the lifts of the interiors).
+We should check that the blowup eq. (60) is well-defined, which concretely means that, for each
+d, the blow-ups in the step in which we blow up the lifts of the elements of Fℓ,m,n;d commute. This
+can be done via a somewhat tedious inductive argument, which we only sketch.
+When the time has come to blow up the facets f ̸= f′ in the lifted Fℓ,m,n;d, their intersection is –
+if nonempty – either a point (which we denote K0,0,0) or else an associahedron Kℓ∩,m∩,n∩ (which
+will not change upon performing further blowups) of dimension < N, and a neighborhood thereof is
+diffeomorphic to
+[0, 1)N−d
+t
+× Kℓ∩,m∩,n∩ × [0, 1)N−d
+t′
+,
+(61)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+13
+with f corresponding to {t = 0} and f′ corresponding to {t′ = 0}; the blowups of these two faces in
+the product above commute, with the result being naturally diffeomorphic to
+[[0, 1)N−d
+t
+, {0}] × Kℓ∩,m∩,n∩ × [[0, 1)N−d
+t′
+; {0}].
+(62)
+In order to prove the claimed decomposition, eq. (61), it is first useful to note when f ∩ f′ = ∅. If
+I, I′ satisfy |I| = N − d + 1 = |I′| and I ∩ I′ ̸= ∅, then the corresponding facets
+f = cl[△ℓ,m,n;Fℓ,m,n,0;··· ;Fℓ,m,n,d−1]f◦
+{I}
+(63)
+f′ = cl[△ℓ,m,n;Fℓ,m,n,0;··· ;Fℓ,m,n,d−1]f◦
+{I′}
+(64)
+of [△ℓ,m,n; Fℓ,m,n,0; · · · ; Fℓ,m,n,d−1] satisfy f ∩ f′ = ∅. Indeed, I ∩ I′ ̸= ∅ implies
+f{I} ∩ f{I′} = f{I∪I′} ∈ Fℓ,m,n(△),
+(65)
+and since this is blown up in an earlier stage of the construction, f and f′ cannot intersect.
+So, if our two facets f, f′ to be blown up have nonempty intersection, then they must be the
+lifts of f{I} and f{I′} for I, I′ satisfying I ∩ I′ = ∅. The intersection f ∩ f′ lies in the preimage of
+f{I} ∩ f{I′} = f{I,I′}. This facet of △ℓ,m,n is of the form △ℓ∩,m∩,n∩ for ℓ∩ + m∩ + n∩ = 2d − N ≥ 0.
+As seen inductively, the lift of this facet after performing the blow-ups so far is Kℓ∩,m∩,n∩, although
+this is not crucial for the proof that the construction is well-defined. Since this has dimension
+2d − N, a neighborhood of this facet in our partially blown-up manifold automatically has the form
+[0, 1)2N−2d × Kℓ∩,m∩,n∩,
+(66)
+so it just needs to be checked that f, f′ sit inside of this in the expected way. Indeed, the projections
+of f, f′ onto the [0, 1)2N−2d factor are necessarily transversal. With the fact that they both have
+dimension d, this implies that we can decompose
+[0, 1)2N−2d = [0, 1)N−d
+t
+× [0, 1)N−d
+t′
+(67)
+such that f corresponds to {t = 0} and f′ corresponds to {t′ = 0}. This completes our sketch.
+We now discuss the combinatorial structure of Kℓ,m,n. All of the faces of △ℓ,m,n are in Fℓ,m,n;N−1,
+so every face of Kℓ,m,n is the front face of one of our blowups. So, the faces of Kℓ,m,n are in bijection
+with the elements of Fℓ,m,n and thus with I as above. Such a subset is uniquely specified by its
+endpoints j, k ∈ Z/(N + 3)Z, since only two consecutive subsets of Z/(N + 3)Z have the same
+endpoints as I, namely I itself and I∁ ∪ {j, k}, and the latter contains two of 0, ℓ + 1, ℓ + m + 2.
+Let Jℓ,m,n denote the set of unordered pairs {j, k} arising in this way. For {j, k} ∈ Jℓ,m,n, let
+I(j, k) = I(k, j) denote the unique consecutive subset of Z/(N + 3)Z having these endpoints and
+containing at most one member of {0, ℓ + 1, ℓ + m + 2}. For such j, k, let Fj,k = Fk,j denote the
+corresponding face of Kℓ,m,n, and let xFj,k = xFk,j denote a bdf of that face constructed inductively
+as in the introduction to this section. (Note that these bdfs may depend on the particular order in
+which the elements of the Fℓ,m,n;d are blown up.)
+There are 2−1N(N + 3) faces in Kℓ,m,n.
+Example. Consider the case N = 2. Then, up to essential equivalence, the cases to consider are
+K1,1,0 and K0,2,0. These are depicted in Figure 3. The mwc K1,1,0 is identical to A1,1,0; in §2.2
+we introduce notation for labeling the faces of the Aℓ,m,n, and this notation appears in Figure 4
+alongside that used for the Kℓ,m,n. We have introduced an additional notation for the faces of
+Kℓ,m,n, indicating I in the subscript using the following conventions:
+• The elements 0, ℓ + 1, ℓ + m + 2 ∈ Z/5Z are depicted using a ‘◦,’ and 0 is omitted if not
+included in I.
+• The other elements of Z/5Z are depicted using a ‘•.’
+• Except for 0, the elements of Z/5Z are depicted in order. If 0 is to be depicted, it is listed
+either first or last.
+The elements included in I are enclosed in parentheses.
+■
+
+14
+ETHAN SUSSMAN
+F3,4 = F•◦(•◦)
+F2,3 = F•(◦•)◦
+F0,1 = F(◦•)◦•◦
+F1,2 = F(•◦)•◦
+F1,3 = F(•◦•)◦
+x2
+w1
+F1,2 = F(◦•)•◦
+F3,4 = F◦•(•◦)
+F2,3 = F◦(••)◦
+F1,3 = F(◦••)◦
+F2,4 = F◦(••◦)
+1 − x2
+x1
+Figure 3. The associahedra K1,1,0 (left) and K0,2,0 (right), realized as polyhedra
+roughly in accordance with the blowup procedure. In the first figure, the horizontal
+axis is roughly w1 = 1/(1 − x1), increasing to the right. In the second figure, it is
+just (roughly) x1. In both figures, the vertical axis is (roughly) x2.
+F4,5 = F•◦•(◦•) = F{3},∅;1
+F0,1 = F(◦•)◦•◦• = F{1},∅;∞
+F1,5 = F◦•)◦•◦(• = F{1},{3};∞
+F2,3 = F•(◦•)◦• = F{2},∅;0
+x2
+y3
+w1
+F1,3 = F(•◦•)◦• = F{1},{2};0
+F3,4 = F•◦(•◦)• = F{2},{∅};1
+F3,5 = F•◦(•◦•) = F{2},{3};1
+F1,2 = F(•◦)•◦• = F{1},{∅};0
+F0,5 = F5,6 = F•◦•◦(•◦) = F∅,{3};∞
+Figure 4. The mwc K1,1,1, with labeled faces, realized as a polyhedron roughly in
+accordance with the blowup procedure. Here w1 = 1/(1 − x1) and y3 = (x3 − 1)/x3.
+The faces in the line of sight are F1,2 = F(•◦)•◦•, F1,3 = F(•◦•)◦•, F3,4 = F•◦(•◦)•,
+F3,5 = F•◦(•◦•), and F0,5 = F•◦•◦(•◦).
+Example. Consider the case N = 3. Then, up to essential equivalence, the cases to consider are
+K1,1,1, K1,2,0, and K0,3,0. These are depicted in Figure 4, Figure 5, Figure 6. The mwc K1,1,1 is
+identical to A1,1,1.
+We have modified the “•” notation from the previous example and used it to label the faces in
+the figures, alongside the notation used in the rest of this section. For instance, when considering
+K0,3,0, “◦(• • •)◦” denotes {2, 3, 4} ⊂ Z/6Z. When considering K1,2,0, “◦(• ◦ •)• ◦” denotes {1, 2, 3}.
+When considering K1,1,1, “•) ◦ • ◦ (•◦” denotes {0, 1, 5}.
+■
+The Kℓ,m,n satisfy the following “universal property:”
+• For any subsets S ⊆ {1, . . . , ℓ}, Q ⊆ {ℓ + 1, . . . , ℓ + m}, R ⊆ {ℓ + m + 1, . . . , N} that are not
+all empty, let forg : △ℓ,m,n → △|S|,|Q|,|R| denote the forgetful map forgetting the variables xj
+for j /∈ S ∪ Q ∪ R. Then, forg lifts to a smooth b-map [Mel]
+forg : Kℓ,m,n → K|S|,|Q|,|R|.
+(68)
+Given any face F of K|S|,|Q|,|R|, forg
+∗xF vanishes to first order at each face in forg
+−1(F).
+This can be proven by inducting on the number of blowups.
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+15
+Figure 5. The mwc K1,2,0, with
+labeled faces, realized as a polyhe-
+dron roughly in accordance with
+the blowup procedure. As above,
+w1 = 1/(1 − x1).
+The faces
+in the line of site are F1,2 =
+F(•◦)••◦, F1,3
+= F(•◦•)•◦, and
+F4,5 = F•◦•(•◦).
+F3,4 = F•◦(••)◦
+F2,3 = F•(◦•)•◦
+F0,1 = F(◦•)◦••◦
+F3,5 = F•◦(••◦)
+F1,4 = F(•◦••)◦
+F2,4 = F•(◦••)◦
+x2
+x3 − x2
+w1
+F4,5 = F•◦•(•◦)
+F1,2 = F(•◦)••◦
+F1,3 = F(•◦•)•◦
+Figure 6. The mwc K0,3,0, with
+labeled faces, realized as a polyhe-
+dron roughly in accordance with
+the blowup procedure. The faces
+in the line of sight are F4,5 =
+F◦••(•◦) and F3,5 = F◦•(••◦). Cf.
+[KT86a, Fig. 5.2], where the full
+blowup procedure is depicted.
+F1,4 = F(◦•••)◦
+F3,4 = F◦•(••)◦
+F1,3 = F(◦••)•◦
+F2,4 = F◦(•••)◦
+F1,2 = F(◦•)••◦
+F2,3 = F◦(••)•◦
+F2,5 = F◦(•••◦)
+x2 − x1
+x3 − x2
+x1
+F4,5 = F◦••(•◦)
+F3,5 = F◦•(••◦)
+Proposition 2.1. Suppose that µ ∈ C∞(△ℓ,m,n; Ω△ℓ,m,n) is a strictly positive smooth density on
+△ℓ,m,n. Then, the lift of µ to Kℓ,m,n has the form
+�
+�
+{j,k}∈Jℓ,m,n
+x|j−k|−1
+Fj,k
+�
+µ
+(69)
+for a strictly positive µ ∈ C∞(Kℓ,m,n; ΩKℓ,m,n). Here, for j, k ∈ Z/(N + 3)Z, we use the notation
+|j − k| = min{|j0 − k0|, |k0 − j0| : j0, k0 ∈ Z : j0 ≡ j mod (N + 3), k0 ≡ k mod (N + 3)}.
+■
+In the product, each unordered pair is counted only once.
+Proof. We recall the following lemma:
+• Suppose that M is a mwc and µ ∈ C∞(M; ΩM) is a strictly positive smooth density on M.
+Then, if f is a facet of M of codimension d ∈ N+, the lift of µ to [M; f] has the form xd−1
+ff
+ν
+and ν a strictly positive smooth density on [M; f].
+Working in local coordinates, this follows from the case of blowing up a facet in [0, ∞)N. In this
+case, we can use cylindrical coordinates (that is, spherical coordinates if the facet we are blowing up
+is the corner). The result follows from the form of the Lebesgue measure in cylindrical coordinates.
+The proposition follows from an inductive application of the lemma, once we note that |j − k| is
+the codimension of Fj,k.
+□
+Proposition 2.2. The Lebesgue measure on RN, which defines a strictly positive smooth density
+on △◦
+ℓ,m,n, has the form
+�
+ℓ
+�
+j=1
+(1 − xj)2��
+N
+�
+j=ℓ+m+1
+x2
+j
+�
+µ
+(70)
+for µ ∈ C∞(△ℓ,m,n; Ω△ℓ,m,n) a strictly positive smooth density on △ℓ,m,n.
+■
+
+16
+ETHAN SUSSMAN
+Proof. It is the case that the 1-form dxj ∈ Ω1△◦
+ℓ,m,n defines an extendable 1-form on △ℓ,m,n if
+j ∈ {ℓ + 1, · · · , ℓ + m}, and the extension is nonvanishing. The same holds for
+• dwj = (1 − xj)−2 dxj for wj = 1/(1 − xj) if j ∈ {1, . . . , ℓ} and
+• dyj = x−2
+j
+dxj for yj = (xj − 1)/xj if j ∈ {ℓ + m + 1, · · · , N},
+since △ℓ,m,n is a submanifold of RN. The µ in eq. (70) can therefore be taken to be |dw1 ∧ · · · ∧
+dwℓ ∧ dxℓ+1 ∧ · · · ∧ dxℓ+m ∧ dyℓ+m+1 ∧ · · · ∧ dyN|, which lies in C∞(△ℓ,m,n; Ω△ℓ,m,n) and is strictly
+positive.
+□
+We now record the results of lifting the factors xi, 1 − xi, and xj − xk comprising the Selberg
+integrand to Kℓ,m,n. Beginning with the first two cases:
+• If i ∈ {1, . . . , ℓ}, then
+−xi ∈
+�
+N+3
+�
+j=ℓ+m+3
+ℓ
+�
+k=i
+x−1
+Fj,k
+��
+i�
+j=1
+ℓ+m+1
+�
+k=ℓ+1
+xFj,k
+�
+C∞(Kℓ,m,n; R+),
+(71)
+(1 − xi) ∈
+�
+N+3
+�
+j=ℓ+m+3
+ℓ
+�
+k=i
+x−1
+Fj,k
+�
+C∞(Kℓ,m,n; R+).
+(72)
+• If i ∈ {ℓ + 1, . . . , ℓ + m}, then
+xi ∈
+� ℓ+1
+�
+j=1
+ℓ+m+1
+�
+k=i+1
+xFj,k
+�
+C∞(Kℓ,m,n; R+),
+(73)
+(1 − xi) ∈
+�
+i+1
+�
+j=ℓ+2
+N+2
+�
+k=ℓ+m+2
+xFj,k
+�
+C∞(Kℓ,m,n; R+).
+(74)
+• If i ∈ {ℓ + m + 1, . . . , N}, then
+xi ∈
+�
+i+2
+�
+j=ℓ+m+3
+ℓ
+�
+k=0
+x−1
+Fj,k
+�
+C∞(Kℓ,m,n; R+),
+(75)
+−(1 − xi) ∈
+�
+i+2
+�
+j=ℓ+m+3
+ℓ
+�
+k=0
+x−1
+Fj,k
+�� ℓ+m+2
+�
+j=ℓ+2
+N+2
+�
+k=i+2
+xFj,k
+�
+C∞(Kℓ,m,n; R+).
+(76)
+If N = 1, then these are all trivial to prove. By applying the universal property of the associahedra,
+the N ≥ 2 case follows from the N = 1 case.
+In a similar manner, by working out the case of K0,2,0 in detail and applying the universal
+property, we get, for k > j:
+• If j, k ∈ {ℓ + 1, . . . , ℓ + m}, then
+(xk − xj) ∈
+� ℓ+1
+�
+j0=1
+ℓ+m+1
+�
+k0=k+1
+xFj0,k0
+��
+j+1
+�
+j0=ℓ+2
+N+2
+�
+k0=k+1
+xFj0,k0
+�
+C∞(Kℓ,m,n; R+).
+(77)
+Indeed, in the case of ℓ, n = 0 and m = 2, this says that x2 − x1 ∈ xF1,3xF2,3xF2,4C∞(K0,2,0; R+).
+Indeed, if we construct K0,2,0 by first blowing up F1,3 and then blowing up F2,4, we get
+xF1,3 = x2,
+xF2,3 =
+x2 − x1
+2x2 − x2
+2 − x1
+,
+xF2,4 = 2x2 − x2
+2 − x1
+x2
+,
+(78)
+so that xF1,3xF2,3xF2,4 = x2 − x1, on the nose. On the other hand, if we reverse the order of the
+blowups, then we get
+xF1,3 = x2 − x2
+1
+1 − x1
+,
+xF2,3,0 = x2 − x1
+x2 − x2
+1
+,
+xF2,4 = 1 − x1,
+(79)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+17
+so we still get xF1,3xF2,3xF2,4 = x2 − x1.
+From this, we can deduce the following.
+• If j, k ∈ {1, . . . , ℓ}, then, in terms of wi = −xi/(1−xi), (xk−xj) = (1−wj)−1(1−wk)−1(wj −
+wk), so,
+(xk − xj) ∈
+�
+ℓ
+�
+j0=j
+N+3
+�
+k0=ℓ+m+3
+x−1
+Fj0,k0
+��
+j�
+j0=1
+ℓ+m+1
+�
+k0=k
+xFj0,k0
+�
+C∞(Kℓ,m,n; R+).
+(80)
+• If j, k ∈ {ℓ + m + 1, . . . , N}, then, in terms of yi = 1/xi, (xk − xj) = y−1
+j y−1
+k (yj − yk), so
+(xk − xj) ∈
+�
+j+2
+�
+j0=ℓ+2
+N+2
+�
+k0=k+2
+xFj0,k0
+��
+k+2
+�
+j0=ℓ+m+3
+ℓ
+�
+k0=0
+x−1
+Fj0,k0
+�
+C∞(Kℓ,m,n; R+).
+(81)
+The next three follow from the K1,1,0, K1,0,1, and K0,1,1 cases. We illustrate the K1,1,0 case, and
+the others are similar.
+• If j ∈ {1, . . . , ℓ} and k ∈ {ℓ + 1, . . . , ℓ + m}, then (xk − xj) = (1 − wj)−1(wj + xk − xkwj), so
+(xk − xj) ∈
+�
+ℓ
+�
+j0=j
+N+3
+�
+k0=ℓ+m+3
+x−1
+Fj0,k0
+��
+j�
+j0=1
+ℓ+m+1
+�
+k0=k+1
+xFj0,k0
+�
+C∞(Kℓ,m,n; R+).
+(82)
+In the case ℓ, m = 1, n = 0, this says that (x2 − x1) ∈ x−1
+F1,5xF1,3C∞(K1,1,0; R+). Indeed,
+the bdf xF1,3 of F1,3 in K1,1,0 is defined by
+xF1,3 = (1 − w1) + x2 = −
+x1
+1 − x1
++ x2,
+(83)
+and xF1,5 = xF0,1 = w1 = 1/(1 − x1). So,
+x−1
+F1,5xF1,3 = x2 − x1 − x1x2.
+(84)
+The supposed C∞(K1,1,0; R+) term above is therefore (x2 − x1)(x2 − x1 − x1x2)−1 =
+(1 − x2x1/(x2 − x1))−1. One way (besides checking in a system of local coordinate charts)
+to see that this is smooth (and positive) on K1,1,0 is the identity
+−
+x2x1
+x2 − x1
+=
+xF1,2xF1,3xF2,3
+xF1,2 + xF2,3xF0,1
+.
+(85)
+The faces F0,1, F2,3 are disjoint from F1,2 (see Figure 3), so the denominator on the right-hand
+side of eq. (85) is nonvanishing, so the quotient is indeed smooth.
+Likewise:
+• If j ∈ {ℓ + 1, . . . , ℓ + m} and k ∈ {ℓ + m + 1, . . . , N}, then (xk − xj) = y−1
+k (1 − xjyk), so
+(xk − xj) ∈
+�
+j+1
+�
+j0=ℓ+2
+N+2
+�
+k0=k+2
+xFj0,k0
+��
+ℓ
+�
+j0=0
+k+2
+�
+k0=ℓ+m+3
+x−1
+Fj0,k0
+�
+C∞(Kℓ,m,n; R+).
+(86)
+• If j ∈ {1, . . . , ℓ} and k ∈ {ℓ + m + 1, . . . , N}, then (xk − xj) = y−1
+k (1 − wj)−1(1 − wj + wjyk),
+so
+(xk − xj) ∈
+�
+ℓ
+�
+j0=j
+N+3
+�
+k0=k+3
+x−1
+Fj0,k0
+��
+ℓ
+�
+j0=0
+k+2
+�
+k0=ℓ+m+3
+x−1
+Fj0,k0
+�
+C∞(Kℓ,m,n; R+).
+(87)
+We associate to each face F• ∈ F(Kℓ,m,n) an affine functional
+ρ• : C2N+N(N−1)/2 ∋ (α, β, γ) �→ ρ•(α, β, γ) ∈ C.
+(88)
+Suppose that we are given some α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2. If one of
+(I) j, k ∈ {1, . . . , ℓ}
+
+18
+ETHAN SUSSMAN
+(II) j, k ∈ {ℓ + 2, . . . , ℓ + m + 1},
+(III) j, k ∈ {ℓ + m + 3, . . . , N + 2}
+holds, then, letting k denote the larger of {j, k},
+ρj,k = k − j + 2
+�
+j′≤j0<k0≤k′
+γj0,k0,
+(89)
+where, for each i ∈ {j, k}, i′ = i if i ∈ {1, . . . , ℓ}, i′ = i − 1 if i ∈ {ℓ + 2, . . . , ℓ + m + 1}, and i′ = i − 2
+if i ∈ {ℓ + m + 3, . . . , N + 2}. The other cases are:
+• If j ∈ {1, . . . , ℓ + 1} and k ∈ {ℓ + 1, . . . , ℓ + m + 1} and j ̸= k, then
+ρj,k = k − j − 1 +
+ℓ
+�
+i=j
+αi +
+k
+�
+i=ℓ+2
+αi−1 + 2
+�
+j≤j0<k0≤k−1
+γj0,k0.
+(90)
+• If j ∈ {ℓ + 2, . . . , ℓ + m + 2} and k ∈ {ℓ + m + 2, . . . , N + 2} and j ̸= k, then
+ρj,k = k − j − 1 +
+ℓ+m+1
+�
+i=j
+βi−1 +
+k
+�
+i=ℓ+m+3
+βi−2 + 2
+�
+j−1≤j0<k0≤k−2
+γj0,k0.
+(91)
+• If j ∈ {0, . . . , ℓ} and k ∈ {ℓ + m + 3, . . . , N + 3} and at least one of j ̸= 0, k ̸= N + 3 holds,
+then
+ρj,k = k − j − N − 4 −
+j
+�
+i=1
+αi −
+j
+�
+i=1
+βi −
+N+2
+�
+i=k
+αi−2 −
+N+2
+�
+i=k
+βi−2 − 2
+j
+�
+j′=1
+N
+�
+i=1,i̸=j′
+γi,j′
+− 2
+N
+�
+k′=k−2
+N
+�
+i=1,i̸=k′−2
+γi,k′ + 2
+�
+1≤j′<k′≤j
+γj′,k′ + 2
+�
+k−2≤j′<k′≤N
+γj′,k′ + 2
+j
+�
+j′=1
+N
+�
+k′=k−2
+γj′,k′.
+(92)
+Proposition 2.3. Given any α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2, the Selberg-
+like integrand
+N
+�
+i=1
+|xi|αi|1 − xi|βi
+�
+1≤j<k≤N
+(xk − xj)2γj,k|dx1 · · · dxN| ∈ C∞(△◦
+ℓ,m,n; Ω△◦
+ℓ,m,n)
+(93)
+lifts, via the blowdown map bd : Kℓ,m,n → △ℓ,m,n, to an extendable density of the form
+�
+�
+{j,k}∈Jℓ,m,n
+xρj,k
+Fj,k
+�
+µℓ,m,n(α, β, γ),
+(94)
+for some strictly positive smooth density µℓ,m,n(α, β, γ) ∈ C∞(Kℓ,m,n; ΩKℓ,m,n), depending entirely
+on α, β, γ.
+■
+Proof. Each ρj,k is an affine function of α, β, γ, so it suffices to check 2N + N(N − 1)/2 + 1 cases,
+the case when all three of α, β, γ are zero and 2N + N(N − 1)/2 cases where the triple (α, β, γ)
+ranges over a basis of C2N+N(N−1)/2. Write
+ρj,k(α, β, γ) = ρ(0)
+j,k + ρ(1)
+j,k(α, β, γ),
+(95)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+19
+where ρ(0)
+j,k = ρj,k(0, 0, 0) and ρ(1)
+j,k(α, β, γ) = ρj,k(α, β, γ) − ρ(0)
+j,k is the linear part of ρj,k. Thus, we
+want to show that, upon lifting to Kℓ,m,n,
+|dx1 · · · dxN| ∈
+�
+�
+{j,k}∈Jℓ,m,n
+x
+ρ(0)
+j,k
+Fj,k
+�
+C∞(Kℓ,m,n; ΩKℓ,m,n),
+(96)
+N
+�
+i=1
+|xi|αi|1 − xi|βi
+�
+1≤j<k≤N
+(xk − xj)2γj,k ∈
+�
+�
+{j,k}∈Jℓ,m,n
+x
+ρ(1)
+j,k(α,β,γ)
+Fj,k
+�
+C∞(Kℓ,m,n; R+),
+(97)
+with it sufficing to check eq. (97) on a basis of C2N+N(N−1)/2.
+• Equation (96) is simply a restatement of Proposition 2.2.
+• For a basis of C2N+N(N−1)/2, we look at (α, β, γ) such that all of the components α1, . . . , αN,
+β1, . . . , βN, γ1,2, · · · of α, β, γ are all 0 except for one, which we set to 1. The result then
+follows, via a bit of algebra, from eq. (71) through eq. (87).
+□
+Let T(ℓ, m, n) denote the collection of maximal families I of consecutive subsets I ⊊ Z/(N + 3)Z
+such that
+• 2 ≤ |I| ≤ N + 1 for all I ∈ I,
+• no two of 0, ℓ + 1, ℓ + m + 2 are in any I ∈ I together, and
+• if I, I′ ∈ I satisfy I ∩ I′ ̸= ∅, then either I ⊆ I′ or I′ ⊆ I.
+The elements of T(ℓ, m, n) can be thought of as specifying valid ways of adding parentheses to group
+together the elements of Z/(N + 3)Z without grouping any of 0, ℓ + 1, ℓ + m + 2 together. The
+minimal facets of Kℓ,m,n are in bijective correspondence with the elements of T(ℓ, m, n), with
+fI =
+�
+I(j,k)∈I
+Fj,k
+(98)
+the facet corresponding to I.
+2.2. The Associahedra Aℓ,m,n. We now define the mwc Aℓ,m,n for ℓ, m, n ∈ N not all zero. We
+begin with the N = ℓ + m + n cube □N = [0, 1]N
+t , which we identify with
+[−∞, 0]ℓ
+x1,...,xℓ × [0, 1]m
+xℓ+1,...,xℓ+m × [1, ∞]n
+xℓ+m+1,...,xN
+(99)
+via the coordinate changes ti = 1/(1 − xi) for xi ∈ [−∞, 0] and i ∈ {1, . . . , ℓ} and ti = (xi − 1)/xi
+for xi ∈ [1, ∞] and i ∈ {ℓ + m + 1, . . . , N}.
+The facets of □N we label by sextuples (S, Q, S′, Q′, S′′, Q′′) consisting of (possibly empty)
+subsets S, Q ⊆ {1, . . . , ℓ}, S′, Q′ ⊆ {ℓ + 1, . . . , ℓ + m}, and S′′, Q′′ ⊆ {ℓ + m + 1, . . . , N} such that
+S ∩ Q = S′ ∩ Q′ = S′′ ∩ Q′′ = ∅. Let
+FS,Q,S′,Q′,S′′,Q′′ =
+�
+(t1, . . . , tN) ∈ □N :
+s ∈ S ∪ S′ ∪ S′′ ⇒ ts = 0
+q ∈ Q ∪ Q′ ∪ Q′′ ⇒ tq = 1
+�
+.
+(100)
+For instance, □N = F∅,∅,∅,∅,∅,∅.
+Now let Fℓ,m,n = Fℓ,m,n(□) denote the family of facets defined by
+Fℓ,m,n = ({FS,∅,∅,∅,∅,Q′′}S,Q′′ ∪ {F∅,Q,S′,∅,∅,∅}Q,S′ ∪ {F∅,∅,∅,Q′,S′′,∅}Q′,S′′)\{□N}
+(101)
+where S, S′, S′Q, Q′, Q′′ range over all subsets as above. For each d ∈ {0, . . . , N − 1}, let Fℓ,m,n;d
+denote the set of elements of Fℓ,m,n of dimension d. Then, Aℓ,m,n is defined by the iterated blowup
+Aℓ,m,n = [□N; Fℓ,m,n] = [□N; Fℓ,m,n;0; · · · ; Fℓ,m,n;N−1].
+(102)
+
+20
+ETHAN SUSSMAN
+t2
+t3
+t1
+A1,1,1
+A1,2,0
+A0,3,0
+Figure 7. The 3-cube □3 and the three blowups A1,1,1 = K1,1,1, A1,2,0, A0,3,0
+thereof. The □ℓ,m,n(S, S′, S′′) are eight subcubes corresponding to the eight vertices
+of □3. One such cube is depicted in red.
+As in the previous section, we should check that, for each d = 1, . . . , N, having already blown up
+Fℓ,m,n;d0 for d0 < d, the blowups of the closures of the lifts of the interiors of all of the F ∈ Fℓ,m,n;d
+all commute. One way to see this is to split
+□N =
+�
+S,S′,S′′
+□ℓ,m,n(S, S′, S′′),
+(103)
+where S varies over all subsets of {1, . . . , ℓ}, S′ varies over all subsets of {ℓ + 1, . . . , ℓ + m}, and S′′
+varies over all subsets of {ℓ + m + 1, . . . , N}, and
+□ℓ,m,n(S, S′, S′′) =
+�
+(t1, . . . , tN) ∈ □N : i ∈ S ∪ S′ ∪ S′′ ⇒ ti ∈ [0, 2/3)
+i /∈ S ∪ S′ ∪ S′′ ⇒ ti ∈ (1/3, 1]
+�
+.
+(104)
+Once we have established that blowing up Fℓ,m,n;0, · · · , Fℓ,m,n;d−1 is fine, then
+[□N; Fℓ,m,n;0; · · · ; Fℓ,m,n;d−1] =
+�
+S,S′,S′′
+[□ℓ,m,n(S, S′, S′′); Fℓ,m,n;0; · · · ; Fℓ,m,n;d−1]
+(105)
+naturally, with the left-hand side being well-defined if the right-hand side is. Thus, it suffices to
+check that the blowups [□ℓ,m,n(S, S′, S′′); Fℓ,m,n;0; · · · ; Fℓ,m,n;d] are all well-defined. To see this,
+identify
+□ℓ,m,n(S, S′, S′′) =
+��
+0, 2
+3
+�S
+{ti}i∈S
+×
+�1
+3, 1
+�(S′′)∁
+{ti}i∈(S′′)∁
+�
+×
+��
+0, 2
+3
+�S′
+{ti}i∈S′ ×
+�1
+3, 1
+�S∁
+{ti}i∈S∁
+�
+×
+��
+0, 2
+3
+�S′′
+{ti}i∈S′′ ×
+�1
+3, 1
+�(S′)∁
+{ti}i∈(S′)∁
+�
+(106)
+and note that the blowup prescription is just that of performing the total boundary blowup [HMM97]
+on each of the three factors. (Note that this is not the same as the total boundary blowup of the
+product of the factors.) Here,
+• S∁ = {1, . . . , ℓ}\S,
+• (S′)∁ = {ℓ + 1, . . . , ℓ + m}\S′,
+• and (S′′)∁ = {ℓ + m + 1, . . . , N}\S′′.
+Thus,
+Aℓ,m,n =
+�
+S,S′,S′′
+�
+��
+0, 2
+3
+�S
+×
+�1
+3, 1
+�(S′′)∁�
+tb ×
+��
+0, 2
+3
+�S′
+×
+�1
+3, 1
+�S∁�
+tb ×
+��
+0, 2
+3
+�S′′
+×
+�1
+3, 1
+�(S′)∁�
+tb
+�
+.
+(107)
+The faces of Aℓ,m,n are in bijection with the elements of Fℓ,m,n. We label the faces of Aℓ,m,n as
+follows:
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+21
+F∅,{3};0
+F∅,{1};∞
+F∅,{2};0
+F∅,{2,3};0
+F∅,{1};0
+F∅,{2};1
+F∅,{3};1
+F∅,{2,3};1
+F{1},{2,3};0
+F{1},{3};0
+F{1},{2};0
+F∅,{3};0
+F∅,{1};0
+F∅,{2};0
+F∅,{1,2};0
+F∅,{2,3};0
+F∅,{1,3};0
+F∅,{1,2,3};0
+F∅,{3};1
+F∅,{1,2,3};1
+F∅,{1,2};1
+F∅,{2};1
+F∅,{1};1
+F∅,{1,3};1
+F∅,{2,3};1
+Figure 8. The eleven faces of A1,2,0 and the fourteen faces of A0,3,0.
+• for S ⊆ {1, . . . , ℓ} and Q ⊆ {ℓ + m + 1, . . . , N}, the face corresponding to FS,∅,∅,∅,∅,Q is
+labeled as FS,Q;∞ = FQ,S;∞,
+• for Q ⊆ {1, . . . , ℓ} and S ⊆ {ℓ + 1, . . . , ℓ + m}, the face corresponding to F∅,Q,S,∅,∅,∅ is
+labeled as as FS,Q;0 = FQ,S;0, and
+• for S ⊆ {ℓ+m+1, . . . , N} and Q ⊆ {ℓ+1, . . . , ℓ+m}, the face corresponding to F∅,∅,∅,Q,S,∅
+is labeled as FS,Q;1 = FQ,S;1.
+Here, S, Q are not allowed to both be empty.
+For any subsets S ⊆ {1, . . . , ℓ}, Q ⊆ {ℓ + 1, . . . , ℓ + m}, R ⊆ {ℓ + m + 1, . . . , N} that are not
+all empty, let forg : □ℓ,m,n → □|S|,|Q|,|R| denote the forgetful map forgetting the coordinates xi for
+i /∈ S ∪ Q ∪ R, Then, forg lifts to a smooth b-map
+forg : Aℓ,m,n → A|S|,|Q|,|R|,
+(108)
+and given any face F of A|S|,|Q|,|R|, the pullback forg
+∗xF vanishes to first order at each face F0
+satisfying
+F0 ⊆ forg
+−1(F).
+(109)
+This is the “universal property” of the Aℓ,m,n. Via the decomposition in eq. (107), it follows from
+the corresponding universal property of the total boundary blowup of a product, which is essentially
+given by Proposition B.2.
+Proposition 2.4. Suppose that µ is a strictly positive smooth density on □ℓ,m,n. Then, the lift of
+µ to Aℓ,m,n has the form
+�
+�
+S⊆{1,...,ℓ}
+Q⊆{ℓ+m+1,...,N}
+x|S∪Q|−1
+FS,Q;∞
+��
+�
+S⊆{1,...,ℓ}
+Q⊆{ℓ+1,...,ℓ+m}
+x|S∪Q|−1
+FS,Q;0
+��
+�
+S⊆{ℓ+1,...,ℓ+m}
+Q⊆{ℓ+m+1,...,N}
+x|S∪Q|−1
+FS,Q;1
+�
+µ
+(110)
+for a strictly positive smooth density µ ∈ C∞(Aℓ,m,n; ΩAℓ,m,n) on Aℓ,m,n.
+■
+As a notational convenience, we are setting xF∅,∅;x0 = 1 for each x0 ∈ {0, 1, ∞}.
+Proof. Follows via induction on the number of blowups, as in the proof of Proposition 2.1.
+□
+Proposition 2.5. The Lebesgue measure on RN, which defines a strictly positive smooth density
+on □◦
+ℓ,m,n, has the form
+�
+ℓ
+�
+j=1
+(1 − xj)2��
+N
+�
+j=ℓ+m+1
+x2
+j
+�
+µ
+(111)
+
+22
+ETHAN SUSSMAN
+for some strictly positive smooth density µ ∈ C∞(□ℓ,m,n; Ω□ℓ,m,n) on □ℓ,m,n.
+■
+Proof. Follows from the same computation as in Proposition 2.2.
+□
+Proposition 2.6. For each pair of distinct i, j ∈ {1, . . . , N} such that either i, j ∈ {1, . . . , ℓ},
+i, j ∈ {ℓ + 1, . . . , ℓ + m}, or i, j ∈ {ℓ + m + 1, . . . , N}, the set Hj,k = clAℓ,m,n{p ∈ □◦
+ℓ,m,n : xj = xk}
+is a p-submanifold of Aℓ,m,n.
+■
+See [MS08, §1.2] for the definition of “p-submanifold.”
+Proof. Consider the neighborhood bd−1(□ℓ,m,n(S, S′, S′′)) ⊆ Aℓ,m,n. If one of j, k is in S ∪ S′ ∪ S′′
+and the other is not, then the intersection of Hj,k with bd−1(□ℓ,m,n(S, S′, S′′)) is a submanifold
+disjoint from the boundary and therefore a p-submanifold. It therefore suffices to consider the case
+when j, k ∈ S ∪ S′ ∪ S′′ (and the case when neither are in S ∪ S′ ∪ S′′ is similar). For notational
+simplicity, we only consider the case when j, k ∈ S′. Then,
+Hj,k ∩ bd−1(□ℓ,m,n(S, S′, S′′)) =
+��
+0, 2
+3
+�S
+×
+�1
+3, 1
+�(S′′)∁�
+tb ×
+� ˜Hj,k ∩
+��
+0, 2
+3
+�S′
+×
+�1
+3, 1
+�S∁�
+tb
+�
+×
+��
+0, 2
+3
+�S′′
+×
+�1
+3, 1
+�(S′)∁�
+tb,
+(112)
+where ˜Hj,k is the closure of {xj = xk} in ([0, 2/3)S′ × (1/3, 1]S∁)tb, which is a p-submanifold [MS08]
+(this also follows from Proposition B.1). Thus, Hj,k ∩ bd−1(□ℓ,m,n(S, S′, S′′)) is a p-submanifold of
+bd−1(□ℓ,m,n(S, S′, S′′)). As the neighborhoods bd−1(□ℓ,m,n(S, S′, S′′)) cover Aℓ,m,n, the conclusion
+follows.
+□
+This result is illustrated in Figure 9.
+We now record the results of lifting xi and 1−xi to Aℓ,m,n, these being derivable via the universal
+property.
+• If i ∈ {1, . . . , ℓ}, then
+−xi ∈
+�
+�
+i∈Q⊆{1,...,ℓ}
+S⊆{ℓ+m+1,...,N}
+x−1
+FS,Q;∞
+��
+�
+i∈S⊆{1,...,ℓ}
+Q⊆{ℓ+1,...,ℓ+m}
+xFS,Q;0
+�
+C∞(Aℓ,m,n; R+),
+(113)
+(1 − xi) ∈
+�
+�
+i∈Q⊆{1,...,ℓ}
+S⊆{ℓ+m+1,...,N}
+x−1
+FS,Q;∞
+�
+C∞(Aℓ,m,n; R+).
+(114)
+• If i ∈ {ℓ + 1, . . . , ℓ + m}, then
+xi ∈
+�
+�
+S⊆{1,...,ℓ}
+i∈Q⊆{ℓ+1,...,ℓ+m}
+xFS,Q;0
+�
+C∞(Aℓ,m,n; R+),
+(115)
+(1 − xi) ∈
+�
+�
+i∈S⊆{ℓ+1,...,ℓ+m}
+Q⊆{ℓ+m+1,...,N}
+xFS,Q;1
+�
+C∞(Aℓ,m,n; R+).
+(116)
+• If i ∈ {ℓ + m + 1, . . . , N}, then
+xi ∈
+�
+�
+i∈S⊆{ℓ+m+1,...,N}
+Q⊆{1,...,ℓ}
+x−1
+FS,Q;∞
+�
+C∞(Aℓ,m,n; R+),
+(117)
+−(1 − xi) ∈
+�
+�
+S⊆{ℓ+1,...,ℓ+m}
+i∈Q⊆{ℓ+m+1,...,N}
+xFS,Q;1
+��
+�
+i∈S⊆{ℓ+m+1,...,N}
+Q⊆{1,...,ℓ}
+x−1
+FS,Q;∞
+�
+C∞(Aℓ,m,n; R+).
+(118)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+23
+Let I1 = {1, . . . , ℓ}, I2 = {ℓ + 1, . . . , ℓ + m}, and I3 = {ℓ + m + 1, . . . , N}. For j, k ∈ I• for the
+same • ∈ {1, 2, 3}, let yj,k denote a defining function of Hj,k, with the sign chosen so as to have the
+same sign as xj − xk. Then, for all distinct j, k ∈ {1, . . . , N},
+(xj − xk) ∈ Yj,kXj,kC∞(Aℓ,m,n; R+),
+(119)
+where Xj,k = Xk,j is given by
+Xj,k =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+� �
+S⊆I3
+j∈Q⊆I1 or k∈Q⊆I1
+x−1
+FS,Q;∞
+�� �
+j,k∈S⊆I1
+Q⊆I2
+xFS,Q;0
+�
+(j, k ∈ I1),
+� �
+S⊆I1
+j,k∈Q⊆I2
+xFS,Q;0
+�� �
+j,k∈S⊆I2
+Q⊆I3
+xFS,Q;1
+�
+(j, k ∈ I2),
+� �
+S⊆I2
+j,k∈Q⊆I3
+xFS,Q;1
+�� �
+j∈S⊆I3 or k∈S⊆I3
+Q⊆I1
+x−1
+FS,Q;∞
+�
+(j, k ∈ I3),
+� �
+j∈S⊆I1
+Q⊆I3
+x−1
+FS,Q;∞
+�� �
+j∈S⊆I1
+k∈Q⊆I2
+xFS,Q;0
+�
+(j ∈ I1, k ∈ I2),
+� �
+S⊆I1
+k∈Q⊆I3
+x−1
+FS,Q;∞
+�� �
+j∈S⊆I2
+k∈Q⊆I3
+xFS,Q;1
+�
+(j ∈ I2, k ∈ I3),
+� �
+S⊆I1,Q⊆I3
+{j,k}∩S∪Q̸=∅
+x−1
+FS,Q;∞
+�
+(j ∈ I3, k ∈ I1),
+(120)
+and Yj,k = yj,k if, for some • ∈ {1, 2, 3}, we have j, k ∈ I•, and Yj,k = 1 otherwise.
+We associate to each face F• ∈ F(Aℓ,m,n) an affine functional
+ϱ• : C2N+N(N−1)/2 ∋ (α, β, γ) �→ ϱF•(α, β, γ) ∈ C.
+(121)
+Suppose that we are given some α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2. Then,
+ϱ•(α, β, γ) is defined as follows:
+• For S ⊆ {1, . . . , ℓ} and Q ⊆ {ℓ + 1, . . . , ℓ + m},
+ϱS,Q;0 = |S| + |Q| − 1 +
+�
+j∈S∪Q
+αj + 2
+�
+j,k∈S∪Q
+j>k
+γj,k.
+(122)
+• For S ⊆ {ℓ + 1, . . . , ℓ + m} and Q ⊆ {ℓ + m + 1, . . . , N},
+ϱS,Q;1 = |S| + |Q| − 1 +
+�
+j∈S∪Q
+βj + 2
+�
+j,k∈S∪Q
+j>k
+γj,k.
+(123)
+• For S ⊆ {ℓ + m + 1, . . . , N} and Q ⊆ {1, . . . , ℓ},
+ϱS,Q;∞ = −|S| − |Q| − 1 −
+�
+j∈S∪Q
+αj −
+�
+j∈S∪Q
+βj − 2
+�
+j>k
+j∈S∪Q or k∈S∪Q
+γj,k.
+(124)
+Then, letting ∆ ⊂ □ℓ,m,n be defined by ∆ = ∪3
+•=1 ∪j̸=k,j,k∈I• {xj = xk}:
+Proposition 2.7. Given any α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2,
+N
+�
+i=1
+|xi|αi|1−xi|βi
+�
+1≤j<k≤N
+(xk −xj +i0)2γj,k|dx1 · · · dxN| ∈ C∞(□◦
+ℓ,m,n\∆; C⊗Ω(□◦
+ℓ,m,n\∆)) (125)
+lifts, via the blowdown map bd : Aℓ,m,n → □ℓ,m,n, to
+�
+�
+S⊆{1,...,ℓ}
+Q⊆{ℓ+1,...,ℓ+m}
+xϱS,Q;0
+FS,Q;0
+��
+�
+S⊆{ℓ+1,...,ℓ+m}
+Q⊆{ℓ+m+1,...,N}
+xϱS,Q;1
+FS,Q;1
+��
+�
+S⊆{ℓ+m+1,...,N}
+Q⊆{1,...,ℓ}
+xϱS,Q;∞
+FS,Q;∞
+��
+�
+1≤j<k≤ℓ
+(yk,j + i0)2γj,k
+�
+�
+�
+ℓ+1≤j<k≤ℓ+m
+(yk,j + i0)2γj,k
+��
+�
+ℓ+m+1≤j<k≤N
+(yk,j + i0)2γj,k
+�
+µℓ,m,n
+(126)
+
+24
+ETHAN SUSSMAN
+A1,2,0
+A0,3,0
+Figure 9. The sets Hj,k ∈ P in A1,2,0 and A0,3,0. Pictured are H1,2, H1,3, and H2,3,
+where the axes are oriented as in Figure 7. In A0,3,0, the intersection H1,2∩H1,3∩H2,3
+has been indicated with an extra dashed line.
+for some strictly positive smooth density µℓ,m,n ∈ C∞(Aℓ,m,n; ΩAℓ,m,n) on Aℓ,m,n, depending entirely
+on α, β, γ.
+■
+Proof. Follows from the preceding computations, along with Proposition 2.5.
+□
+If M is an orientable mwc, we say that a collection P of interior p-submanifolds each of codimension
+one is consistently orientable if we can choose an orientation on each such that, for any p ∈ M, the
+subset
+�
+P∈P,p∈P
+++N∗
+p P ⊂ T ∗
+p M
+(127)
+does not contain zero, where ++N∗P ⊂ +N∗P ⊂ T ∗M is the induced positively oriented conormal
+bundle, sans the zero section, and T ∗M is the extendable cotangent bundle of M. Whether or
+not this holds does not depend on the choice of orientation of M. Choosing defining functions
+{yP }P∈P ⊂ C∞(M; R) for the P ∈ P such that
+dyP (p) ∈ +N∗
+p P
+(128)
+for each p ∈ P, we say that the {yP }P∈P are consistently oriented defining functions.
+Example. In □◦
+0,3,0 = (0, 1)3, consider P = {H◦
+1,2, H◦
+2,3, H◦
+3,1}. The functions x2 −x1, x3 −x2, x1 −x3
+are not consistently oriented defining functions, as
+0 = d(x2 − x1) + d(x3 − x2) + d(x1 − x3),
+(129)
+but x2 − x1, x3 − x2, and x3 − x1 are.
+■
+Let
+P = {Hj,k}j,k∈I1,j̸=k ∪ {Hj,k}j,k∈I2,j̸=k ∪ {Hj,k}j,k∈I3,j̸=k.
+(130)
+Proposition 2.8. The collection P defined by eq. (130) is consistently orientable, and {yk,j}j<k is
+a set of consistently oriented defining functions.
+■
+Proof. We will show that, for any p ∈ Aℓ,m,n and {λj,k}p∈Hj,k∈P ∈ [0, ∞), if the 1-form
+�
+Hj,k∈P s.t. p∈Hj,k
+λj,kdyk,j ∈ Ω1(Aℓ,m,n)
+(131)
+vanishes at p, then λj,k = 0 for all Hj,k ∈ P such that p ∈ Hj,k. Put differently, we want to
+show that if P is any partition of {1, . . . , N} into nonempty subsets S ⊂ I1, I2, I3, then, given any
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+25
+{λj,k}j,k∈S∈P,j<k ⊂ R≥0 not all zero, then
+�
+j,k∈S∈P
+j<k
+λj,k dyk,j
+(132)
+is nonvanishing on ∩j,k∈S∈P,j<kHj,k. If P consists only of singletons, then this is vacuously true, so
+it suffices to consider the case when at least one member of P has cardinality > 1.
+This is certainly true for p ∈ □◦
+ℓ,m,n, as dyk,j ∝ dxk − dxj on □◦
+ℓ,m,n ∩ Hj,k, where the coefficient
+of proportionality is positive. Indeed, by the results above,
+xk − xj = fj,kyk.j
+(133)
+for some fj,k ∈ C∞(Aℓ,m,n; R≥0) that is nonvanishing in the interior, so
+dyk,j = f−1
+j,k (dxk − dxj) − f−1
+j,k yk,j dfj,k
+(134)
+in □◦
+ℓ,m,n, which is equal to f−1
+j,k (dxk − dxj) on Hj,k ∩ □◦
+ℓ,m,n, as claimed. This argument does not
+work for p ∈ ∂Aℓ,m,n, as fj,k may vanish there.
+A homogeneity argument can be used to show that, for any p ∈ ∂Aℓ,m,n, there exists a tubular
+neighborhood T : U → U0 of a neighborhood U0 ⊂ F0 of p in F0, where F0 is the smallest facet
+containing p, such that the intersections U ∩ P of this neighborhood with the P ∈ P are all vertical
+subsets, meaning of the form T −1(B) for some B ⊂ U0. This implies that if the 1-form above
+vanishes at p, then it also vanishes on the fiber of the tubular neighborhood over p and hence
+somewhere in □◦
+ℓ,m,n ∩Hj,k∋p Hj,k.
+□
+We illustrate the preceding argument with an example. Consider the case when the only one
+of ℓ, m, n that is nonzero is m, and consider p ∈ ∩Hj,k∈PHj,k. The set ∩Hj,k∈PHj,k ⊂ A0,N,0 (the
+“small diagonal”) is a p-submanifold located away from all but the very first two blowups involved
+in the construction of A0,N,0. Near this p-submanifold, A0,N,0 is canonically diffeomorphic to
+[□0,N,0, {0}, {1}], the result of blowing up two opposite corners of the N-cube. We consider the
+situation near the blowup of
+{0} = F∅,∅,{1,...,N},∅,∅,∅,
+(135)
+and the situation near the opposite corner is similar. In the interior of the front face of that blowup,
+we can use ϱ = x1 as a bdf and coordinates ˆxj = xj/x1 for j = 2, . . . , N as parametrizing the face
+itself. In terms of these coordinates,
+∩Hj,k∈P Hj,k = {ˆx2, · · · , ˆxN = 1}
+(136)
+locally, and, for 1 ≤ j < k ≤ N, we can write yk,j = ˜yk,jC∞(A0,N,0; R+) for ˜yk,j given locally by
+˜yk,j = ϱ−1(xk − xj) = ˆxk − ˆxj, where ˆx1 = 1. This satisfies
+d˜yk,j =
+�
+dˆxk
+(j = 1),
+dˆxk − dˆxj
+(j ̸= 1).
+(137)
+So, if λk,j ≥ 0, then �
+1≤j<k≤N λj,kd˜yk,j = 0 ⇒ λj,k = 0 for all k, j. Since the yk,j differ from the
+˜yk,j by a (smooth) positive factor, the yk,j have the same property on ∩Hj,k∈PHj,k.
+There is a more direct argument using the coordinates in Proposition B.1 (with the decomposition
+eq. (107)). Namely, using eq. (107), the result follows from the analogous result for [0, 1)N
+tb. Given
+any σ ∈ SN, consider the coordinates ϱ, ˆxσ(2), · · · , ˆxσ(N) defined in Proposition B.1, these giving a
+C∞-atlas as σ varies over all permutations. In these coordinate systems, the relevant p-submanifolds
+are, locally,
+Hj,k = {ˆxj+1 · · · ˆxk = 1} ⊂ [0, 1)N
+tb,
+(138)
+
+26
+ETHAN SUSSMAN
+so have defining functions yk,j = −1 + ˆxj+1 · · · ˆxk. This satisfies
+dyk,j =
+k
+�
+i=j+1
+dˆxi
+ˆxi
+(139)
+on Hj,k. The 1-forms, ωj,k = �k
+i=j+1 ˆx−1
+i
+dˆxi, defined by the right-hand side of eq. (139) satisfy
+{λj,k}1≤j<k≤N and j,k∈S∈P ⊂ [0, ∞) and
+�
+1≤j<k≤N and j,k∈S∈P
+λj,kωj,k = 0
+⇒ λj,k = 0 for all j < k in S ∈ P,
+(140)
+from which the result follows.
+Let ΣT(ℓ, m, n) denote the collection of maximal families I of pairs (x0, S) of x0 ∈ {0, 1, ∞} and
+nonempty S ⊆ {1, . . . , N} such that
+• if (x0, S), (x0, Q) ∈ I, either S ⊆ Q or Q ⊆ S,
+•
+(x0, S) ∈ I ⇒
+�
+�
+�
+�
+�
+�
+�
+S ∩ I3 = ∅
+(x0 = 0),
+S ∩ I1 = ∅
+(x0 = 1),
+S ∩ I2 = ∅
+(x0 = ∞).
+(141)
+The minimal facets of Aℓ,m,n are in bijective correspondence with the elements of ΣT(ℓ, m, n), with
+fI =
+�
+(x0,S)∈I, S∪Q⊆S
+FS,Q;x0
+(142)
+the facet corresponding to I.
+3. Meromorphic continuation
+We now turn to the analytic extension of Selberg-like integrals to dense, open subsets of the space
+of possible exponents. As discussed in the introduction, the results in this section are apparently
+sharp for generic Selberg-like integrals, but for e.g. symmetric Selberg-like integrals they are only
+preliminary. Nevertheless, the results we prove here will be useful in establishing the sharp results
+later. For our discussion of the symmetric and DF-symmetric cases, it is useful to consider somewhat
+more general integrals than eq. (2). Let ℓ, m, n ∈ N satisfy ℓ + m + n = N ∈ N+. Fix a finite
+collection D of indexed sets
+{dF}F∈F(Kℓ,m,n) ⊆ R.
+(143)
+Define
+Sℓ,m,n[F](α, β, γ) =
+�
+△ℓ,m,n
+� N
+�
+i=1
+|xi|αi|1 − xi|βi
+��
+�
+1≤j<k≤N
+(xk − xj)2γj,k
+�
+F dx1 · · · dxN,
+(144)
+for (α, β, γ) ∈ Ωℓ,m,n[D], where
+• Ωℓ,m,n[D] denotes the set of (α, β, γ) ∈ CN
+α × CN
+β × CN(N−1)/2
+γ
+such that
+� N
+�
+i=1
+|xi|αi|1 − xi|βi
+��
+�
+1≤j<k≤N
+(xk − xj)2γj,k
+��
+�
+F∈F(Kℓ,m,n)
+xdF
+F
+�
+∈ L1(△ℓ,m,n, dx1 · · · dxN)
+(145)
+for all {dF}F∈F(Kℓ,m,n) ∈ D, and
+• F has the form
+F =
+�
+{dF}F∈F(Kℓ,m,n)
+�
+�
+F∈F(Kℓ,m,n)
+xdF
+F
+�
+F{dF}F∈F(Kℓ,m,n)
+(146)
+for some F{dF}F∈F(Kℓ,m,n) ∈ C∞(Kℓ,m,n).
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+27
+We denote the set of such F by AD(Kℓ,m,n). Observe that Ωℓ,m,n[D] is a nonempty, open, and
+connected subset of C2N+N(N−1)/2. In the case N = 1, we consider Ωℓ,m,n[D] as a subset of C2
+α,β.
+We write Ω[F] to denote Ωℓ,m,n[D] for arbitrary D such that F ∈ AD(Kℓ,m,n). Let
+Ωℓ,m,n = Ωℓ,m,n[{{0}F∈F(Kℓ,m,n)}].
+(147)
+If ϕ ∈ C∞(△ℓ,m,n), then we can consider ϕ as an element of C∞(Kℓ,m,n), so Sℓ,m,n[ϕ] : Ωℓ,m,n → C
+is well-defined, and Ωℓ,m,n[ϕ] ⊇ Ωℓ,m,n. If f ∈ C[x1, . . . , xN], then the lift of fϕ is also a classical
+symbol on Kℓ,m,n (it is smooth if ℓ, n = 0, but not necessarily otherwise), so
+Sℓ,m,n[fϕ] : Ωℓ,m,n[f] → C
+(148)
+is well-defined, except now we may have Ωℓ,m,n[fϕ] ̸⊆ Ωℓ,m,n if ℓ ̸= 0 or n ̸= 0.
+In the special case when ℓ, n = 0 and m = N, we use the abbreviations Ω0,N,0 = ΩN, Ω0,N,0[•] =
+ΩN[•], and
+S0,N,0[F](α, β, γ) = SN[F](α, β, γ),
+(149)
+this being consistent with our earlier notation.
+As in the introduction, when α, β, γ are constant, we just write ‘α’ in place of ‘α,’ ‘β’ in place of
+‘β,’ and ‘γ’ in place of ‘γ.’ Let Uℓ,m,n[•] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ Ωℓ,m,n[•]
+holds when α = α, β = β, and γ = γ.
+Similar abbreviations will be used throughout the rest of this paper.
+In addition to the general Selberg-like integral above, we have the following general integral of
+Dotsenko–Fateev type:
+Iℓ,m,n[F](α, β, γ) =
+�
+□ℓ,m,n
+� N
+�
+i=1
+|xi|αi|1 − xi|βi
+��
+�
+1≤j<k≤N
+(xk − xj + i0)2γj,k
+�
+F dx1 · · · dxN
+(150)
+for (α, β, γ) ∈ Vℓ,m,n[D], where now D denotes a finite collection of indexed sets {dF}F∈F(Aℓ,m,n) ⊆ C,
+• Vℓ,m,n[D] denotes the set of (α, β, γ) ∈ C2N+N(N−1)/2 for which the integrand in eq. (150)
+lies in L1(□ℓ,m,n, dx1 · · · dxN) – that is the set of (α, β, γ) such that
+� N
+�
+i=1
+|xi|αi|1 − xi|βi
+��
+�
+1≤j<k≤N
+|xk − xj|2γj,k
+��
+�
+F∈F(Kℓ,m,n)
+xdF
+F
+�
+∈ L1(□ℓ,m,n, dx1 · · · dxN)
+(151)
+for all {dF}F∈F(Aℓ,m,n) ∈ D, and
+• F has the form eq. (146) for F{dF}F∈F(Aℓ,m,n) ∈ C∞(Aℓ,m,n).
+In eq. (150), xγ = eπiγeγ log |x| if x < 0 and xγ = eγ log x if x > 0. We apply abbreviations for
+Dotsenko–Fateev-like integrals that are analogous to those used for Selberg-like integrals.
+Let Wℓ,m,n[•] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ Vℓ,m,n[•] holds when α = α,
+β = β, and γ = γ. Let W DF0
+ℓ,m,n[F] denote the set of (α−, α+, β−, β+, γ−, γ0, γ+) ∈ C7 such that
+(αDF0, βDF0, γDF0) ∈ Vℓ,m,n[F]. Let
+IDF0;S
+ℓ,m,n [F](α−, α+, β−, β+, γ−, γ0, γ+) = Iℓ,m,n[F](αDF0, βDF0, γDF0).
+(152)
+This section is split into many short subsections. The general analytic framework in which the
+extension is performed is discussed in §3.1, and the specific application to Selberg-like integrals
+is contained in §3.2. We prove a family of identities relating Iℓ,m,n, Iℓ,n,m, In,ℓ,m, · · · in §3.3. As
+preparation for our discussion of singularity removal in the DF-symmetric case, we discuss in §3.4 an
+alternative regularization procedure suggested by Dotsenko–Fateev that works for some suboptimal
+range of parameters (in particular allowing γ0 = −1, but not allowing the real parts of α−, α+, β−, β+
+to be too negative). The Iℓ,m,n are related to the Selberg-like integrals Sℓ,m,n in §3.5. A key lemma
+used in the removal of singularities is in §3.6. This lemma is a generalization of a result proven by
+Aomoto [Aom87] and discussed heuristically by Dotsenko–Fateev [DF85a]. For completeness and
+
+28
+ETHAN SUSSMAN
+later convenience, we record in §3.7 the symmetric and DF-symmetric cases of the results in §3.2
+regarding the Dotsenko–Fateev integrals.
+Let Sℓ,m,n = Sℓ × Sm × Sn, which we consider as the subgroup of SN leaving each of I1, I2, I3
+invariant, where I1, I2, I3 are as in the previous section. Given a permutation σ ∈ Sℓ,m,n, let
+Iℓ,m,n[F](α, β, γ)σ =
+�
+□ℓ,m,n
+� N
+�
+i=1
+|xi|αi|1−xi|βi
+��
+�
+1≤j<k≤N
+(xσ(k)−xσ(j)+i0)2γσ(j),σ(k)
+�
+F dx1 · · · dxN,
+(153)
+defined for (α, β, γ) ∈ Vℓ,m,n[F]. If we define ασ, βσ, γσ by ασ
+j = ασ(j), βσ
+j = βσ(j), and γσ
+j,k =
+γσ(j),σ(k), and
+F σ(y1, . . . , yN) = F(yσ−1(1), . . . , yσ−1(N)),
+(154)
+then Iℓ,m,n[F](α, β, γ)σ = Iℓ,m,n[F σ](ασ, βσ, γσ). This relation will be very useful below. More
+generally, for any σ ∈ SN, let
+Iℓ,m,n[F](α, β, γ)σ = Iℓ,m,n[F σ](ασ, βσ, γσ)
+(155)
+Sℓ,m,n[F](α, β, γ)σ = Sℓ,m,n[F σ](ασ, βσ, γσ),
+(156)
+defined for (α, β, γ) ∈ Vℓ,m,n[F] in the former case or for
+(α, β, γ) ∈ Ωℓ,m,n[F]σ = {(α, β, γ) ∈ C2N+N(N−1)/2 : (ασ, βσ, γσ) ∈ Ωℓ,m,n[F σ]}
+(157)
+in the latter case. We will use similar notation for other subsets of C2N+N(N−1)/2 below, as well as
+for the meromorphic extensions of Sℓ,m,n[F] and Iℓ,m,n[F].
+3.1. Some Generalities. Let N ∈ N be arbitrary. For a Fréchet space X, let O(CN; X) denote the
+Fréchet space of entire X-valued functions on CN, where the topology is that of uniform convergence
+in compact subsets, as measured with respect to each Fréchet seminorm on X, and similarly for X
+an LF-space. Let E′(RN) denote the LCTVS of compactly supported distributions on RN. By the
+Schwartz representation theorem,
+E′(RN) = ∪m∈RHm,s
+c
+(RN),
+(158)
+where Hm
+c (RN) is the set of compactly supported elements of Hm(RN).
+Let N ∈ N+, k ∈ {0, . . . , N}, and κ ∈ N. For any
+ψ ∈ C∞
+c (Rk
+t1,··· ,tk; E′(RN−k
+tk+1,··· ,tN )) =
+�
+m,s∈R
+C∞
+c (Rk
+t1,··· ,tk; Hm,s
+sc,c (RN−k))
+(159)
+let, for ρ = (ρ1, . . . , ρk),
+IN,k,κ[ψ](ρ) =
+� ∞
+0
+· · ·
+� ∞
+0
+tρ1
+1 · · · tρk
+k ⟨1, ψ(t1, . . . , tk, −)⟩ dt1 · · · dtk,
+(160)
+which we abbreviate as
+IN,k,κ[ψ](ρ) =
+�
+RN
+k
+tρ1
+1 · · · tρk
+k ψ(t) dNt.
+(161)
+Here, RN
+k = [0, ∞)k
+t1,··· ,tk × Rn−k
+tk+1,··· ,tN, and IN,k,κ[ψ](ρ) is defined initially for ℜρ1, · · · , ℜρk > −1,
+for which the right-hand side of eq. (160) is a well-defined integral.
+Let
+O(Ck × Cκ; C∞
+c (Rk
+t1,··· ,tk; E′(RN−k
+tk+1,··· ,tN ))) =
+�
+m,s∈R
+O(Ck × Cκ; C∞
+c (Rk
+t1,··· ,tk; Hm,s
+sc,c (RN−k))), (162)
+endowed with the strongest topology such that the inclusions
+O(Ck × Cκ; C∞
+c (Rk
+t1,··· ,tk; Hm,s
+sc,c (RN−k))) �→ O(Ck × Cκ; C∞
+c (Rk
+t1,··· ,tk; E′(RN−k
+tk+1,··· ,tN )))
+(163)
+are all continuous.
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+29
+Proposition 3.1. Suppose that, for each ρ ∈ Ck and δ ∈ Cκ, we are given some ψ(−; ρ, δ) as in
+eq. (159), depending entirely on ρ, δ in the sense that the map
+Ck × Cκ ∋ (ρ, δ) �→ ψ ∈ C∞
+c (Rk; E′(RN−k))
+(164)
+is entire, i.e. lies in O(Ck × Cκ; C∞
+c (Rk
+t1,··· ,tk; E′(RN−k
+tk+1,··· ,tN ))). Define
+IN,k,κ[ψ](ρ, δ) = IN,k,κ[ψ(ρ, δ)](ρ).
+(165)
+Then, the function JN,k,κ[ψ] defined by
+IN,k,κ[ψ](ρ, δ) =
+�
+k
+�
+j=1
+Γ(ρj + 1)
+�
+JN,k,κ[ψ](ρ, δ)
+(166)
+extends to an entire function on Ck
+ρ × Cκ
+δ. Moreover, the function
+JN,k,κ[−] : O(Ck × Cκ; C∞
+c (Rk
+t1,...,tk; E′(RN−k
+tk+1,...,tN ))) ∋ ψ �→ JN,k,κ[ψ] ∈ O(Ck × Cκ)
+(167)
+is continuous.
+■
+Cf. [GS64][Var95, Lemma 10.7.9].
+Proof. The k = 0 case is essentially tautologous.
+We now proceed inductively on k. Let k ≥ 1, and assume that we have proven the result for
+smaller k. Expanding ψ in Taylor series around t1 = 0, there exist
+ψ(j) ∈ O(Ck × Cκ; C∞
+c (Rk−1
+t2,...,tk; E′(RN−k
+tk+1,...,tN )))
+(168)
+E(j) ∈ O(Ck × Cκ; C∞(Rt1; C∞
+c (Rk−1
+t2,...,tk; E′(RN−k
+tk+1,...,tN ))))
+(169)
+such that
+ψ(t1, · · · , tN; ρ, δ) =
+J
+�
+j=0
+tj
+1ψ(j)(t2, · · · , tN; ρ, δ) + tJ+1
+1
+E(J+1)(t1, · · · , tN; ρ, δ)
+(170)
+for all J ∈ N. Let K ⊂ Ck+κ be an arbitrary nonempty compact set. There exists some T > 0
+such that supp ψ(−; ρ, δ) ⊆ {−T ≤ t1 ≤ T} for all (ρ, δ) ∈ K. Then, if ℜρ1, · · · , ℜρk > −1 and
+(ρ, δ) ∈ K,
+IN,k,κ[ψ](ρ, δ) =
+J
+�
+j=0
+IN−1,k−1,κ[ψ(j)]( ˆρ, δ)
+ρ1 + j + 1
+T ρ1+j+1 +
+� T
+0
+tρ1+J+1
+1
+IN−1,k−1[E(J+1)(t1, −)]( ˆρ, δ) dt1,
+(171)
+where ˆρ = (ρ2, · · · , ρk). We now define JN,k,κ[ψ](ρ, δ) : {ℜρ1 > −2 − J} × Cκ
+δ → C by
+JN,k,κ[ψ](ρ, δ) =
+1
+Γ(ρ1 + 1)
+J
+�
+j=0
+JN−1,k−1,κ[ψ(j)]( ˆρ, δ)
+ρ1 + j + 1
+T ρ1+j+1
++
+1
+Γ(ρ1 + 1)
+� T
+0
+tρ1+J+1
+1
+JN−1,k−1[E(J+1)(t1, −)]( ˆρ, δ) dt1.
+(172)
+By construction, eq. (166) holds when ℜρ1, · · · , ℜρk > −1.
+By the continuity clause of the
+inductive hypothesis, the integral in eq. (171) is a well-defined Bochner integral, for each individual
+(ρ, δ) ∈ {ℜρ1 > −2 − J} × Cκ. Moreover, the right-hand side of eq. (172) depends analytically on
+(ρ, δ) ∈ {ℜρ1 > −1 − J} × Cκ. By the inductive hypothesis, this is true for the sum on the first line
+(multiplied by Γ(ρ1 + 1)−1), as the simple poles due to the factors of 1/(ρ1 + j + 1) cancel with those
+
+30
+ETHAN SUSSMAN
+of Γ(ρ1 + 1). So, in order to show that the whole right-hand side of eq. (172) depends analytically
+on (ρ, δ) in this domain, we can show it for
+� T
+0
+tρ1+J+1
+1
+JN−1,k−1[E(J+1)(t1, −)]( ˆρ, δ) dt1.
+(173)
+Justifying differentiation under the integral sign, this is a C1-function of (ℜρ1, ℑρ1) ∈ {(u, v) ∈
+R2, u > −1 − J}, and it satisfies the Cauchy-Riemann equations, so it follows that the integral in
+eq. (173) is analytic as a function of ρ1 ∈ {ℜρ1 > −1 − J}, for each fixed ˆρ ∈ Ck−1 and δ ∈ Cκ.
+Adding ˆρ, δ-dependence does not change the argument.
+So, the formula eq. (171) yields an analytic extension of IN,k,κ, and we can take a union over all
+J ∈ N, the various partial extensions agreeing with each other via analyticity. The continuity clause
+is evident from the formula eq. (172) and the inductive hypothesis.
+□
+Consequently, IN,k,κ[ψ] admits an analytic continuation ˙IN,k,κ[ψ] : Ω → C to the set Ω =
+(Ck
+ρ\ �
+j∈{1,...,k}{ρj ∈ Z≤−1}) × Cκ
+δ, and the map
+˙IN,k,κ[−] : O(Ck × Cκ; C∞
+c (Rk
+t1,...,tk; E′(RN−k
+tk+1,...,tN ))) ∋ ψ �→ ˙IN,k,κ[ψ] ∈ O(Ω)
+(174)
+is continuous.
+If P is a consistently orientable collection of codimension-1 interior p-submanifolds on a mwc M,
+then, letting xF for F ∈ F(M) denote a bdf of the face F, it is the case that, for any δ ∈ CP and
+ρ ∈ CF(M), the product
+ω(ρ, δ) =
+�
+F∈F(M)
+xρF
+F
+�
+P∈P
+(yP + i0)δP : ˙C∞
+c (M; ΩM) ∋ µ �→ lim
+ε→0+
+�
+M
+�
+F∈F(M)
+�
+P∈P
+xρF
+F (yP + iε)δP µ
+(175)
+is a well-defined classical distribution on M, where {yP }P∈P are consistently oriented defining
+functions. (Here, ˙C∞
+c (M; ΩM) is the set of compactly supported smooth densities on M that are
+Schwartz at each boundary hypersurface.) That is, ω is an extendable distribution on M and defines,
+for small ϵ > 0, an element of C∞([0, ϵ)xF; D′(F)) for each face F. We write the right-hand side of
+eq. (175) as
+�
+M ω(ρ, δ)µ. More generally, if µ ∈ C∞
+c (M; ΩM), then
+lim
+ε→0+
+�
+M
+�
+F∈F(M)
+�
+P∈P
+xρF
+F (yP + iε)δP µ =
+�
+M
+ω(ρ, δ)µ
+(176)
+exists whenever ρF > −1 for all F ∈ F(M).
+Let κ ∈ N. Suppose that we are given some entire family
+µ : CF(M) × CP × Cκ → C∞
+c (M; ΩM)
+(177)
+of compactly supported smooth densities µ(ρ, δ, λ) ∈ C∞
+c (M; ΩM) on M. Consider the function
+I[M, µ](ρ, δ, λ) : {(ρ, δ, λ) ∈ CF(M) × CP × Cκ : ρF > −1 for all F ∈ F(M)} → C
+(178)
+defined by
+I[M, µ](ρ, δ, λ) =
+�
+M
+ω(ρ, δ)µ(ρ, δ, λ).
+(179)
+Proposition 3.2. Suppose that, for some N0 ∈ N+, we are given an affine map L = (L1, L2, L3) :
+CN0
+ϱ
+→ CF(M)
+ρ
+× CP
+δ × Cκ
+λ such that, for each F ∈ F(M), the affine functional
+(L•)F : CN0 ∋ ϱ �→ (L1ϱ)F ∈ C
+(180)
+is nonconstant. Then, there exist entire functions Ireg,f[M, µ](L•) : CN0
+ϱ
+→ C associated to the
+minimal facets f of M such that
+I[M, µ](Lϱ) =
+�
+f
+�
+�
+F∈F(M),F⊇f
+Γ(1 + (Lϱ)F)
+�
+Ireg,f[M, µ](Lϱ)
+(181)
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+31
+for all ϱ ∈ CN0 for which the left-hand side is defined by eq. (179).
+■
+Proof. Pass to a partition of unity subordinate to a system of coordinate charts on M and apply
+Proposition 3.1 locally.
+□
+Then, letting L = {(L•)F : F ∈ F(M)},
+� �
+Λ∈L
+1
+Γ(1 + Λ(ϱ))#Λ
+�
+I[M, µ](Lϱ)
+(182)
+extends to an entire function CN0
+ϱ
+→ C, where #Λ ∈ N+ is the maximum size of any set S ⊆ F(M)
+of faces such that ∩F∈SF ̸= ∅ and (L•)F = Λ for all F ∈ S. Indeed, this follows from the proposition
+above since, for each facet f,
+� �
+Λ∈L
+1
+Γ(1 + Λ(ϱ))#Λ
+�
+�
+F∈F(M),F⊇f
+Γ(1 + (Lϱ)F)
+(183)
+is entire.
+3.2. Specialization to Generic Selberg- and DF-like Integrals. We now apply the results of
+the previous section to the specific case of the integrals eq. (144) and eq. (150). Fix ℓ, m, n ∈ N
+satisfying ℓ + m + n = N, N ∈ N+.
+3.2.1. The Selberg case. Fix F ∈ AD(Kℓ,m,n). Let ρj,k = ρj,k(α, β, γ) be defined by eq. (89), eq. (90),
+eq. (91), and eq. (92). Recalling the definition of T(ℓ, m, n) given in §2.1:
+Proposition 3.3. There exist entire functions
+Sℓ,m,n;reg,I,{dF}F∈F(Kℓ,m,n)[F] : C2N+N(N−1)/2
+α,β,γ
+→ C,
+(184)
+associated to pairs of minimal facets f of Kℓ,m,n and collections {dF}F∈F(ℓ,m,n) ∈ D of weights such
+that
+Sℓ,m,n[F](α, β, γ) =
+�
+I∈T(ℓ,m,n)
+�
+{dF}F∈F(Kℓ,m,n)∈D
+�
+�
+I(j,k)∈I
+Γ(1 + ρj,k + dFj,k)
+�
+× Sℓ,m,n;reg,I,{dF}F∈F(Kℓ,m,n)[F](α, β, γ)
+(185)
+for all (α, β, γ) ∈ Ωℓ,m,n[D].
+■
+Proof. This is a corollary of Proposition 2.3 and Proposition 3.2, using the fact that the minimal
+facets of Kℓ,m,n are in correspondence with the elements of T(ℓ, m, n) via eq. (98).
+□
+Consequently, there exists an analytic extension ˙Sℓ,m,n[F] :
+˙Ωℓ,m,n[D] → C of Sℓ,m,n[F] :
+Ωℓ,m,n[D] → C, where
+˙Ωℓ,m,n[D] = C2N+N(N−1)/2
+α,β,γ
+��
+�
+{dF}F∈F(Kℓ,m,n)∈D
+�
+�
+{j,k}∈Jℓ,m,n
+{ρj,k + dFj,k ∈ Z≤−1}
+��
+.
+(186)
+This is an open and connected subset of full measure; namely, it is the complement of a locally finite
+collection of complex (affine) hyperplanes in C2N+N(N−1)/2. In the case m = N, this agrees with
+eq. (13).
+As a corollary of the previous proposition, there exists an entire function
+Sℓ,m,n;reg[F] : C2N+N(N−1)/2
+α,β,γ
+→ C
+(187)
+such that
+Sℓ,m,n[F](α, β, γ) =
+�
+�
+{j,k}∈Jℓ,m,n
+Γ(1 + ρj,k + dmin
+Fj,k)
+�
+Sℓ,m,n;reg[F](α, β, γ)
+(188)
+
+32
+ETHAN SUSSMAN
+holds for all (α, β, γ) ∈ Ωℓ,m,n[D], where dmin
+F
+= min{dF : {dF0}F0∈F(Kℓ,m,n) ∈ D}.
+The case of the proposition above where m = N gives Theorem 1.1. Indeed, if F ∈ C∞(△N), F
+lifts to an element of C∞(K0,N,0), and the orders of vanishing of F at the relevant facets of △N
+imply the same order of vanishing at the lift in K0,N,0.
+3.2.2. The Dotsenko–Fateev case. Fix F ∈ AD(Aℓ,m,n), where D is now a collection of orders for
+the faces of Aℓ,m,n. Recalling the definition of ΣT(ℓ, m, n) given in §2.2:
+Proposition 3.4. There exist entire functions
+Iℓ,m,n;reg,I,{dF}F∈F(Aℓ,m,n)[F] : C2N+N(N−1)/2
+α,β,γ
+→ C
+(189)
+associated to the I ∈ ΣT(ℓ, m, n) such that
+Iℓ,m,n[F](α, β, γ) =
+�
+I∈ΣT(ℓ,m,n)
+�
+{dF}F∈F(Aℓ,m,n)∈D
+��
+�
+(x0,S)∈I
+Γ(1 + ϱS,Q;x0 + dFS,Q;x0)
+�
+× Iℓ,m,n;reg,I{dF}F∈F(Aℓ,m,n)[F](α, β, γ)
+�
+(190)
+for all (α, β, γ) ∈ Vℓ,m,n[D], where we have abbreviated I1 ∩ S, I2 ∩ S, and I3 ∩ S as S or Q as
+appropriate.
+■
+Proof. Follows from Proposition 2.7 and Proposition 3.2.
+□
+Consequently, Iℓ,m,n[F] : Vℓ,m,n[D] → C admits an analytic continuation ˙Iℓ,m,n[F] : ˙Vℓ,m,n[D] → C,
+where
+˙Vℓ,m,n[D] = C2N+N(N−1)/2
+α,β,γ
+�
+�
+{dF}F∈F(Aℓ,m,n)
+�
+x0∈{0,1,∞}
+�
+S,Q
+{ϱS,Q;x0 + dFS,Q;x0 ∈ Z≤−1}.
+(191)
+Note that ˙Vℓ,m,n[F] ⊇ ∩σ∈Sℓ,m,n ˙Ωℓ,m,n[F]σ, as every functional (α, β, γ) �→ ϱS,Q;x0(α, β, γ) has the
+form ρj,k(ασ, βσ, γσ) for some σ ∈ Sℓ,m,n and {j, k} ∈ Jℓ,m,n.
+As a corollary of the previous proposition, there exists a function
+Iℓ,m,n;reg[F] : C2N+N(N−1)/2
+α,β,γ
+→ C
+(192)
+such that, for all (α, β, γ) ∈ Vℓ,m,n[D],
+Iℓ,m,n[F](α, β, γ) =
+�
+�
+x0∈{0,1,∞}
+�
+S,Q
+Γ(1 + ϱS,Q;x0 + dmin
+FS,Q;x0)
+�
+Iℓ,m,n;reg[F](α, β, γ),
+(193)
+where S, Q vary over subsets of I1 = {1, . . . , ℓ}, I2 = {ℓ+1, . . . , ℓ+m}, and I3 = {ℓ+m+1, . . . , N},
+depending on x0.
+The m = N case of the previous proposition is Theorem 1.3.
+3.3. A simple identity. For each permutation σ of {0, 1, ∞}. Let
+(ℓ′, m′, n′) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(ℓ, m, n)
+(σ = 1),
+(n, m, ℓ)
+(σ = (0 1)),
+(ℓ, n, m)
+(σ = (0 ∞)),
+(m, ℓ, n)
+(σ = (1 ∞)),
+(n, ℓ, m)
+(σ = (0 1 ∞)),
+(m, n, ℓ)
+(σ = (1 0 ∞)).
+(194)
+In other words, if the elements of {0, 1, ∞} label the vertices of a triangle and the edges are labeled
+accordingly – that is, ‘ℓ’ labels the edge between 0 and ∞, ‘m’ labels the edge between 0 and 1, and
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+33
+‘n’ labels the edge between 1 and ∞ – then (ℓ′, m′, n′) is the permutation of (ℓ, m, n) resulting from
+applying σ to the triangle and reading off the new labels.
+Let Tσ : CP 1 → CP 1 denote the unique automorphism acting on {0, 1, ∞} via σ. These are
+T1(z) = z,
+T(0 1)(z) = 1 − z,
+T(0 ∞)(z) = 1
+z ,
+T(1 ∞)(z) = −
+z
+1 − z ,
+(195)
+T(0 1 ∞)(z) =
+1
+1 − z ,
+T(0 ∞ 1)(z) = z − 1
+z
+.
+(196)
+Let σparam : C2N+N(N−1)/2 → C2N+N(N−1)/2 denote the affine map
+σparam(α, β, γ) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(α, β, γ)
+(σ = 1),
+(β, α, γ)
+(σ = (0 1)),
+(−2 − α − β − 2γ⌟1, β, γ)
+(σ = (0 ∞)),
+(α, −2 − α − β − 2γ⌟1, γ)
+(σ = (1 ∞)),
+(−2 − α − β − 2γ⌟1, α, γ)
+(σ = (0 1 ∞)),
+(β, −2 − α − β − 2γ⌟1, γ)
+(σ = (1 0 ∞)),
+(197)
+where γ⌟1 ∈ CN has jth component �
+k̸=j γj,k.
+Let rev ∈ Sℓ′,m′,n′ denote the permutation
+that reverses the order of the elements in each of the sets {1, . . . , ℓ′}, {ℓ′ + 1, . . . , ℓ′ + m′}, and
+{ℓ′ + m′ + 1, . . . , N}. Let |σ| denote the order of σ.
+Proposition 3.5. If (α, β, γ) ∈ ˙Vℓ,m,n, then σparam(α, β, γ) ∈ ˙Vℓ′,m′,n′, and if (α, β, γ) ∈ ˙Ωℓ,m,n,
+then σparam(α, β, γ) ∈ ˙Ωrev|σ|
+ℓ′,m′,n′, and
+˙Iℓ,m,n[1](α, β, γ) = ˙Iℓ′,m′,n′[1](σparam(α, β, γ))rev|σ|,
+˙Sℓ,m,n[1](α, β, γ) = ˙Sℓ′,m′,n′[1](σparam(α, β, γ))rev|σ|
+(198)
+for all (α, β, γ) ∈ ˙Ωℓ,m,n.
+■
+Proof. It can be checked case-by-case that
+{ϱS,Q;• ◦ σparam : • ∈ {0, 1, ∞}, S, Q as above} = {ϱS,Q;• : • ∈ {0, 1, ∞}, S, Q as above},
+(199)
+where on the left-hand side (S, Q) varies over appropriate pairs of subsets of {1, . . . , ℓ′}, {ℓ′ +
+1, . . . , ℓ′ + m′}, and {ℓ′ + m′ + 1, . . . , N} and on the right-hand side (S, Q) varies over appropriate
+pairs of subsets {1, . . . , ℓ}, {ℓ + 1, . . . , ℓ + m}, and {ℓ + m + 1, . . . , N}, depending on •. It can be
+seen from eq. (199) that
+˙Vℓ,m,n = (σparam)−1( ˙Vℓ′,m′,n′).
+(200)
+The case of ˙Ωℓ,m,n is similar but more complicated.
+Equation (198) can be proven for (α, β, γ) ∈ Ωℓ,m,n by way of a change-of-variables by substituting
+x = Tσ−1(y). The full result follows via analytic continuation.
+□
+3.4. An imperfect alternative. For I ∈ {(−∞, 0], [0, 1], [1, ∞)} and r > 0, let ΓI,±,r : (0, 1) → C
+be defined by
+Γ[0,1],±,r(t) =
+�
+�
+�
+�
+�
+�
+�
+t ± irt
+(t ∈ (0, 1/3)),
+t ± ir/3
+(t ∈ [1/3, 2/3]),
+t ± ir/3 ∓ ir(t − 2/3)
+(t ∈ (2/3, 1)),
+(201)
+Γ[1,∞),±,r(t) = Γ[0,1],∓,r(1 − t)−1, and Γ(−∞,0],±,r(t) = 1 − Γ[1,∞),∓,r(1 − t). Note that the images of
+these contours are permuted amongst themselves by the transformations Tσ above.
+
+34
+ETHAN SUSSMAN
+ℑz
+ℜz
+Γ[0,1],+,1
+Γ[0,1],+,4
+0
+1
+Γ[1,∞),+,1
+Γ(−∞,0],+,1
+Figure 10. The contours Γ(−∞,0],+,1, Γ[0,1],+,1, Γ[0,1],+,4, Γ[1,∞),+,1. Cf. [DF85a,
+Figure 16]. (For our purposes, the contours drawn by Dotsenko & Fateev approach
+±∞ with imaginary part too small. This is why our ΓI,±,r look different for I ̸= [0, 1].
+Suppose that F ∈ C[x1, x−1
+1 , . . . , xN, x−1
+N ]. For any compact K ⋐ C with nonempty interior, let
+O = O[F, K] denote the set, which depends on ℓ, m, n ∈ N, though we suppress this dependence
+notationally, of (α, β) ∈ C2N such that
+�
+Γ(−∞,0],+,0
+· · ·
+�
+Γ(−∞,0],+,ℓ−1
+� �
+Γ[0,1],+,0
+· · ·
+�
+Γ[0,1],+,m−1
+� �
+Γ[1,∞),+,0
+· · ·
+�
+Γ[1,∞),+,n−1
+� N
+�
+j=1
+zαj
+j (1 − zj)βj
+�
+�
+1≤j<k≤N
+(zk − zj)2γj,kF0 dzN · · · dzℓ+m+1
+�
+dzℓ+m · · · dzℓ+1
+�
+dzℓ · · · dz1
+(202)
+is an absolutely convergent Lebesgue integral whenever γj,k ∈ K for all j, k ∈ {1, . . . , N} with j < k,
+for every monomial F0 in F. In the definition of the integral above we are defining the integrand
+such that the branch cuts are not encountered. For such (α, β, γ),
+(α, β, γ) ∈ ˙Vℓ,m,n[F],
+(203)
+and the integral in eq. (202) is equal to ˙Iℓ,m,n(α, β, γ)[F], assuming that we choose our branches
+appropriately. The latter part of this statement can be proven by checking that the integral defined
+above depends analytically on its parameters and agrees with Iℓ,m,n(α, β, γ)[F] for (α, β, γ) ∈
+Vℓ,m,n[F], which in turn is proven via a contour deformation argument.
+The set O is nonempty, open, and contains an affine cone. If
+• αj has sufficiently large real part for j ∈ I1 ∪ I2 and sufficiently negative real part for j ∈ I3,
+and
+• βj has sufficiently large real part for j ∈ I2 ∪ I3 and sufficiently negative real part for j ∈ I1,
+then (α, β) ∈ O[F, K], where what “sufficiently large” means depends on K. Consequently, given
+any subset S ⊆ Sℓ × Sm × Sn, the set OS∩ defined by
+OS∩ = {(α, β) ∈ C2N : (ασ, βσ) ∈ O[F σ, Kσ] for all σ ∈ S}
+(204)
+is open and nonempty. If K contains e.g. −1, then O[F, K] contains some (α, β) such that (α, β, γ) /∈
+Vℓ,m,n[F]. So, eq. (202) gives us an alternative definition of ˙Iℓ,m,n(α, β, γ)[F] for some range of
+parameters.
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+35
+Proposition 3.6. Consider • ∈ {1, 2, 3} and j, k ∈ I• with j < k and |j − k| = 1. Suppose that
+γj,k ∈ Z. Let τ ∈ Sℓ,m,n denote the transposition swapping j, k. Then,
+Iℓ,m,n[F](α, β, γ) − Iℓ,m,n[F](α, β, γ)τ =
+�
+Γ(−∞,0],+,0;1
+· · ·
+�
+Γ(−∞,0],+,ℓ−1;ℓ
+�
+�
+Γ[0,1],+,0;ℓ+1
+· · ·
+�
+Γ[0,1],+,m−1;ℓ+m
+� �
+Γ[1,∞),+,0;ℓ+m+1
+· · ·
+�
+Γ[1,∞),+,n−1;N
+�
+N
+�
+j0=1
+z
+αj0
+j0 (1 − zj0)βj0
+�
+×
+�
+�
+1≤j0<k0≤N
+(zk0 − zj0)2γj0,k0
+�
+F dzN · · · dzℓ+m+1
+�
+dzℓ+m · · · dzℓ+1
+�
+dzℓ · · · dz1,
+(205)
+whenever (α, β, γ) ∈ O∩{1,τ}, where ΓI,+,r;i = ΓI,+,r;i unless i = j, in which case ΓI,+,r;i =
+ΓI,+,r;i({zi0}i0̸=j) is a small counterclockwise circle around zk not winding around any of the other
+z’s or 0, 1.
+■
+Proof. It suffices to consider the case F = 1. Indeed, if F is a monomial, then we can simply absorb
+it into a redefinition of α. The set O∩{1,τ} is decreasing with the set of monomials in F, so once the
+result has been proven for monomials, it follows for all Laurent polynomials.
+For • = 2, the proposition follows via a straightforward countour deformation argument. The
+case • ∈ {1, 3} can be reduced to • = 3 via Proposition 3.5.
+□
+3.5. Symmetrization. Let F ∈ AD(Aℓ,m,n).
+Proposition 3.7. For any (α, β, γ) ∈ ∩σ∈Sℓ,m,n ˙Ωℓ,m,n[F]σ,
+˙Iℓ,m,n[F](α, β, γ) =
+�
+σ∈Sℓ,m,n
+eiΘ(σ−1) ˙Sℓ,m,n[F](α, β, γ)σ,
+(206)
+where Θ(σ) = 2π �
+1≤j<k≤N 1σ(j)>σ(k)γj,k.
+■
+Proof. By analyticity, it suffices to prove the result when the quantities above are well-defined
+Lebesgue integrals. Decomposing □ℓ,m,n into ℓ!m!n! copies of △ℓ,m,n,
+Iℓ,m,n[F](α, β, γ) =
+�
+σ∈Sℓ,m,n
+�
+△ℓ,m,n
+N
+�
+j=1
+|xj|ασ(j)|1 − xj|βσ(j)
+�
+1≤j<k≤N
+(xσ−1(k) − xσ−1(j) + i0)2γj,k
+F(xσ−1(1), · · · , xσ−1(N)) dNx.
+(207)
+The right-hand side is
+�
+σ∈Sℓ,m,n
+eiΘ(σ−1)
+�
+△ℓ,m,n
+N
+�
+j=1
+|xj|ασ(j)|1 − xj|βσ(j)
+�
+1≤j<k≤N
+(xk − xj)2γσ(j),σ(k)(F σ) dNx,
+(208)
+which is the right-hand side of eq. (206).
+□
+Proposition 3.8. Suppose that α, β, γ are invariant under all σ ∈ Sℓ,m,n, and suppose now that
+F ∈ C[x1, . . . , xN]Sℓ,m,n. Then, for all (α, β, γ) ∈ ˙Ωℓ,m,n[F],
+˙Iℓ,m,n[F](α, β, γ) =
+�
+ℓ
+�
+k=1
+1 − e2πikγ1
+1 − e2πiγ1
+�� m
+�
+k=1
+1 − e2πikγ2
+1 − e2πiγ2
+��
+n
+�
+k=1
+1 − e2πikγ3
+1 − e2πiγ3
+� ˙Sℓ,m,n[F](α, β, γ),
+(209)
+where, for each • ∈ {1, 2, 3}, γ• = γj,k for all distinct j, k ∈ I•.
+■
+Here, we are treating (1 − e2πiγ)−1(1 − e2πikγ) as an entire function.
+
+36
+ETHAN SUSSMAN
+Proof. Applying the previous proposition,
+Iℓ,m,n[F](α, β, γ) =
+� �
+σ∈Sℓ
+eπi o.o.(σ)γ�� �
+σ∈Sm
+eπi o.o.(σ)γ�� �
+σ∈Sn
+eπi o.o.(σ)γ�
+Sℓ,m,n[F](α, β, γ),
+(210)
+where o. o.(σ) is the number of out-of-order pairs in σ. We appeal to the algebraic identity [Bón12]
+Z[ζ] ∋
+�
+σ∈SN
+ζo.o.(σ) =
+N−1
+�
+n=0
+n
+�
+m=0
+ζm =
+N
+�
+n=1
+1 − ζn
+1 − ζ ,
+(211)
+which holds for all N ∈ N and encodes the bijection between SN and the set of possible runs of the
+bubble sort algorithm. Plugging in ζ = eπiγ, eq. (210) becomes eq. (209).
+□
+3.6. The Aomoto-Dotsenko–Fateev relations. Fix N ∈ N+ and F ∈ C[x1, x−1
+1 , . . . , xN, x−1
+N ].
+For each j ∈ {1, . . . , N}, let σj ∈ SN be the permutation that takes 1 and inserts it in the jth
+position while maintaining the relative order of the other terms. That is, σj = (1 j j − 1 · · · 2).
+For any ℓ ∈ N+ and m, n ∈ N with ℓ + m + n = N, let
+˙Λℓ,m,n[F] = ˙Vℓ,m,n[F] ∩ ˙Vℓ−1,m+1,n[F]σℓ ∩ ˙Vℓ−1,m,n+1[F]σℓ+m
+= ˙Vℓ,m,n[F] ∩ ˙Vℓ−1,m+1,n[F]σℓ+m ∩ ˙Vℓ−1,m,n+1[F]σN ,
+(212)
+˙℧ℓ,m,n[F] = (∩ℓ
+j=1 ˙Ωℓ,m,n[F]σj) ∩ (∩ℓ+m
+j=ℓ ˙Ωℓ−1,m+1,n[F]σj) ∩ (∩N
+j=ℓ+m ˙Ωℓ−1,m,n+1[F]σj).
+(213)
+Note that ˙℧ℓ,m,n[F], ˙Λℓ,m,n[F] are open, dense, and connected subsets of C2N+N(N−1)/2, being the
+complements of locally finite unions of complex affine hyperplanes.
+Proposition 3.9. For any (α, β, γ) ∈ ˙℧ℓ,m,n[F],
+0 =
+ℓ
+�
+j=1
+e±iθj ˙Sℓ,m,n[F](α, β, γ)σj +
+ℓ+m
+�
+j=ℓ
+e±iϑj ˙Sℓ−1,m+1,n[F](α, β, γ)σj
++
+N
+�
+j=ℓ+m
+e±iϕj ˙Sℓ−1,m,n+1[F](α, β, γ)σj
+(214)
+holds for each choice of sign, where θj = 2π �
+2≤j0≤j γ1,j0, ϑj = πα + 2π �
+2≤j0≤j γ1,j0, and ϕj =
+πα + πβ + 2π �
+2≤j0≤j γ1,j0.
+■
+Proof. Without loss of generality, we may assume F = 1. Let
+℧ℓ,m,n =
+�
+�
+�
+�
+�
+�
+�
+(α, β, γ) ∈ C2N+N(N−1)/2 : (ασj, βσj, γσj) ∈
+�
+�
+�
+�
+�
+�
+�
+Ωℓ,m,n
+(j ∈ {1, . . . , ℓ})
+Ωℓ−1,m+1,n
+(j ∈ {ℓ, . . . , ℓ + m})
+Ωℓ−1,m,n+1
+(j ∈ {ℓ + m, . . . , N})
+�
+�
+�
+�
+�
+�
+�
+.
+(215)
+(This is to be read such that, whenever (α, β, γ) ∈ ℧ℓ,m,n, (ασℓ, βσℓ, γσℓ) ∈ Ωℓ,m,n ∩ Ωℓ−1,m+1,n and
+(ασℓ+m, βσℓ+m, γσℓ+m) ∈ Ωℓ−1,m+1,n ∩ Ωℓ−1,m,n+1.)
+Let ε > 0. For each ϝ1, ϝ2, ϝ3 > 0 and γ, γ ∈ (−(N −1)−1, 0) with γ < γ, let ℧0,ϝ,γ,γ (suppressing
+the ℓ, m, n dependence for brevity) denote the set of (α, β, γ) ∈ C2N+N(N−1)/2 such that
+• γ < ℜγj,k < γ for all j, k ∈ {1, . . . , N} with j ̸= k,
+• ϝ1 < ℜαj < ϝ2 for each j ∈ {2, . . . , ℓ}, ℜαj > ϝ1 for each j ∈ {ℓ + 1, . . . , ℓ + m}, and
+ℜαj < −ϝ3 for each j ∈ {ℓ + m + 1, . . . , N},
+• ϝ1 < ℜβj < ϝ2 for each j ∈ {ℓ + m + 1, . . . , N}, ℜβj > ϝ1 for each j ∈ {ℓ + 1, . . . , ℓ + m},
+and ℜβj < −ϝ3 for j ∈ {2, . . . , ℓ},
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+37
+where ϝ = (ϝ1, ϝ2, ϝ3). The set ℧0,ϝ,γ,γ is open and nonempty. By eq. (7) and the analogue of
+eq. (7) for the m < N case, there exist ϝ00, ϝ0, ϝ01 > 0 (depending on ℓ, m, n, γ, γ) such that
+℧ϝ,γ,γ
+def
+= {(α, β, γ) ∈ ℧0,ϝ,γ,γ and (α1, β1) ∈ Ω1,0,0 ∩ Ω0,1,0 ∩ Ω0,0,1} ⊂ ℧ℓ,m,n
+(216)
+whenever ϝ2 > ϝ1 > ϝ0 and ϝ3 > ϝ01ϝ2 + ϝ00. Observe that Ω1,0,0 ∩ Ω0,1,0 ∩ Ω0,0,1 is the subset
+of C2
+α,β defined by the inequalities −1 < ℜα, ℜβ and ℜα + ℜβ < −1. The set
+{(r1, r2) ∈ R2 : −1 < r1, r2 and r1 + r2 < −1}
+(217)
+is a nonempty triangle. So, ℧ϝ,γ,γ is an open and nonempty subset of C2N+N(N−1)/2 and moreover
+of ˙℧ℓ,m,n.
+For such ϝ and (α, β, γ) ∈ ℧ϝ,γ,γ, eq. (214) (with F = 1) just reads
+0 =
+ℓ
+�
+j=1
+e±iθjSℓ,m,n[1](α, β, γ)σj +
+ℓ+m
+�
+j=ℓ
+e±iϑjSℓ−1,m+1,n[1](α, β, γ)σj
++
+N
+�
+j=ℓ+m
+e±iϕjSℓ−1,m,n+1[1](α, β, γ)σj
+(218)
+(note the absence of the dots over the S’s). By the analyticity of all of the functions in eq. (214) on
+˙℧ℓ,m,n, it suffices to prove that eq. (218) holds for such (α, β, γ).
+By Fubini’s theorem, the right-hand side of eq. (218) is
+�
+△ℓ−1,m,n
+ω(x2, . . . , xN)
+� � +∞
+−∞
+(−x1 ± i0)α1(1 − x1 ± i0)β1� N
+�
+j=2
+(xj − x ± i0)2γ1,j
+�
+dx
+�
+dx2 · · · dxN,
+(219)
+where ω(x2, . . . , xN) = [�N
+j=2 |xj|αj|1 − xj|βj] �
+2≤j<k≤N(xk − xj)2γj,k. The claim then follows from
+0 =
+� +∞
+−∞
+(−x ± i0)α(1 − x ± i0)β� N
+�
+j=2
+(xj − x ± i0)2γj
+�
+dx,
+(220)
+which holds for every (x2, . . . , xN) ∈ (R\{0, 1})N−1 such that x2, . . . , xN are pairwise distinct and
+all α, β, γ2, . . . , γN ∈ C for which
+• the integrand of eq. (220) lies in L1(R) and
+• ℜγj ∈ (−1, 0) for all j ∈ {2, . . . , N}.
+Denote the right-hand side of eq. (220) by I± = I±(x2, . . . , xN; α, β, γ2, . . . , γN).
+For R >
+max{|x1|, . . . , |xN−1|},
+0 =
+�
+Γ∓(R)
+(−z ± i0)α(1 − z ± i0)β
+N
+�
+j=2
+(xj − z ± i0)2γj dz,
+(221)
+where Γ±(R) = Γ±(R)(x2, . . . , xN) ⊂ C is the semicircular contour (with N + 1 semicircular insets
+placed so that the contour avoids x2, . . . , xN) connecting −R and +R, with the arc and semicircular
+insets in the half-plane {z ∈ C : ±ℑz ≥ 0}. See Figure 11. In eq. (221), the integrand is defined
+taking the branch cut along the negative real axis, so
+(x − z ± i0)2γj =
+�
+exp(2γj(log |x − z| + i arg(x − z)))
+(+ case, ℑz ≤ 0),
+exp(2γj(log |x − z| − 2πi + i arg(x − z)))
+(− case, ℑz ≥ 0),
+(222)
+for any x ∈ R, where arg(x − z) ∈ [0, 2π). We orient Γ+ counter-clockwise and Γ− clockwise.
+
+38
+ETHAN SUSSMAN
+ℑz
+ℜz
+0
+1
+x2
+x1
+x3
+−R
+R
+Figure 11. The contour Γ+(R) in the case ℓ = 2, m = 1, n = 0.
+Let Γ++(R) denote the large arc of Γ+(R) and Γ+0(R) denote the rest, and likewise let Γ−−(R)
+denote the large arc of Γ−(R) and Γ−0(R) denote the rest. Then,
+I± = lim
+R→∞
+�
+Γ∓0(R)
+(−z ± i0)α(1 − z ± i0)β
+N
+�
+j=2
+(xj − z ± i0)2γj dz.
+(223)
+On the other hand, for R sufficiently large,
+���
+�
+Γ∓∓(R)
+(−z ± i0)α(1 − z ± i0)β
+N
+�
+j=2
+(xj − z ± i0)2γj dx
+��� ≤ π(2R)1+ℜα+ℜβ = O(R−ε)
+(224)
+for some ε > 0 depending on (α, β) ∈ Ω1,0,0 ∩ Ω0,1,0 ∩ Ω0,0,1. Combining eq. (221), eq. (223), and
+eq. (224), we get I± = 0.
+□
+Proposition 3.10. For any F ∈ C[x1, x−1
+1 , . . . , xN, x−1
+N ],
+0 = ˙Iℓ,m,n[F](α, β, γ) + e+πi(α+2�ℓ
+j=2 γ1,j) ˙Iℓ−1,m+1,n[F](α, β, γ)σℓ
++ e+πi(α+β+2�ℓ+m
+j=2 γ1,j) ˙Iℓ−1,m,n+1[F](α, β, γ)σℓ+m
+(225)
+0 = ˙Iℓ,m,n[F](α, β, γ)σℓ + e−πi(α+2�ℓ
+j=2 γ1,j) ˙Iℓ−1,m+1,n[F](α, β, γ)σℓ+m
++ e−πi(α+β+2�ℓ+m
+j=2 γ1,j) ˙Iℓ−1,m,n+1[F](α, β, γ)σN
+(226)
+both hold, for all (α, β, γ) ∈ ˙Λℓ,m,n[F].
+■
+Proof. Via analyticity, it suffices to prove this for all (α, β, γ) ∈ ∩σ∈Sℓ,m,n s.t. σ(1)=1 ˙℧ℓ,m,n[F]σ. For
+such (α, β, γ), we can cite the previous proposition to get
+0 =
+�
+σ∈Sℓ,m,n s.t. σ(1)=1
+eπiΘ(σ−1)�
+ℓ
+�
+j=1
+e±iθσ
+j ˙Sℓ,m,n[F](α, β, γ)σjσ
++
+ℓ+m
+�
+j=ℓ
+e±iϑσ
+j ˙Sℓ−1,m+1,n[F](α, β, γ)σjσ +
+N
+�
+j=ℓ+m
+e±iϕσ
+j ˙Sℓ−1,m,n+1[F](α, β, γ)σjσ�
+,
+(227)
+where θσ
+j = 2π �
+2≤j0≤j γ1,σ(j0), ϑσ
+j = πα+2π �
+2≤j0≤j γ1,σ(j0), and ϕσ
+j = πα+πβ+2π �
+2≤j0≤j γ1,σ(j0).
+The order of multiplication is such that σjσ is a permutation satisfying (σjσ)(1) = j. In eq. (227),
+Θ is defined as in Proposition 3.7.
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+39
+Every σ0 ∈ Sℓ,m,n has the form σ0 = σjσ for some j ∈ {1, . . . , N} and σ ∈ Sℓ,m,n satisfying
+σ(1) = 1. It can be seen that
+Θ(σ−1
+0 ) = Θ(σ−1) + θσ
+j .
+(228)
+Using Proposition 3.7, we check that the two cases of eq. (227) yield the two results, eq. (225) and
+eq. (226). For instance,
+�
+σ∈Sℓ,m,n s.t. σ(1)=1
+eπiΘ(σ−1)
+ℓ
+�
+j=1
+e+iθσ
+j ˙Sℓ,m,n[F](α, β, γ)σjσ =
+�
+σ∈Sℓ,m,n
+eπiΘ(σ−1) ˙Sℓ,m,n[F](α, β, γ)σ
+= ˙Iℓ,m,n[F](α, β, γ).
+(229)
+Similar statements apply to the other two sums in eq. (227) in the ‘+’ case, thus yielding eq. (225).
+Similar computations apply to the ‘−’ case.
+□
+3.7. The symmetric and DF-symmetric cases. Fix F ∈ AD(Aℓ,m,n), not necessarily symmetric.
+We assume that dFS,Q;• ∈ Z for all FS,Q;• ∈ F(Aℓ,m,n). Let
+δk = min{dFS,Q;0 : S ⊆ I1, Q ⊆ I2, |S ∪ Q| = k}
+(230)
+for each k ∈ {1, . . . , ℓ + m},
+δ
+k = min{dFS,Q;1 : S ⊆ I2, Q ⊆ I3, |S ∪ Q| = k}
+(231)
+for each k ∈ {1, . . . , m + n}, and
+dk = − min{dFS,Q;∞ : S ⊆ I3, Q ⊆ I1, |S ∪ Q| = k}
+(232)
+for each k ∈ {1, . . . , ℓ + n}. Here, we are ranging over all {dF}F∈F(Aℓ,m,n) ∈ D.
+Let ˙Wℓ,m,n[D] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ ˙Vℓ,m,n[D] whenever α, β, γ
+have components given by αj = α and βj = β for all indices j ∈ {1, . . . , N} and γj,k = γ for all
+j, k ∈ {1, . . . , N} with j < k.
+Proposition 3.11. There exists an entire function Iℓ,m,n;Reg[F] : C3 → C such that
+˙Iℓ,m,n[F](α, β, γ) =
+� ℓ+m
+�
+k=1
+Γ(δk + k(1 + α + (k − 1)γ))
+�� m+n
+�
+k=1
+Γ(
+δ
+k + k(1 + β + (k − 1)γ))
+�
+×
+� ℓ+n
+�
+k=1
+Γ(−dk − k(1 + α + β + (2N − k − 1)γ))
+�
+Iℓ,m,n;Reg[F](α, β, γ)
+(233)
+for all (α, β, γ) ∈ ˙Wℓ,m,n[D].
+■
+Proof. Follows from Proposition 3.4.
+□
+For later reference, consider the special case F ∈ C[x1, . . . , xN]SN . Referring to eq. (14), eq. (15),
+and eq. (28), set dFS,Q;0 = δj[F], dFS,Q;1 =
+δ
+j[F], and dFS,Q;∞ = degj[F], for S, Q ⊆ {1, . . . , N} as
+usual, where, for each S and Q, j = |S ∪ Q|. Then, as follows straightforwardly from eq. (71),
+eq. (73), eq. (75),
+F ∈
+�
+F∈F(Aℓ,m,n)
+xdF
+F C∞(Aℓ,m,n).
+(234)
+Thus, letting D denote the collection of the integers above, F ∈ AD(Aℓ,m,n). We can therefore apply
+the results above, with δj = δj[F],
+δ
+j =
+δ
+j[F], and dj = − degj[F].
+We now turn to the “DF0-symmetric” case. For any S ⊆ {1, . . . , N}, let
+˙W DF0,S
+ℓ,m,n [F] = {(α−, α+, β−, β+, γ−, γ0, γ+) ∈ C7 : (αDF0, βDF0, γDF0) ∈ ˙Vℓ,m,n[F]}.
+(235)
+
+40
+ETHAN SUSSMAN
+This is a dense, open, and connected subset of C7 and depends on S only through the numbers
+|S ∩ Ij|. Actually, we need a slightly refined version of this later; let
+˙W DF1,S
+ℓ,m,n [F] = {(α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ−, γ0, γ+) ∈ C9
+: (αDF1, βDF1, γDF0) ∈ ˙Vℓ,m,n[F]},
+(236)
+where αDF1, βDF1 are defined as their DF0-counterparts, but defining the jth component using
+α+,ν in place of α+ and β+,ν in place of β+ for ν ∈ Iν.
+For (α−, α+, β−, β+, γ−, γ0, γ+) ∈ ˙W DF0,S
+ℓ,m,n [F], let
+˙IDF0;S
+ℓ,m,n [F](α−, α+, β−, β+, γ−, γ0, γ+) = ˙Iℓ,m,n[F](αDF0, βDF0, γDF0).
+(237)
+Let ℓ+ = S ∩ I1, ℓ− = ℓ − ℓ+, m+ = S ∩ I2, m− = m − m+, n+ = S ∩ I3, and n− = n − n+. Set
+N+ = |S| and N− = N − N+.
+Suppose now that F ∈ AD(Aℓ,m,n) is symmetric in the variables {xi}i∈S and {xi}i/∈S separately.
+Let
+δj−,j+ = min{dFS,Q;0 : S ⊆ I1, Q ⊆ I2, |(S ∪ Q)\S| = j−, |(S ∪ Q) ∩ S| = j+}
+(238)
+for j− ∈ {1, . . . , ℓ− + m−} and j+ ∈ {1, . . . , ℓ+ + m+},
+δ
+j−,j+ = min{dFS,Q;1 : S ⊆ I2, Q ⊆ I3, |(S ∪ Q)\S| = j−, |(S ∪ Q) ∩ S| = j+}
+(239)
+for j− ∈ {1, . . . , m− + n−} and j+ ∈ {1, . . . , m+ + n+}, and
+dj−,j+ = − min{dFS,Q;∞ : S ⊆ I3, Q ⊆ I1, |(S ∪ Q)\S| = j−, |(S ∪ Q) ∩ S| = j+}
+(240)
+for j− ∈ {1, . . . , ℓ− + n−} and j+ ∈ {1, . . . , ℓ+ + n+}. A similar argument to that above yields:
+Proposition 3.12. There exists an entire function IDF0;S
+ℓ,m,n;Reg[F] : C7 → C such that
+˙IDF0
+ℓ,m,n[F](α−, α+, β−, β+, γ−, γ0, γ+) = IDF0
+ℓ,m,n;Reg[F](α−, α+, β−, β+, γ−, γ0, γ+)
+×
+� ℓ−+m−
+�
+j−=1
+ℓ++m+
+�
+j+=1
+Γ(δj−,j+ + j−(1 + α− + (j− − 1)γ−) + j+(1 + α+ + (j+ − 1)γ+) + 2γ0j−j+)
+�
+×
+� m−+n−
+�
+j−=1
+m++n+
+�
+j+=1
+Γ(
+δ
+j−,j+ + j−(1 + β− + (j− − 1)γ−) + j+(1 + β+ + (j+ − 1)γ+) + 2γ0j−j+)
+�
+×
+� ℓ−+n−
+�
+j−=1
+ℓ++n+
+�
+j+=1
+Γ(−dj−,j+ − j−(1 + α− + β− + (2N− − j− − 1)γ−)
+− j+(1 + α+ + β+ + (2N+ − j+ − 1)γ+) − 2γ0j−j+)
+�
+(241)
+holds whenever (α−, α+, β−, β+, γ−, γ0, γ+) ∈ ˙W DF0
+ℓ,m,n[F].
+■□
+4. Removing singularities
+As in previous sections, fix ℓ, m, n ∈ N not all zero, and let N = ℓ + m + n and I1 = {1, . . . , ℓ},
+I2 = {ℓ + 1, . . . , ℓ + m}, and I3 = {ℓ + m + 1, . . . , N}. For k ∈ N, let
+ϝk : Cγ\{kγ ∈ Z≤−1 and γ /∈ Z} → C
+(242)
+denote the analytic function given by ϝk(γ) = Γ(1 + γ)−1Γ(1 + kγ) for kγ /∈ Z≤−1. We can consider
+ϝ−1
+k
+as an entire function.
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+41
+4.1. The symmetric case. Fix F ∈ C[x1, . . . , xN]SN , and let δj,
+δ
+j, dj ∈ N be as above.
+Let ˙Uℓ,m,n[F] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ ˙Ωℓ,m,n[F] whenever α, β, γ
+have components given by αj = α and βj = β for all indices j ∈ {1, . . . , N} and γj,k = γ for all
+j < k. Thus, we can define
+˙Sℓ,m,n[F](α, β, γ) = ˙Sℓ,m,n[F](α, β, γ)
+(243)
+for any (α, β, γ) ∈ ˙Uℓ,m,n[F].
+Proposition 4.1. The function Sreg
+ℓ,m,n[F] : ˙Uℓ,m,n[F] → C defined by
+Sreg
+ℓ,m,n[F](α, β, γ) =
+� ℓ+m
+�
+k=1
+Γ(δk + k(1 + α + (k − 1)γ))
+�−1
+� m+n
+�
+k=1
+Γ(
+δ
+k + k(1 + β + (k − 1)γ))
+�−1� ℓ+n
+�
+k=1
+Γ(−dk − k(1 + α + β + (N + k − 2)γ))
+�−1
+�
+ℓ
+�
+k=1
+1
+ϝk(γ)
+�� m
+�
+k=1
+1
+ϝk(γ)
+��
+n
+�
+k=1
+1
+ϝk(γ)
+� ˙Sℓ,m,n[F](α, β, γ)
+(244)
+extends to an entire function C3
+α,β,γ → C.
+■
+Proof.
+• Since the prefactor on the right-hand side of eq. (244) consisting of all of the Γ-
+function reciprocals is entire, Sreg
+ℓ,m,n[F] extends to an analytic function on ˙Uℓ,m,n[F], the
+domain of ˙Sℓ,m,n[F](α, β, γ).
+• For all (α, β, γ) ∈ ˙Uℓ,m,n[F], we have
+�
+ℓ
+�
+k=1
+1 − e2πikγ
+1 − e2πiγ ϝk(γ)
+�� m
+�
+k=1
+1 − e2πikγ
+1 − e2πiγ ϝk(γ)
+��
+n
+�
+k=1
+1 − e2πikγ
+1 − e2πiγ ϝk(γ)
+�
+Sreg
+ℓ,m,n[F](α, β, γ)
+= Iℓ,m,n;Reg[F](α, β, γ)
+(245)
+by Proposition 3.8. By Proposition 3.11, this extends to an entire function C3
+α,β,γ → C.
+The product ϝk(γ)(1 − e2πikγ)(1 − e2πiγ)−1, with its removable singularities removed,
+vanishes if and only if kγ ∈ N and γ /∈ N. Thus, Sreg
+ℓ,m,n[F] extends to an analytic function on
+C3
+α,β,γ\ ∪M
+k=2 {kγ ∈ N, γ /∈ N},
+(246)
+where M = max{ℓ, m, n}.
+Combining these two observations, Sreg
+ℓ,m,n[F] extends to an analytic function on ˙Uℓ,m,n[F] ∪
+(C3
+α,β,γ\ ∪M
+k=2 {kγ ∈ N, γ /∈ N}).
+The set ∪M
+k=2{kγ ∈ N, γ /∈ N} is a union of hyperplanes, and it is disjoint from
+N
+�
+k=1
+{k(k + 1)γ ∈ Z≤−k},
+(247)
+so ˙Uℓ,m,n[F]∪(C3
+α,β,γ\∪M
+k=2 {kγ ∈ N, γ /∈ N}) is the complement in C3
+α,β,γ of a locally finite collection
+of complex codimension-2 affine subspaces of C3. The result therefore follows from Hartog’s extension
+theorem.
+□
+For any ℓ ∈ N+ and m, n ∈ N,
+{(α, β, γ) ∈ C3 : (α, β, γ) ∈ ˙℧ℓ,m,n[F]} = ˙Uℓ,m,n[F] ∩ ˙Uℓ−1,m+1,n[F] ∩ ˙Uℓ−1,m,n+1[F].
+(248)
+
+42
+ETHAN SUSSMAN
+The symmetric case of Proposition 3.9 reads, after multiplying through by 1 − e±2iγ,
+0 = (1 − e±2πiℓγ) ˙Sℓ,m,n[F](α, β, γ) + e±πi(α+2(ℓ−1)γ)(1 − e±2πi(m+1)γ) ˙Sℓ−1,m+1,n[F](α, β, γ)
++ e±πi(α+β+2(ℓ−1+m)γ)(1 − e±2πi(n+1)γ) ˙Sℓ−1,m,n+1[F](α, β, γ)
+(249)
+for all (α, β, γ) in the set defined by eq. (248). Define
+ON;0 = {(α, β, γ) ∈ C3 : α + mγ /∈ Z for any j ∈ {0, . . . , N − 1}},
+(250)
+ON;1 = {(α, β, γ) ∈ C3 : β + mγ /∈ Z for any j ∈ {0, . . . , N − 1}}.
+(251)
+Proposition 4.2 (Cf. [DF85a][Aom87][FW08]).
+• For all (α, β, γ) ∈ ˙UN,0,0[F] ∩ ˙U0,N,0[F] ∩ ON;1,
+˙S0,N,0[F](α, β, γ) = (−1)N� N−1
+�
+m=0
+sin(π(α + β + (N + m − 1)γ))
+sin(π(β + mγ))
+� ˙SN,0,0[F](α, β, γ).
+(252)
+• For all (α, β, γ) ∈ ˙U0,N,0[F] ∩ ˙U0,0,N[F] ∩ ON;0,
+˙S0,N,0[F](α, β, γ) = (−1)N� N−1
+�
+m=0
+sin(π(α + β + (N + m − 1)γ))
+sin(π(α + mγ))
+� ˙S0,0,N[F](α, β, γ).
+(253)
+■
+Proof. We prove the second claim, and the proof of the first is similar. Suppose that
+(α, β, γ) ∈
+N
+�
+n=0
+˙U0,N−n,n[F] ∩
+N−1
+�
+n=0
+˙U1,N−1−n,n[F].
+(254)
+We can apply eq. (249) for ℓ = 1 and all pairs of m, n ∈ {0, . . . , N − 1} such that m + n = N − 1.
+Combining the plus and minus cases of eq. (249) to eliminate the ˙S1,N−n−1,n[F] term,
+1
+2i
+�
+e+πiα 1 − e+2πi(m+1)γ
+1 − e+2πiγ
+− e−πiα 1 − e−2πi(m+1)γ
+1 − e−2πiγ
+� ˙S0,N−n,n[F](α, β, γ)
+= − 1
+2i
+�
+e+πi(α+β+2(N−n−1)γ) 1 − e+2πi(n+1)γ
+1 − e+2πiγ
+− e−πi(α+β+2(N−n−1)γ) 1 − e−2πi(n+1)γ
+1 − e−2πiγ
+�
+× ˙S0,N−n−1,n+1[F](α, β, γ)
+(255)
+if γ /∈ Z. We calculate:
+1
+2i
+�
+e+πiα 1 − e+2πi(N−n)γ
+1 − e+2πiγ
+−e−πiα 1 − e−2πi(N−n)γ
+1 − e−2πiγ
+�
+= 2s(γ)s(α + (N − n − 1)γ)s((N − n)γ)
+1 − cos(2πγ)
+(256)
+and
+1
+2i
+�
+e+πi(α+β+2(N−n−1)γ) 1 − e+2πi(n+1)γ
+1 − e+2πiγ
+− e−πi(α+β+2(N−n−1)γ) 1 − e−2πi(n+1)γ
+1 − e−2πiγ
+�
+=
+2s(γ)
+1 − cos(2πγ)s(α + β + (2N − n − 2)γ)s((n + 1)γ),
+(257)
+where s(t) = sin(πt). So, for (α, β, γ) as above such that none of the trigonometric factors on the
+right-hand side of eq. (256) vanish,
+˙S0,N−n,n[F](α, β, γ) = −s(α + β + (2N − n − 2)γ)s((n + 1)γ)
+s(α + (N − n − 1)γ)s((N − n)γ)
+˙S0,N−1−n,n+1[F](α, β, γ)
+(258)
+Applying this recursively for n = 0, . . . , N − 1, we end up with eq. (253).
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+43
+Figure 12. The sets in S1, S2, S3 in R3
+α,β,γ ∩ {β = 1/5} in the case N = 2.
+In summary, eq. (253) holds for a nonempty, open subset of (α, β, γ) ∈ ˙U0,N,0[F]∩ ˙U0,0,N[F]∩ON;0.
+By analyticity, the result follows.
+□
+Proposition 4.3. The function SN;Reg[F](α, β, γ) defined by
+SN;Reg[F](α, β, γ) =
+� N
+�
+j=1
+Γ(2 + ¯dj + α + β + (N + j − 2)γ)
+Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯
+δ
+j + β + (j − 1)γ)ϝj(γ)
+�
+SN[F](α, β, γ)
+(259)
+extends to an entire function SN;Reg[F] : C3
+α,β,γ → C.
+■
+Proof. We begin by defining the following open (and dense) subsets of C3:
+QN;0 = {(α, β, γ) ∈ C3 : ¯δj + α + (j − 1)γ /∈ N for any j ∈ {1, . . . , N}},
+QN;1 = {(α, β, γ) ∈ C3 : ¯
+δ
+j + β + (j − 1)γ /∈ N for any j ∈ {1, . . . , N}},
+QN;∞ = {(α, β, γ) ∈ C3 : ¯dj + α + β + (N + j − 2)γ /∈ Z≤−2 for any j ∈ {1, . . . , N}},
+UN;0 = {(α, β, γ) ∈ C3 : δj + j(α + (j − 1)γ) /∈ Z≤−j for any j ∈ {1, . . . , N}},
+UN;1 = {(α, β, γ) ∈ C3 :
+δ
+j + j(β + (j − 1)γ) /∈ Z≤−j for any j ∈ {1, . . . , N}},
+UN;∞ = {(α, β, γ) ∈ C3 : −dj − j(1 + α + β + (N + j − 2)γ) /∈ Z≤0 for any j ∈ {1, . . . , N}}
+= {(α, β, γ) ∈ C3 : dj + j(1 + α + β + (N + j − 2)γ) /∈ N for any j ∈ {1, . . . , N}}.
+(260)
+We write
+SN;Reg[F](α, β, γ) = Υ0(α, β, γ)Υ1(α, β, γ)
+×
+� N
+�
+j=1
+Γ(δj + j(1 + α + (j − 1)γ))Γ(
+δ
+j + j(1 + β + (j − 1)γ))ϝj(γ)
+�−1
+SN[F](α, β, γ)
+(261)
+for
+Υ0(α, β, γ) =
+� N
+�
+j=1
+Γ(δj + j(1 + α + (j − 1)γ))Γ(
+δ
+j + j(1 + β + (j − 1)γ))
+Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯
+δ
+j + β + (j − 1)γ)
+�
+,
+(262)
+Υ1(α, β, γ) =
+� N
+�
+j=1
+Γ(2 + ¯dj + α + β + (N + j − 2)γ)
+�
+.
+(263)
+
+S1
+4
+2
+0
+-4
+-2
+0
+2
+4
+QS2
+4
+2
+0
+-2
+-4
+-4
+-2
+0
+2
+4S3
+4
+2
+0
+-2
+-4
+-4
+-2
+0
+2
+444
+ETHAN SUSSMAN
+By Proposition 4.1, the second line on the right-hand side of eq. (261) defines an entire function.
+Since Υ0 extends to an analytic function on UN;0 ∩ UN;1 and Υ1 extends to an analytic function on
+QN;∞, SN;Reg[F] extends to an analytic function on UN;0 ∩ UN;1 ∩ QN;∞.
+In ON;0 ∩ ˙U0,N,0 ∩ ˙U0,0,N, Proposition 4.2 gives
+SN;Reg[F](α, β, γ) = (−1)NΥ2(α, β, γ)Υ3(α, β, γ)
+×
+� N
+�
+j=1
+Γ(−dj − j(1 + α + β + (N + j − 2)γ))Γ(
+δ
+j + j(1 + β + (j − 1)γ))ϝj(γ)
+�−1 ˙S0,0,N[F](α, β, γ),
+(264)
+where
+Υ2 =
+� N
+�
+j=1
+Γ(
+δ
+j + j(1 + β + (j − 1)γ))
+s(α + (j − 1)γ)Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯
+δ
+j + β + (j − 1)γ)
+�
+(265)
+Υ3 =
+� N
+�
+j=1
+s(α + β + (N + j − 2)γ)Γ(2 + ¯dj + α + β + (N + j − 2)γ)
+× Γ(−dj − j(1 + α + β + (N + j − 2)γ))
+�
+.
+(266)
+By Proposition 4.1, the function on the second line of eq. (264) extends to an entire function of
+α, β, γ. On the other hand, Υ2 extends to an analytic function on QN;0 ∩ UN;1, and Υ3 extends to
+an analytic function on UN;∞. Combining these observations, SN;Reg[F] analytically continues to
+QN;0 ∩ UN;1 ∩ UN;∞.
+Likewise, SN;Reg[F] extends analytically to UN;0 ∩ QN;1 ∩ UN;∞, using ON;1 in place of ON;0 and
+the other part of Proposition 4.2.
+So, SN;Reg[F](α, β, γ) analytically continues to
+U = (UN;0 ∩ UN;1 ∩ QN;∞) ∪ (UN;0 ∩ QN;1 ∩ UN;∞) ∪ (QN;0 ∩ UN;1 ∩ UN;∞).
+(267)
+This is
+U = C3���
+�
+H1∈S1,H2∈S2,H3∈S3
+H1 ∩ H2 ∩ H3)
+�
+,
+(268)
+where
+• S1 is the set of hyperplanes that are contained in the complement of one of UN;0, UN;1, QN;∞,
+• S2 is the set of hyperplanes that are contained in the complement of one of UN;0, QN;1, UN;∞,
+and
+• S3 is the set of hyperplanes that are contained in the complement of one of QN;0, UN;1, UN;∞.
+Let
+H = {H1 ∩ H2 ∩ H3 ̸= ∅ : H1 ∈ S1, H2 ∈ S2, H3 ∈ S3},
+(269)
+so that SN;Reg[F] defines an analytic function on U = C3\∪H∈HH. Observe that every H ∈ H is
+an affine subspace of C3 of complex codimension two or three (since S1 ∩ S2 ∩ S3 = ∅), and the
+collection H is locally finite.
+Hartog’s theorem therefore implies that SN;Reg[F] analytically continues to the entirety of C3.
+□
+This completes the proof of Theorem 1.2.
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+45
+4.2. The DF-symmetric case. Given γ+ ∈ C\{0, 1} and α+, β+ ∈ C, let γ− = γ−1
++ , α− = −γ−α+,
+and β− = −γ−β+ as in the introduction. Fix S ⊆ {1, . . . , N}.
+Given γ+ ̸= 0, 1 and F ∈ DFSym(N; S, λ) for λ = γ−1
++ (γ+ − 1), let ˙W DF,S
+ℓ,m,n[F; γ+] denote the set
+of (α+, β+) ∈ C2 such that
+(α−, α+, β−, β+, γ−, −1, γ+) ∈ ˙W DF0,S
+ℓ,m,n [F].
+(270)
+For (α+, β+) ∈ ˙W DF,S
+ℓ,m,n[F; γ+], let
+˙IDF;S
+ℓ,m,n[F](α+, β+, γ+) = IDF0;S
+ℓ,m,n [F](α−, α+, β−, β+, γ−, −1, γ+).
+(271)
+Then, as adumbrated by Dotsenko and Fateev:
+Proposition 4.4. For any σ ∈ Sℓ,m,n,
+˙IDF;S
+ℓ,m,n[F](α+, β+, γ+) = ˙IDF;S
+ℓ,m,n[F](α+, β+, γ+)σ
+(272)
+for all (α+, β+) ∈ ˙W DF,S
+ℓ,m,n[F; γ+].
+■
+Since ˙W DF,S
+ℓ,m,n[F] depends only on S through |S ∩ I1|, |S ∩ I2|, |S ∩ I3|,
+˙W DF,S
+ℓ,m,n[F; γ+] = ˙W DF,S
+ℓ,m,n[F; γ+]σ,
+(273)
+so the right-hand side of eq. (272) is defined for any (α+, β+) ∈ ˙W DF,S
+ℓ,m,n[F; γ+].
+Proof. Since Sℓ,m,n is generated by transpositions τ of adjacent elements of I1, I2, I3, it suffices to
+consider the case when σ is such a transposition, τ. For notational simplicity, we consider the case
+when τ is a transposition of some j, j + 1 ∈ I2 and j ∈ S. The other cases are similar but involve
+some notational changes.
+Let ˙W DF,1,S
+ℓ,m,n [F; γ+] ⊆ C6 denote the set of (α1,+, α2,+, α3,+, β1,+, β2,+, β3,+) ∈ C6 such that
+(α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ−, −1, γ+) ∈ ˙W DF1,S
+ℓ,m,n [F],
+(274)
+where α−,ν = −γ−α+,ν and β−,ν = −γ−β+,ν. It suffices to prove that, for any σ ∈ Sℓ,m,n,
+˙IDF,1,;S
+ℓ,m,n [F](α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ+)
+= ˙IDF,1;S
+ℓ,m,n [F](α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ+)σ
+(275)
+for all (α1,+, α2,+, α3,+, β1,+, β2,+, β3,+) ∈ ˙W DF,1,S
+ℓ,m,n [F; γ+], where
+˙IDF,1;S
+ℓ,m,n [F](α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ+) =
+˙Iℓ,m,n[F](αDF,1, βDF,1, γDF),
+(276)
+where αDF,1, βDF,1 are defined as αDF1, βDF1, using α−,ν = −γ−α+,ν and β−,ν = −γ−β+,ν.
+First observe that there exists a nonempty, open subset
+O ⊂ ˙W DF,1,S
+ℓ,m,n [F; γ+]
+(277)
+(containing an affine cone) such that (αDF,1, βDF,1, γDF) ∈ O{1,τ} whenever (α+,1, . . . , β+,3) ∈ O,
+where O{1,τ} is defined as in §3.4. We can choose O such that ℜα±,2, ℜβ±,2 > 0 everywhere in O.
+Since ˙W DF,S
+ℓ,m,n[F; γ+] is connected, it suffices via analyticity to prove the result for
+(α1,+, α2,+, α3,+, β1,+, β2,+, β3,+) ∈ O.
+(278)
+We write α± in place of α±,2 and β± in place of β±,2 below.
+
+46
+ETHAN SUSSMAN
+We can apply Proposition 3.6 for (α+,1, . . . , β+,3) ∈ O. By Proposition 3.6, it suffices to check
+that, whenever all of the zk’s besides zj and zj+1 are somewhere in the interior of the corresponding
+contour in eq. (205),
+�
+Γ[0,1],+,j−ℓ
+�
+zα+
+j
+(1 − zj)β+zα−
+j+1(1 − zj+1)β−�
+�
+1≤j0<k0≤N
+{j0,k0}∩{j,j+1}̸=∅
+(zk0 − zj0)2γj0,k0
+�
+F dzj dzj+1 = 0,
+(279)
+where the inner integral is taken over a small circle around zj+1, for each zj+1 ∈ Γ[0,1],+,j−ℓ\{0, 1}.
+Since the integrand is holomorphic in zj in a punctured neighborhood of zj+1, we apply the
+Cauchy residue theorem to deduce that the left-hand side is proportional to
+�
+Γ[0,1],+,j−ℓ
+G0
+∂G
+∂zj
+���
+zj=zj+1
+dzj+1,
+(280)
+where
+G0(z1, . . . , zN) = zα−
+j+1(1 − zj+1)β−�
+�
+j0∈([N]\S)\{j+1}
+(zj+1 − zj0)2γ−��
+�
+j0∈S\{j}
+(zj+1 − zj0)−2�
+, (281)
+G(z1, . . . , zN) = zα+
+j
+(1 − zj)β+�
+�
+j0∈([N]\S)\{j+1}
+(zj − zj0)−2��
+�
+j0∈S\{j}
+(zj − zj0)2γ+�
+F.
+(282)
+We are choosing branch cuts such that we do not encounter any as zj, zj+1 are integrated along
+Γ[0,1],+,j−ℓ (except at the endpoints). Other than that, it is not important what the precise choice
+of branch cuts are.
+The integrand in eq. (280) is computed to be
+G0
+∂G
+∂zj
+���
+zj=zj+1
+=
+� α+
+zj+1
+−
+β+
+1 − zj+1
++ 2γ+
+�
+j0∈S\{j}
+1
+zj+1 − zj0
+− 2
+�
+j0∈([N]\S)\{j+1}
+1
+zj+1 − zj0
++ ∂zjF
+F
+���
+zj=zj+1
+�
+H,
+(283)
+where H = G0G|zj=zj+1. On the other hand,
+∂H
+∂zj+1
+=
+�α− + α+
+zj+1
+− β− + β+
+1 − zj+1
++(2γ+−2)
+�
+j0∈S\{j}
+1
+zj+1 − zj0
++(2γ−−2)
+�
+j0∈([N]\S)\{j+1}
+1
+zj+1 − zj0
++ ∂zj+1(F|zj=zj+1)
+F|zj=zj+1
+�
+H.
+(284)
+Since 1 − γ− = α−1
++ (α− + α+) = β−1
++ (β− + β+) = γ−1
++ (γ+ − 1) = λ, and since F ∈ DFSym(N, S, λ),
+G0
+∂G
+∂zj
+���
+zj=zj+1
+= 1
+λ
+∂H
+∂zj+1
+.
+(285)
+Consequently,
+�
+Γ[0,1],+,j−ℓ
+G0
+∂G
+∂zj
+���
+zj=zj+1
+dzj+1 = 1
+λ
+�
+Γ[0,1],+,j−ℓ
+∂H
+∂zj+1
+dzj+1.
+(286)
+The right-hand side is proportional to
+�
+p
+∂H
+∂zj+1
+dzj+1
+(287)
+if α+, β+ /∈ Z, where p is a Pochhammer contour in C2\{0, 1} staying sufficiently close to Γ[0,1],+,j−ℓ.
+Lifting to a cover of a neighborhood of Γ[0,1],+,j−ℓ on which H lifts to a single-valued analytic
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+47
+function, we can conclude (using analyticity) that the integral in eq. (287) is zero. By analyticity,
+we can remove the nonintegrality constraint on α+, β+ to conclude that
+�
+Γ[0,1],+,j−ℓ
+∂H
+∂zj+1
+dzj+1 = 0
+(288)
+for all (α+,1, . . . , β+,3) ∈ O.
+□
+Proposition 4.5 (Cf. [DF85a]). Given the setup above, for arbitrary S:
+• For all (α+, β+) ∈ ˙W DF,S
+N,0,0[F; γ+] ∩ ˙W DF,S
+0,N,0[F; γ+] such that β± + m±γ± /∈ Z for any m± ∈
+{0, . . . , N± − 1}
+˙IDF,S
+0,N,0[F](α+, β+, γ+) = (−1)N ˙IDF,S
+N,0,0[F](α+, β+, γ+)
+� N+−1
+�
+m+=0
+sin(π(α+ + β+ + (N+ + m+ − 1)γ+))
+sin(π(β+ + m+γ+))
+�� N−−1
+�
+m−=0
+sin(π(α− + β− + (N− + m− − 1)γ−))
+sin(π(β− + m−γ−))
+�
+.
+(289)
+• For all (α+, β+) ∈ ˙W DF,S
+0,N,0[F; γ+] ∩ ˙W DF,S
+0,0,N[F; γ+] such that α± + m±γ± /∈ Z for any m± ∈
+{0, . . . , N± − 1},
+˙IDF,S
+0,N,0[F](α+, β+, γ+) = (−1)N ˙IDF,S
+0,0,N[F](α+, β+, γ+)
+� N+−1
+�
+m+=0
+sin(π(α+ + β+ + (N+ + m+ − 1)γ+))
+sin(π(α+ + m+γ+))
+�� N−−1
+�
+m−=0
+sin(π(α− + β− + (N− + m− − 1)γ−))
+sin(π(α− + m−γ−))
+�
+.
+(290)
+For S = {1, . . . , N+}, we also have:
+• For all (α+, β+) ∈
+˙W DF,S
+N+,N−,0[F; γ+] ∩ ˙W DF,S
+0,N,0[F; γ+] such that β+ + m+γ+ /∈ Z for any
+m+ ∈ {0, . . . , N+ − 1}
+˙IDF,S
+0,N,0[F](α+, β+, γ+) = (−1)N+ ˙IDF,S
+N+,N−,0[F](α+, β+, γ+)
+� N+−1
+�
+m+=0
+sin(π(α+ + β+ + (N+ + m+ − 1)γ+))
+sin(π(β+ + m+γ+))
+�
+.
+(291)
+• For all (α+, β+) ∈
+˙W DF,S
+0,N,0[F; γ+] ∩ ˙W DF,S
+0,N+,N−[F; γ+] such that α− + m−γ− /∈ Z for any
+m− ∈ {0, . . . , N− − 1},
+˙IDF,S
+0,N,0[F](α+, β+, γ+) = (−1)N− ˙IDF,S
+0,N+,N−[F](α+, β+, γ+)
+� N−−1
+�
+m−=0
+sin(π(α− + β− + (N− + m− − 1)γ−))
+sin(π(α− + m−γ−))
+�
+.
+(292)
+Similarly, for S = {N − N+ + 1, . . . , N}, we have:
+• For all (α+, β+) ∈
+˙W DF,S
+N−,N+,0[F; γ+] ∩ ˙W DF,S
+0,N,0[F; γ+] such that β− + m−γ− /∈ Z for any
+m− ∈ {0, . . . , N− − 1}
+˙IDF,S
+0,N,0[F](α+, β+, γ+) = (−1)N− ˙IDF,S
+N−,N+,0[F](α+, β+, γ+)
+� N−−1
+�
+m−=0
+sin(π(α− + β− + (N− + m− − 1)γ−))
+sin(π(β− + m−γ−))
+�
+.
+(293)
+
+48
+ETHAN SUSSMAN
+• For all (α+, β+) ∈
+˙W DF,S
+0,N,0[F; γ+] ∩ ˙W DF,S
+0,N−,N+[F; γ+] such that α+ + m+γ+ /∈ Z for any
+m+ ∈ {0, . . . , N+ − 1},
+˙IDF,S
+0,N,0[F](α+, β+, γ+) = (−1)N+ ˙IDF,S
+0,N−,N+[F](α+, β+, γ+)
+� N+−1
+�
+m+=0
+sin(π(α+ + β+ + (N+ + m+ − 1)γ+))
+sin(π(α+ + m+γ+))
+�
+.
+(294)
+■
+Proof. Follows from a repeated application of Proposition 3.10, as in the proof of Proposition 4.2.
+The only difference with the proof of Proposition 4.2 is that we appeal to Proposition 4.4 to show
+(instead of it being an automatic consequence of symmetry) that
+˙Iℓ,m,n[F](αDF,S, βDF,S, γDF,S) = ˙Iℓ,m,n[F](αDF,S, βDF,S, γDF,S)σℓ
+(295)
+˙Iℓ−1,m+1,n[F](αDF,S, βDF,S, γDF,S)σℓ = ˙Iℓ−1,m+1,n[F](αDF,S, βDF,S, γDF,S)σℓ+m
+(296)
+˙Iℓ−1,m,n+1[F](αDF,S, βDF,S, γDF,S)σℓ+m = ˙Iℓ−1,m,n+1[F](αDF,S, βDF,S, γDF,S)σN .
+(297)
+□
+Proposition 4.6. For γ+ ∈ C\{0, 1}, the functions IDF,S
+N;Reg[F](α+, β+, γ+) defined by
+˙IDF,S
+N
+[F](α+, β+, γ+) =
+� �
+±
+N±
+�
+j=1
+sin(π(α± + β± + (N± + j − 2)γ±))
+sin(π(α± + (j − 1)γ±)) sin(π(β± + (j − 1)γ±))
+�
+× IDF,S
+N;Reg[F](α+, β+, γ+)
+(298)
+extend to entire functions IDF,S
+N;Reg[F] : C2
+α+,β+ → C.
+■□
+Proof. The proof is very similar to that used to prove Proposition 4.3. Using the previous proposition
+with Proposition 4.4, it suffices to note that the union of all nine of the sets
+˙W DF,S−
+N,0,0 [F; γ+], ˙W DF,S−
+0,N,0 [F; γ+], ˙W DF,S−
+0,0,N [F; γ+], ˙W DF,S−
+N−,N+,0[F; γ+], ˙W DF,S−
+N−,0,N+[F; γ+],
+˙W DF,S−
+0,N−,N+[F; γ+], ˙W DF,S+
+N+,N−,0[F; γ+], ˙W DF,S+
+N+,0,N−[F; γ+], ˙W DF,S+
+0,N+,N−[F; γ+] ⊂ C2
+α+,β+,
+(299)
+where S+ = {1, . . . , N+} and S− = S∁, is the complement of locally finite set of points, and then the
+result follows via Hartog’s theorem.
+□
+Appendix A. The N = 2 case
+We now consider the N = 2 case in some detail, beginning with the formula
+S2(α, β, γ) = Γ(1 + α1)Γ(1 + β2)Γ(2 + 2γ1,2 + α1 + α2)Γ(1 + 2γ1,2)
+Γ(2 + α1 + 2γ1,2)Γ(3 + α1 + α2 + β2 + 2γ1,2)
+· 3F2(a, b; 1),
+(300)
+a = (a1, a2, a3) = (1 + α1, −β1, 2 + 2γ1,2 + α1 + α2) and b = (b1, b2) = (2 + α1 + 2γ1,2, 3 + α1 + α2 +
+β2 + 2γ1,2). This is asymmetric in the role of the α’s and β’s, but there is an analogous formula
+with the α’s and β’s on the right-hand side switched. Some of the singularities of S2 are manifest in
+this formula, but others are hidden in the 3F2 factor.
+Consider now the Dotsenko–Fateev-like integral
+IDF0
+2
+(α1, α2, β1, β2, γ) =
+� 1
+0
+� 1
+0
+xα1
+1 xα2
+2 (1 − x1)β1(1 − x2)β2(x2 − x1 + i0)2γ dx1 dx2.
+(301)
+By the previous proposition:
+
+THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+49
+Corollary A.0.1.
+˙IDF0
+2
+(α1, α2, β1, β2, γ) = Γ(2 + 2γ + α1 + α2)Γ(1 + 2γ)
+�
+Γ(1 + α1)Γ(1 + β2)3F2(a, b; 1)
+Γ(2 + α1 + 2γ)Γ(3 + α1 + α2 + β2 + 2γ)
++ e2πiγ
+Γ(1 + α2)Γ(1 + β1)3F2(a′, b′; 1)
+Γ(2 + α2 + 2γ)Γ(3 + α1 + α2 + β1 + 2γ)
+�
+,
+(302)
+where a′ = (a′
+1, a′
+2, a′
+3) = (1 + α2, −β2, 2 + 2γ + α1 + α2) and b′ = (b′
+1, b′
+2) = (2 + α2 + 2γ, 3 + α1 +
+α2 + β1 + 2γ).
+■□
+The formula eq. (302) is not suitable for analytic continuation to γ = −1, for which we instead
+use the method described in §3.4. That yields
+˙I2(α1, α2, β1, β2, γ) = Γ(1 + α1)Γ(1 + β1)
+Γ(2 + α1 + β1)
+�
+Γ
+�
+zα2+2γ(1 − z)β2
+× 2F1
+�
+− 2γ, 1 + α1, 2 + α1 + β1; 1
+z
+��
+dz,
+(303)
+where the Γ is a trapezoidal contour in the upper-half of the complex plane. This formula can be
+used to numerically compute ˙IDF0
+2
+(α1, α2, β1, β2, γ) for γ with large negative real part, as long as
+α1, α2, β1, β2 have sufficiently large positive real part relative to γ.
+We illustrate the method of proof of Theorem 1.1 with the computation of the residues associated
+with α− + α+ + 2γ ∈ Z−2−d. Introducing coordinates ϱ = x2 and λ = x1/x2,
+S2(α1, α2, β1, β2, γ) =
+� 1
+0
+� x2
+0
+xα1
+1 xα2
+2 (1 − x1)β1(1 − x2)β2(x2 − x1)2γ dx1 dx2
+=
+� 1
+0
+� 1
+0
+ϱ1+α1+α2+2γλα1(1 − λϱ)β1(1 − ϱ)β2(1 − λ)2γ dλ dϱ.
+(304)
+Expanding (1 − λϱ)β1(1 − ϱ)β2 in Taylor series around ϱ = 0, we have
+(1 − λϱ)β1(1 − ϱ)β2 =
+∞
+�
+k=0
+k
+�
+κ,κ,κ′=0
+ak,κ,κ,κ′βκ
+1 βκ′
+2 λκϱk
+(305)
+for some ak,κ,κ,κ′ ∈ R. Then, computing the outer integral term by term and using the formula for
+the β-function for the inner integral,
+S2(α1, α2, β1, β2, γ) ∼
+∞
+�
+k=0
+1
+2 + α1 + α2 + 2γ + k
+k
+�
+κ,κ,κ′=0
+ak,κ,κ,κ′βκ
+1 βκ′
+2
+Γ(1 + α1 + κ)Γ(1 + 2γ)
+Γ(2 + α1 + 2γ + κ)
+(306)
+where the ‘∼’ means modulo an error which is not singular at (all but a positive codimension subset
+of) the hyperplane under investigation. The right-hand side of this has an apparent pole whenever
+α1 + α2 + 2γ ∈ Z≤−2.
+We now examine some special cases. Fix d ∈ N. First consider
+S2[xd](α, β, γ) = S2(α, α+d, β, β, γ) =
+� 1
+0
+� x2
+0
+xα
+1 xd+α
+2
+(1−x1)β(1−x2)β(x2−x1)2γ dx1 dx2. (307)
+By Equation (300),
+S2[xd](α, β, γ) = Γ(1 + α)Γ(1 + β)Γ(1 + 2γ)Γ(2 + 2α + 2γ + d)
+Γ(2 + α + 2γ)Γ(3 + 2α + β + 2γ + d)
+· 3F2(a, b; 1),
+(308)
+where now a = (a1, a2, a3) = (1+α, −β, 2+2α+2γ+d) and b = (b1, b2) = (2+α+2γ, 3+2α+β+2γ+d).
+A numerically generated plot of the absolute value of the right-hand side is given in the case d = 2
+
+50
+ETHAN SUSSMAN
+Figure 13. The absolute values of the right-hand sides of eq. (308) (left) and
+eq. (311) (right) plotted against α, for β = 1/2 and γ = 1/3 fixed. The singularities
+predicted in eq. (309), eq. (312) have been drawn as dotted vertical lines, those
+associated with {α ∈ Z} in blue and those associated with {α + γ ∈ 2−1Z} in red.
+It appears that all of the poles that could be present are present. The apparent
+zeroes of S2[F](α, 1/2, 1/3) in the depicted range of α have been marked with dotted
+black lines and numerically computed to be ≈ −2.48503 for F = x2 and ≈ −3.06833,
+−3.57013, and −4.08562 for F = y2.
+in Figure 13. Applying Theorem 1.1, in which α1,∗ = α, α2,∗ = d+2α +2γ, β1,∗ = β, β2,∗ = 2β +2γ,
+and γ1,2,∗ = 2γ, we deduce that S2[xd](α, β, γ) extends to an analytic function on
+C3
+α,β,γ
+��
+{α ∈ Z≤−1}∪{α+γ ∈ 2−1Z≤−2−d}∪{β ∈ Z≤−1}∪{β +γ ∈ 2−1Z≤−2}∪{γ ∈ 2−1Z≤−1}
+�
+.
+(309)
+In the d = 2 case, this can be seen in Figure 13.
+On the other hand, consider
+S2[yd](α, β, γ) = S2(α+d, α, β, β, γ) =
+� 1
+0
+� x2
+0
+xd+α
+1
+xα
+2 (1−x1)β(1−x2)β(x2 −x1)2γ dx1 dx2. (310)
+By Equation (20),
+S2[yd](α, β, γ) = Γ(1 + α + d)Γ(1 + β)Γ(1 + 2γ)Γ(2 + 2α + 2γ + d)
+Γ(2 + α + 2γ + d)Γ(3 + 2α + β + 2γ + d)
+· 3F2(a′, b′; 1),
+(311)
+where a′ = (a′
+1, a2, a3) = (1 + d + α, −β, 2 + d + 2α + 2γ) and b′ = (b′
+1, b2) = (2 + d + α + 2γ, 3 + d +
+2α + β + 2γ). We again apply Theorem 1.1, but now α1,∗ = d + α, α2,∗ = d + 2α + 2γ, in order to
+deduce that S2[yd](α, β, γ) extends analytically to
+C3
+α,β,γ
+��
+{α ∈ Z≤−1−d}∪{α+γ ∈ 2−1Z≤−2−d}∪{β ∈ Z≤−1}∪{β+γ ∈ 2−1Z≤−2}∪{γ ∈ 2−1Z≤−1}
+�
+.
+(312)
+See Figure 13 for a numerically generated plot.
+If we instead pick F(x, y) = xd + yd, which is in some sense the symmetrization of the previous
+two examples, the situation looks very different. Combining the formulas above yields
+S2[xd + yd](α, β, γ) = (S2[xd] + S2[yd])(α, β, γ) = Γ(1 + α)Γ(1 + β)Γ(1 + 2γ)Γ(2 + 2α + 2γ + d)
+Γ(2 + α + 2γ)Γ(3 + 2α + β + 2γ + d)
+×
+�
+3F2(a, b; 1) + Γ(1 + α + d)Γ(2 + α + 2γ)
+Γ(1 + α)Γ(2 + α + 2γ + d) · 3F2(a′, b′; 1)
+�
+.
+(313)
+
+[S2[α-](α, 1/2, 1/3)]
+VS. Qα
+40
+30
+-
+20
+10
+-4
+0S2[y21(α, 1/2, 1/3)
+VS. α
+40
+30
+20
+10
+-2
+4
+αTHE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+51
+Figure 14. The absolute value of the right side of eq. (313), plotted against
+α ∈ (0, −6) (left) and α ∈ (−6.1, −4.75) (right), for β = 1/2 and γ = 1/3 fixed.
+Singularities associated with {α ∈ Z} are marked with blue lines and those with
+{α + γ ∈ Z} with red. The zeroes associated with {α + β + γ ∈ Z} are marked in
+green and those with {α + β + 2γ ∈ Z} in orange. The location of the second plot is
+marked (roughly) as an inset on the left plot.
+On the other hand, by Theorem 1.2, we know that S2[xd + yd] extends analytically to
+C3
+α,β,γ
+��
+{α ∈ Z≤−1−¯δ1} ∪ {α + γ ∈ Z≤−2−¯δ2} ∪ {β ∈ Z≤−1−¯
+δ
+1} ∪ {β + γ ∈ Z≤−1−¯
+δ
+2}
+∪ {γ ∈ 2−1Z≤−1, γ /∈ Z}
+�
+,
+(314)
+where ¯δ1, ¯
+δ
+1, ¯δ2, ¯
+δ
+2 are as in the theorem. In this example, ¯δ1, ¯
+δ
+1, ¯
+δ
+2 = 0, ¯δ2 = ⌈d/2⌉, ¯d2 = d, and
+¯d1 = ⌊d/2⌋. See Figure 14, in which d = 2. In the sum eq. (313), the poles of the individual
+summands at such 2α + 2γ ∈ Z≤−2−d ∩ (2Z + 1) (which we can see from Figure 13 exist) must
+cancel.
+By eq. (306), the residue of S2[xd + yd](α, β, γ) at 2α + 2γ = −2 − d − k for k ∈ N is proportional
+to
+k
+�
+κ,κ,κ′=0
+ak,κ,κ,κ′βκ+κ′�
+Γ(1 + α + κ)
+Γ(2 + α + 2γ + κ) +
+Γ(1 + α + κ + d)
+Γ(2 + α + 2γ + κ + d)
+�
+.
+(315)
+Since this has to vanish whenever −2 − d − k is odd and (α, γ) ∈ {2α + 2γ = −2 − d − k} ⊂ C2
+α,γ is
+such that the function in eq. (315) is well-defined, we conclude that, for such k and (α, γ),
+k
+�
+κ=0
+��
+�
+κ′+κ′′=κ
+ak,κ,κ′,κ′′
+��
+Γ(1 + α + κ)
+Γ(2 + α + 2γ + κ) +
+Γ(1 + α + κ + d)
+Γ(2 + α + 2γ + κ + d)
+��
+= 0
+(316)
+for each κ ∈ {0, . . . , k}.
+A direct algebraic proof of this fact is not entirely trivial, but it is
+straightforward to check case-by-case.
+The function S2;Reg[xd + yd](α, β, γ) is plotted as a function of α ∈ C in Figure 15, still in the
+case d = 2 – for fixed β, γ. As expected, it appears to have no singularities, in accordance with
+Theorem 1.2. Unlike in the case F = 1, where Selberg’s formula shows that S2;Reg[1](α, β, γ) is
+constant, S2;Reg[xd + yd] is nonconstant.
+
+[S2[α² +y²](α, 1/2, 1/3)
+VS. α
+12
+10
+8
+6
+2
+(Inset)
+-2
+0S2α- + y-(α, 1/2, 1/3)/ vs. α (Inset)
+0.25
+0.20
+0.15
+0.10
+0.05
+0.00
+-6.0-5.8
+ -5.6 -5.4 -5.2 -5.0 -4.852
+ETHAN SUSSMAN
+Figure 15. The function S2;Reg[x2 + y2](α, 1/2, 1/3) defined by eq. (29), plotted as
+a function of α ∈ C.
+Consider now ˙IDF
+2
+(α+, β+) = ˙IDF0
+2
+(α+, −α+, β+, −β+, 1), which is a DF-symmetric integral with
+γ± = −1. This is given concretely by
+˙IDF
+2
+(α+, β+) = Γ(1 + α+)Γ(1 + β+)
+Γ(2 + α+ + β+)
+�
+Γ
+�
+zα++2γ(1 − z)β+
+× 2F1
+�
+− 2γ, 1 + α+, 2 + α+ + β+; 1
+z
+��
+dz
+(317)
+when the real parts of α+, β+ are sufficiently large. The Dotsenko–Fateev claim, eq. (48), is, up to
+a sign, that
+˙IDF
+2
+(α+, β+) = −Γ(1 + α+)Γ(1 + β+)Γ(α+)Γ(β+)
+2Γ(1 + α+ + β+)Γ(α+ + β+)
+.
+(318)
+Appendix B. Explicit coordinates on [0, 1)N
+tb
+In this appendix, we discuss the total boundary blowup [0, 1)N
+tb, the mwc constructed by blowing
+up all of the facets of [0, 1)N, starting with those of the lowest dimension. For each nonempty subset
+S ⊆ {1, . . . , N}, let FS denote the face of [0, 1)N
+tb corresponding to the facet {j ∈ S ⇒ xj = 0}
+of [0, 1)N
+x . Tracing through the construction of the total boundary blowup, we have the following
+explicit choice of boundary defining functions (which are different, if N ≥ 3, from the recursively
+defined boundary defining functions discussed in the introduction to §2) of the various faces of
+[0, 1)N
+tb:
+Proposition B.1. The function
+xFS = xFS,N =
+�
+S0⊇S
+� �
+j∈S0
+xj
+�(−1)|S|−|S0|
+(319)
+serves as a bdf of FS. Suppose that I is a (possibly empty) set of nested nonempty subsets of
+{1, . . . , N}. Then,
+fI = {xFS = 0 for all S ∈ I}
+(320)
+is a codimension |I| facet of [0, 1)N
+tb. This defines a bijective correspondence between the set of nested
+nonempty subsets of {1, . . . , N} and the set of facets of [0, 1)N
+tb. If p lies in the interior of fI, then,
+letting σ ∈ SN denote any permutation consistent with I,
+ϱ = xσ(1), ˆxσ(2) = xσ(2)/xσ(1), · · · , ˆxσ(N) = xσ(N)/xσ(N−1)
+(321)
+give a local set of coordinates near p.
+■□
+
+[S2;Reg[α? +y°](α, 1/2, 1/3)]
+VS. Qα
+15
+10
+2
+5
+1
+0
+-6
+0
+-4
+-1
+-2
+-2
+0THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS
+53
+Here, we say that σ ∈ SN is consistent with I if, whenever j < k, σ(j) ∈ S ∈ I ⇒ σ(k) ∈ S. We
+can cover [0, 1)N
+tb with the N! coordinate charts whose restrictions to the interior are of the form
+{0 < xσ(N) < 2xσ(N−1) < · · · < 2Nxσ(1) < 2N} ∩ (0, 1)N
+→ [0, 1)xσ(1) × [0, 2)xσ(2)/xσ(1) × · · · × [0, 2)xσ(N)/xσ(N−1),
+(322)
+for σ ∈ SN.
+The preceding proposition is used to prove:
+Proposition B.2. For any M ∈ {1, . . . , N − 1} and nonempty Q ⊆ {1, . . . , M},
+xFQ,M =
+�
+Q0⊆{M+1,...,N}
+xFQ∪Q0,N
+(323)
+in (0, 1)N
+x .
+■
+Proof. A factor of �
+j∈Q∪Q0 xj appears on the right-hand side of eq. (323) to the power
+�
+Q1⊆Q0
+(−1)|Q0|,
+(324)
+which is, by the binomial theorem, +1 if Q0 = ∅ and 0 otherwise. Thus, �
+Q0⊆{M+1,...,N} xFQ∪Q0,N =
+�
+j∈Q xj.
+□
+The full proof of Proposition B.1 is somewhat incidental to the rest of the paper, so we merely
+illustrate the argument (which generalizes to the N ≥ 3 case, and applies in an even simpler form
+to the N = 2 case) in the case N = 3. The total boundary blowup [0, 1)N
+tb is defined as
+[[0, 1)3; {x, y, z = 0}; {y, z = 0}; {x, z = 0}, {x, y = 0}],
+(325)
+where the first blowup is that of {x, y, z = 0} and must be performed first. The other three blowups
+can be performed in any order, and each order yields a canonically diffeomorphic mwc. The first
+blowup, yielding [[0, 1)3; {x, y, z = 0}], is
+y
+z
+x
+y/(x + y + z)
+z/(x + y + z)
+x/(x + y + z)
+x + y + z
+where we are marking the faces with boundary defining functions (using the Cartesian coordinates
+x, y, z in place of x1, x2, x3). The choice of bdfs on the blowup are in accordance with the prescription
+in the introduction of §2.
+The next blowup, yielding
+[[0, 1)3; {x, y, z = 0}; {y, z = 0}],
+(326)
+is
+
+54
+ETHAN SUSSMAN
+y/(x + y + z)
+z/(x + y + z)
+x/(x + y + z)
+x + y + z
+y/(y + z)
+z/(y + z)
+x/(x + y + z)
+x + y + z
+(y + z)/(x + y + z)
+Again, the choices of bdfs are in accordance with §2.
+Next, we blow up the facet of [[0, 1)3; {x, y, z = 0}; {y, z = 0}] corresponding to the y-axis.
+Because the previous blowup was located away from the facet being blown up now, we can use the
+sum
+x
+x + y + z +
+z
+x + y + z =
+x + z
+x + y + z
+(327)
+of the bdfs of the adjacent faces in [[0, 1)3; {x, y, z = 0}] as a bdf of the front face of the current
+blowup rather than
+x
+x + y + z +
+z
+y + z = xy + 2xz + yz + z2
+(y + z)(x + y + z) ,
+(328)
+which would be the prescription in the introduction §2. The choices in eq. (327), eq. (328) are
+equivalent, in the sense that their quotient is a smooth, nonvanishing function on [[0, 1)3; {x, y, z =
+0}; {y, z = 0}; {x, z = 0}]. Given that eq. (327) serves as a bdf of the front face, the quotient
+x/(x + y + z)
+(x + z)/(x + y + z) =
+x
+x + z
+(329)
+serves as a bdf in [[0, 1)3; {x, y, z = 0}; {y, z = 0}; {x, z = 0}] for the lift of the yz-plane, and
+z/(y + z)
+(x + z)/(x + y + z) = z(x + y + z)
+(x + z)(y + z)
+(330)
+serves as a bdf for the lift of the xy-plane. In summary, the third blowup is
+y/(y + z)
+z/(y + z)
+x/(x + y + z)
+x + y + z
+(y + z)/(x + y + z)
+y/(y + z)
+z(x + y + z)(x + z)−1(y + z)−1
+x/(x + z)
+x + y + z
+(y + z)/(x + y + z)
+(x + z)/(x + y + z)
+The final blowup, yielding [0, 1)3
+tb = [[0, 1)3; {x, y, z = 0}; {y, z = 0}; {x, z = 0}, {x, y = 0}], is
+similar. We use (x + y)/(x + y + z) as a bdf of the blowup of the face corresponding to the z-axis,
+and we can then use
+x/(x + z)
+(x + y)/(x + y + z) = x(x + y + z)
+(x + y)(x + z)
+(331)
+y/(y + z)
+(x + y)/(x + y + z) = y(x + y + z)
+(x + y)(y + z)
+(332)
+
+REFERENCES
+55
+as bdfs of the faces corresponding to the yz- and xz-planes, respectively. Thus, we up with
+y(x + y + z)(x + y)−1(y + z)−1
+z(x + y + z)(x + z)−1(y + z)−1
+x(x + y + z)(x + y)−1(x + z)−1
+x + y + z
+(y + z)/(x + y + z)
+(x + z)/(x + y + z)
+(x + y)/(x + y + z)
+as our final result.
+References
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+0550-3213(89)90568-3 (cit. on pp. 8, 9).
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+Part 1”. Comm. Math. Phys. 333.1 (2015), 389–434. doi: 10.1007/s00220-014-2189-4 (cit. on p. 2).
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+[FK15c]
+. “A solution space for a system of null-state partial differential equations: Part 3”. Comm. Math.
+Phys. 333.2 (2015), 597–667. doi: 10.1007/s00220-014-2190-y (cit. on p. 2).
+[FK15d]
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+G. Felder and R. Silvotti. “Free field representation of minimal models on a Riemann surface”. Phys. Lett.
+B 231.4 (1989), 411–416. doi: 10.1016/0370-2693(89)90685-0 (cit. on p. 1).
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+17–40 (cit. on pp. 1, 2).
+[FW08]
+P. J. Forrester and S. O. Warnaar. “The importance of the Selberg integral”. Bull. Amer. Math. Soc.
+(N.S.) 45.4 (2008), 489–534. doi: 10.1090/S0273-0979-08-01221-4 (cit. on pp. 1, 2, 6, 7, 42).
+
+56
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+Y. Kanie and A. Tsuchiya. “Fock space representations of the Virasoro algebra. Intertwining operators”.
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+11, 15).
+[KT86b]
+. “Fock space representations of Virasoro algebra and intertwining operators”. Proc. Japan Acad.
+Ser. A Math. Sci. 62.1 (1986), 12–15. url: http://projecteuclid.org/euclid.pja/1195514490 (cit. on
+pp. 2, 5, 9).
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+J. Lenells and F. Viklund. “Asymptotic analysis of Dotsenko-Fateev integrals”. Ann. Henri Poincaré 20.11
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+097, front matter+53. doi: 10.1007/jhep08(2017)097. url: https://doi.org/10.1007/jhep08(2017)
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+Email address: ethanws@mit.edu
+Department of Mathematics, Massachusetts Institute of Technology, Massachusetts 02139-4307,
+USA
+
diff --git a/5dE2T4oBgHgl3EQfOgbi/content/tmp_files/load_file.txt b/5dE2T4oBgHgl3EQfOgbi/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..62f7b638005b901ba8708449c18ff5412155777d
--- /dev/null
+++ b/5dE2T4oBgHgl3EQfOgbi/content/tmp_files/load_file.txt
@@ -0,0 +1,2886 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf,len=2885
+page_content='THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE  INTEGRALS  ETHAN SUSSMAN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We discuss the meromorphic continuation of certain hypergeometric integrals modeled  on the Selberg integral, including the 3-point and 4-point functions of BPZ’s minimal models of  2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the “Coulomb gas formalism”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  This is accomplished via a geometric analysis of the singularities of the integrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the case  that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call  “DF-symmetric,” we show that a number of apparent singularities are removable, as required for the  construction of the minimal models via these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Associahedra  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Meromorphic continuation  26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Removing singularities  40  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The N = 2 case  48  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Explicit coordinates on [0, 1)N  tb  52  References  55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Introduction  Let  △N = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) ∈ [0, 1]N : x1 ≤ · · · ≤ xN}  (1)  denote the standard N-simplex, which we consider as a subset of CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We study in this note  Selberg-like integrals, by which we mean definite integrals of the form  SN[F](α, β, γ) =  �  △N  � N  �  j=1  xαj  j (1 − xj)βj  ��  �  1≤j<k≤N  (xk − xj)2γj,k  �  F(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) dx1 · · · dxN, (2)  for N ∈ N+, F ∈ C∞(△N), and α = {αj}N  j=1, β = {βj}N  j=1, γ = {γj,k = γk,j}1≤j<k≤N ⊂ C such  that the integrand above is absolutely integrable on △N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Integrals of this form are relevant to an  array of topics in mathematical physics [FW08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Here, we discuss a particular application to the  Coulomb gas formalism (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' “free field realization,” “Feigin–Fuchs representation,” etcetera) of 2D  CFT [DF84][DF85a][DF85b][FS89][PM97, Chp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 9][FW08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This approach of Dotsenko–Fateev to the  construction of the “minimal models” of Belavin–Polyakov–Zamolodchikov (BPZ) [BPZ84] has been  the subject of substantial interest, but it appears that it has not yet been placed on entirely rigorous  mathematical footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The construction in [FS89][FS92] of the 3-point coefficients of the (1, s)- and  (r, 1)-primary fields and their descendants in the minimal models is satisfactorily rigorous, but it  has remained an open problem to handle the rest of the primary fields at a similarly satisfactory  degree of rigor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' From our perspective, the issue is an insufficient treatment of the meromorphic  Date: January 8th, 2023 (Last update).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Primary 32A20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Secondary 33C60, 33C90, 81T40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='03750v1  [math-ph]  10 Jan 2023  2  ETHAN SUSSMAN  continuation of Selberg-like integrals, which are instead treated somewhat formally in the original  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The issue is that Dotsenko & Fateev (DF) take some of the γ’s to be −1 — see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [DF85a,  Appendix A][FS92, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 27][FW08, §2] — and then the integrand above is, say for F = 1, no longer  integrable over the integral’s domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' As a consequence, the integrals in [DF85a, Appendix A] are  formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' One way of making sense of them is via meromorphic continuation in the exponents of the  integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Dotsenko and Fateev suggest this, but they do not prove that a suitable meromorphic  continuation exists, nor do they discuss the singularities of the extension in sufficient detail to  justify their manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A construction of Kanie–Tsuchiya [KT86a][KT86b] rediscovered by  Mimachi-Yoshida [MY02][MY03][Yos03] yields the existence of some meromorphic continuation  defined for almost all values of the exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' But, this extension is not quite sufficient for our  purposes: it has removable singularities that, while removable, are nontrivial to actually prove  removable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In particular, the Kanie–Tsuchiya construction has an apparent (but isolated, in a  suitable sense) singularity at γ = −1 (see [KT86b, §5, above Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 5]), along with at a few other  problematic affine hyperplanes in the space of possible parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Most of the rigorous work on the analysis of integrals of Dotsenko–Fateev type — see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  [FK15a][FK15b][FK15c][FK15d][LV19] for some recent work — focuses on screened multipoint  correlation functions with at most one screening charge screening per insertion point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Such integrals  are related to the N = 1 case of SN(α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Not much has been done about the N > 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Moreover, while a fair amount of work has gone into the study of general hypergeometric integrals  associated to hyperplane arrangements — the literature on this topic is large, so we just cite  [Var95][AK11] — it does not seem possible to deduce the specific, concrete results below from results  in the current literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We will identify indexed collections of complex numbers (and tuples thereof) with column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For example, we identify γ with an element of CN(N−1)/2 and  (α, β, γ) ∈ CN × CN × CN(N−1)/2  (3)  with an element of C2N+N(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Similar identifications will be made throughout the rest of the  paper without further comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  ΩN =  �  (α, β, γ) ∈ C2N+N(N−1)/2 :  N  �  j=1  xαj  j (1 − xj)βj  �  1≤j<k≤N  (xk − xj)2γj,k ∈ L1(△N)  �  (4)  denote the (open, nonempty) subset of C2N+N(N−1)/2 consisting of the (α, β, γ) ∈ C2N+N(N−1)/2  for which the integrand in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (2) is absolutely integrable on △N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We begin with SN[F] defined as  a function SN[F] : ΩN → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It can be checked – see §2 – that, letting  αj,∗(α, β, γ) =  j  �  j0=1  αj0 + 2  �  j0,k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  1≤j0<k≤j  γj0,k,  βj,∗(α, β, γ) =  N  �  j0=N−j+1  βj0 + 2  �  j0,k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  N−j+1≤j0<k≤N  γj0,k  (5)  for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, and letting  γj,k,∗(α, β, γ) = 2  �  j0,k0∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  j≤j0<k0≤k  γj0,k0  (6)  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  3  for each pair of j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} with j < k,  ΩN =  � N  �  j=1  {ℜαj,∗ > −j}  �  ∩  � N  �  j=1  {ℜβj,∗ > −j}  �  ∩  �  �  1≤j<k≤N  {ℜγj,k,∗ > −(k − j)}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (7)  So, ΩN is nonempty, open, and convex (in particular, connected) and contains all (α, β, γ) ∈  C2N+N(N−1)/2 such that the real parts of the components of α, β, γ are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  To simplify the formula above, let γ0,k,∗ = αk,∗ and γN+1−j,N+1,∗ = βj,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then  ΩN =  �  0≤j<k≤N+1  {(α, β, γ) ∈ C2N+N(N−1)/2 : ℜγj,k,∗ > −(k − j)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (8)  Our first goal is to prove that SN[F] can be analytically continued to a subset  ˙ΩN ⊆ C2N+N(N−1)/2  (9)  having full measure in C2N+N(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In order to describe precisely the structure of the singularity at C2N+N(N−1)/2\\ ˙ΩN, we introduce  some terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let T(N) denote the collection of maximal families I of consecutive subsets  I ⊊ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N + 1} such that  2 ≤ |I| ≤ N + 1 for all I ∈ I and  if I, I′ ∈ I satisfy I ∩ I′ ̸= ∅, then either I ⊆ I′ or I′ ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  “T” stands either for “tree” in “full binary trees” or “Tamari” in Tamari lattice [Tam62][Gey94], and  the elements of T(N) can be thought of as specifying the valid ways of adding a maximal number of  nonredundant parentheses to a string of N + 2 identical characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There are #T(N) = CN+1 such  ways, where CN+1 is the (N + 1)st Catalan number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' To each I ∈ I, we associate the facet  fI = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) ∈ △N : xj = xk for all j, k ∈ I}  (10)  of △N, where x0 = 0 and xN+1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let oI ∈ N denote the order of vanishing of F at fI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (So,  oI = 0 unless F is vanishing identically at fI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=')  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There exist entire functions SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I[F] : C2N+N(N−1)/2 → C associated to the  I ∈ T(N) such that  SN[F](α, β, γ) =  �  I∈T(N)  � �  I∈I  Γ(oI + |I| − 1 + γmin I,max I,∗)  �  SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I[F](α, β, γ)  (11)  for all (α, β, γ) ∈ ΩN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Here, Γ : C\\{−n : n ∈ N} → C is the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' As a consequence of the theorem, there  exists an entire function SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F] : C2N+N(N−1)/2 → C such that  SN[F](α, β, γ) =  �  �  0≤j<k≤N+1  Γ(k − j + γj,k,∗)  �  SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F](α, β, γ)  (12)  for all (α, β, γ) ∈ ΩN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The function SN[F] : ΩN → C admits an analytic continuation ˙SN[F] : ˙ΩN → C  to the domain  ˙ΩN = C2N+N(N−1)/2  α,β,γ  ��� N  �  j=1  {αj,∗ ∈ Z≤−j−δj}  �  ∪  � N  �  j=1  {βj,∗ ∈ Z≤−j−  δ  j}  �  ∪  �  �  1≤j<k≤N  {γj,k,∗ ∈ Z≤−(k−j)−dj,k}  ��  ,  (13)  where Z≤n = {m ∈ Z : m ≤ n} and δj = δj[F] = o{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',j},  δ  j =  δ  j[F] = o{N−j+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N+1}, and  dj,k = dj,k[F] = o{j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■□  4  ETHAN SUSSMAN  The set ˙ΩN contains all elements of C2N+N(N−1)/2 lying outside of a locally finite arrangement  of affine hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Consider F ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Letting [F]d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',dN denote the coefficient of xd1  1 · · · xdN  N  in F, and  letting refl : (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) �→ (1 − x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , 1 − xN), we have  δj[F] = min{d1 + · · · + dj : [F]d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',dj,dj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',dN ̸= 0 for some dj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , dN ∈ N},  (14)  δ  j[F] = min{dN + · · · + dN+1−j : [F ◦ refl]d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',dN−j,dN+1−j,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',dN ̸= 0 for some d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , dN−j ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (15)  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The simplest case is when N = 1 and F = 1 identically, when the Selberg integral is given  by  S1(α, β, γ) = B(α, β) =  � 1  0  xα(1 − x)β dx = Γ(1 + α)Γ(1 + β)  Γ(2 + α + β)  ,  (16)  defined initially for ℜα, ℜβ > −1 via the definite integral and then extended meromorphically via  the formula on the right-hand side above (or via another method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is Euler’s β-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' One  method of meromorphic continuation involves the “Pochhammer contour”  b−1a−1ba ∈ π1(C\\{0, 1}),  (17)  where a, b are the generators of π1(C\\{0, 1}) corresponding to one (say, counterclockwise) circuit  around each of 0, 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  0  1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Pochhammer contour in C\\{0, 1}, up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Then, b−1a−1ba can be lifted to a closed contour p in the cover M of C\\{0, 1} corresponding to  the commutator subgroup of π1(C\\{0, 1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, choosing the basepoint of p appropriately,  B(α, β) =  1  1 − e−2πiα  1  1 − e−2πiβ  �  p  xα(1 − x)β dx,  (18)  where we are now considering xα(1 − x)β as an analytic function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The theorem above tells us  that there exist entire S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,(••)•, S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,•(••) such that  B(α, β) = Γ(1 + α)S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,(••)•(α, β) + Γ(1 + β)S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,•(••)(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (19)  This splitting is not so obvious from the formula B(α, β) = Γ(1 + α)Γ(1 + β)/Γ(2 + α + β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Now consider the case when N = 2 and F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It can be computed that the Selberg-like  integral is then  S2(α, β, γ) = Γ(1 + α1)Γ(1 + β2)Γ(2 + 2γ1,2 + α1 + α2)Γ(1 + 2γ1,2)  Γ(2 + α1 + 2γ1,2)Γ(3 + α1 + α2 + β2 + 2γ1,2)  3F2(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1),  (20)  where a = (a1, a2, a3) = (1 + α1, −β1, 2 + 2γ1,2 + α1 + α2) and b = (b1, b2) = (2 + α1 + 2γ1,2, 3 +  α1 + α2 + β2 + 2γ1,2), where pFq denotes the generalized hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For N = 2, the  theorem above reads  S2(α, β, γ) = Γ(1 + α1)Γ(2 + α1 + α2 + 2γ1,2)S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,((••)•)•(α, β, γ)+  Γ(1 + α1)Γ(1 + β2)S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,(••)(••)(α, β, γ) + Γ(1 + β2)Γ(2 + β1 + β2 + 2γ1,2)S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,•(•(••))(α, β, γ)  + Γ(1 + 2γ1,2)Γ(2 + β1 + β2 + 2γ1,2)S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,•((••)•)(α, β, γ)  + Γ(1 + 2γ1,2)Γ(2 + α1 + α2 + 2γ1,2)S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,(•(••))•(α, β, γ),  (21)  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  5  but once again this splitting is not so obvious from the exact formula eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This example is  explored more in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  The proof below is lower-brow than the twisted homological constructions of [KT86a, §5][KT86b]  and Aomoto [Aom87], as it is based on the method described in [Var95, Chp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This involves  the geometric analysis of the singularities of the Selberg(-like) integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The key observation is  that if the N-simplex is blown up to the N-dimensional associahedron [Sta63][MSS02, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6][Pos09]  (see Figure 2, Figure 6), then the Selberg integrand – which is not polyhomogeneous on △N –  becomes one-step polyhomogeneous (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='a “classical”) on the resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' See §2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This  observation appears, in an essentially equivalent form (albeit with different terminology), already in  [KT86a][KT86b][MY03], though the term “associahedron” does not appear there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Closely related  observations have appeared in the physics literature [Miz17][CKW18][CMT19][Miz20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The application of polyhomogeneity to the proof of the theorem above is given in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In a nutshell,  the classicality of the lift of the Selberg integrand on the associahedron allows us to reduce the  problem to what is essentially a product of one-dimensional cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The faces of the associahedron  are in bijective correspondence with the quantities defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (5), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The correspondence is  depicted in Figure 2 in the case N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The quantities αj,∗, βj,∗, γj,k,∗ are the orders of the Selberg  α1 + α2 + α3 + 2γ1,2 + 2γ1,3 + 2γ2,3  2γ2,3  α1 + α2 + 2γ1,2  2γ1,2 + 2γ1,3 + 2γ2,3  α1  2γ1,2  β1 + β2 + β3 + 2γ1,2 + 2γ1,3 + 2γ2,3  β3  β2 + β3 + 2γ2,3  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The 3-dimensional associahedron, with its faces labeled by the associated  functions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (5), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The C4 = 14 vertices are in correspondence with the 14  elements T(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  integrand at the corresponding faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Each I ∈ T(N) is associated with a minimal facet of the  associahedron, and the I ∈ I are associated with the faces containing that facet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, we have a  geometric interpretation of each of the terms in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Note that ˙ΩN does not contain (α, β, γ) with γj,k = −1 for |j − k| = 1, so the theorem above is  insufficient for the construction of the minimal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Moreover, the theorem cannot be sharpened  while maintaining generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed, the proof of the theorem shows that if F > 0 everywhere in  △N (including the boundary), then  SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F](α, β, γ) ̸= 0  (22)  for any (α, β, γ) ∈ R2N+N(N−1)/2 for which both of  γj,k,∗ ∈ Z≤−(k−j) for precisely one pair of j, k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N + 1} with j < k,  γj,k,∗ > −(k − j) for all other j, k  hold, as for such (α, β, γ) the quantity SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F](α, β, γ) is proportional to a convergent integral of a  positive integrand over the corresponding face of the associahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consequently, SN[F] : ΩN → C  cannot be analytically continued to the complement of any strictly smaller collection of hyperplanes  than that in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  6  ETHAN SUSSMAN  However, for the desired application, we do not need full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Of special importance is the  case when α, β, γ are each “constant,” meaning that, for some α, β, γ ∈ C,  αi = α and βi = β for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, and  γj,k = γ for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} with j < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In this case, we simply write  SN[F](α, β, γ) =  �  △N  � N  �  j=1  xα  j (1 − xj)β��  �  1≤j<k≤N  (xk − xj)2γ�  F(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) dx1 · · · dxN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (23)  We now consider F ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN]SN , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' symmetric polynomial F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This case includes, of course,  Selberg’s original example, in which F = 1, as well as the 3-point coefficients of the (1, s)- and  (r, 1)-primary fields and their descendants in the BPZ minimal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The computation of such  integrals is listed as an open problem in [KT86a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Below, we will introduce a more general notion  of “DF-symmetric” Selberg-like integrals, this including the other 3-point coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' But, for the  purposes of an introductory discussion we focus on the – already interesting – symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The integral eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (23) is defined initially on the subset UN[F] ⊂ C3  α,β,γ given by  UN[F] =  �  (α, β, γ) ∈ C3 : ℜj(α + (j − 1)γ) > −1 − δj[F] and  ℜj(β + (j − 1)γ) > −1 −  δ  j[F] for all j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, and ℜγ > −  1  N − 1  �  ,  (24)  which contains  UN = UN[1] =  �  (α, β, γ) ∈ C3 : min{ℜα, ℜβ}+min{0, (N −1)γ} > −1 and ℜγ > −  1  N − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (25)  An immediate corollary of the theorem above is that the function SN[F] : UN[F] → C defined by  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (23) admits an analytic continuation ˙SN[F](α, β, γ) : ˙UN[F] → C to the domain ˙UN[F] ⊋ UN[F]  given by  ˙UN[F] = C3  α,β,γ  ��� N  �  j=1  {j(α + (j − 1)γ) ∈ Z≤−j−δj}  �  ∪  � N  �  j=1  {j(β + (j − 1)γ) ∈ Z≤−j−  δ  j}  �  ∪  � N−1  �  j=1  {j(j + 1)γ ∈ Z−j}  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (26)  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consider F = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' the Selberg integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In this case, Selberg proved in [Sel44] that  SN(α, β, γ) = SN[1](α, β, γ) is given by  SN(α, β, γ) = 1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  � N  �  j=1  Γ(1 + α + (j − 1)γ)Γ(1 + β + (j − 1)γ)Γ(1 + jγ)  Γ(2 + α + β + (N + j − 2)γ)Γ(1 + γ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (27)  See [FW08] for a review of the history of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  The example of the Selberg integral suggests that, in the symmetric case, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (26) is not the  maximal domain of analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Set  degj[F] = max{d1 + · · · + dj : [F]d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',dj,dj+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',dN ̸= 0 for some dj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , dN ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (28)  (Since F is symmetric, degj[F] = degj[F ◦ refl].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=') Then:  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  7  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any F ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN]SN , there exists an entire function SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] : C3  α,β,γ → C  such that  SN[F](α, β, γ) =  � N  �  j=1  Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯  δ  j + β + (j − 1)γ)Γ(1 + jγ)  Γ(2 + ¯dj + α + β + (N + j − 2)γ)Γ(1 + γ)  �  × SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ)  (29)  for all (α, β, γ) ∈ UN, where ¯δj = ⌈j−1δj[F]⌉, ¯  δ  j = ⌈j−1  δ  j[F]⌉, and ¯dj = ⌊(N − j + 1)−1 degj[F]⌋  for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Thus, SN[F](α, β, γ) admits an analytic continuation ˚SN[F](α, β, γ) : ˚UN[F] → C to the domain  ˚UN[F] ⊋ ˙UN[F] defined by  ˚UN[F] = C3  α,β,γ  ��� N  �  j=1  {α + ¯δj + (j − 1)γ ∈ Z≤−1}  �  ∪  � N  �  j=1  {β + ¯  δ  j + (j − 1)γ ∈ Z≤−1}  �  ∪  � N−1  �  j=1  {(j + 1)γ ∈ Z≤−1, γ /∈ Z}  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (30)  Observe that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (30) allows γ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The Γ(2+ ¯dj+α+β+(N+j−2)γ) term in the denominator of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (29) implies that ˚SN[F](α, β, γ) =  0 for all  (α, β, γ) ∈ ˚UN[F] ∩ {α + β + (N + j − 2)γ ∈ Z≤−2− ¯dj for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (31)  When constructing the 3-point coefficients of the BPZ minimal models, this is one mechanism  preventing the fusion of (0, s)-primary fields (which are not included in the model) with the primary  fields that are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In BPZ’s terminology, this is the truncation of the operator algebras, as  originally derived via the constraint of OPE associativity — see [BPZ84, §6][PM97, Chp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In the case of the original Selberg integral, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2 describes precisely the singularities  and zeroes of the meromorphic continuation of the original integral, and SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg = SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[1] is just  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The functions S2[F] and S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] are explored in §A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The proof of the theorem above consists of three steps:  (1) The first step is the removal of the fictitious singularities of ˙SN[F](α, β, γ) only in γ (as  required e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' in the Coulomb gas formalism with both kinds of screening charges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The basic idea is to employ the relation – which can be found in a heuristic form in  [DF85a, Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A] – between the symmetrization of SN[F](α, β, γ) and the “DF-like” integral  IN[F](α, β, γ) =  �  □N  � N  �  j=1  xαj  j (1 − xj)βj  �  ×  �  �  1≤j<k≤N  (xk − xj + i0)2γj,k  �  F dx1 · · · dxN,  (32)  where □N = [0, 1]N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We can analytically continue IN[F] via an argument similar to that  used to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Unlike that of SN[F](α, β, γ), this extension has no singularities  associated with hyperplanes of constant γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The true singularities of the extension of  SN[F](α, β, γ) associated with hyperplanes of constant γ show up in the relation with the  extension of IN[F](α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (2) The second step removes the other unwanted singularities away from the loci of two or  more unwanted singularities, via some identities proven via Aomoto [Aom87] in the F = 1  case (and [DF85a, Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A], at a physicist’s level of rigor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The use of these identities for  computing the original Selberg integral is sketched in [FW08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It seems there cannot be a  8  ETHAN SUSSMAN  similar computation of SN[F] in the deg F > 1 case, so a statement about the singularities  is the best we can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The simplex △N ⊂ RN can be thought of as a subset of  (C\\{0, 1})N = (CP 1\\{0, 1, ∞})N  (33)  via the embedding R �→ C �→ CP 1, and the rough idea of this step of the proof is to relate  the integrals above to the result of replacing △N with L⊠N△N for L one of the six linear  fractional transformations preserving CP 1\\{0, 1, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Only three of these are essentially  different, and one of these three is just the identity and therefore uninteresting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The other  two integrals each have meromorphic extensions with different manifest singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Using  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2, these functions can be related to each other, and this can be used to  remove most of the apparent singularities that are not present in all three functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Some  singularities are present in the relations between the integrals, and these cannot be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (3) The third step is the application of Hartog’s theorem to remove the remaining removable  singularities, which now lie on a codimension two subvariety of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  This argument is carried out in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The version more relevant to [DF85a] (with the additional  steps needed) is in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We call IN[F] a “DF-like” integral because similar integrals appear, albeit at a somewhat formal  level, in [DF85a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A similar construction appears in [Fel89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let ΣT(N) denote the collection of  maximal collections I of pairs (x0, S) of x0 ∈ {0, 1} and nonempty subsets S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} such  that, given (x0, S), (x0, Q) ∈ I, either S ⊆ Q or Q ⊆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There exist entire functions IN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I[F] : C2N+N(N−1)/2  α,β,γ  → C associated to the  I ∈ ΣT(N) such that  IN[F](α, β, γ) =  �  I∈ΣT(N)  �  �  (1,S)∈I  Γ  �  |S| +  �  j∈S  βj + 2  �  j,k∈S,j<k  γj,k  ��  ×  �  �  (0,S)∈I  Γ  �  |S| +  �  j∈S  αj + 2  �  j,k∈S,j<k  γj,k  ��  IN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I[F](α, β, γ)  (34)  for all (α, β, γ) for which the left-hand side is a well-defined integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  In particular, IN[F](α, β, γ) admits an analytic extension ˙IN[F](α, β, γ) to an open, dense set  ˙VN = C2N+N(N−1)/2  α,β,γ  ���  �  S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  {αS,∗ ∈ Z≤−|S|}  �  ∪  �  �  S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  {βS,∗ ∈ Z≤−|S|}  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (35)  We end this section by discussing the removal of removable singularities for the integrals in  [DF84][DF85a][DF85b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given λ ∈ C and S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, let DFSym(N, S, λ) denote the set of  F ∈ C[x1, x−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN, x−1  N ] such that:  given any σ ∈ SN such that σ : S → S, F = F ◦ σ (where we are identifying σ with the map  CN ∋ {xi}N  i=1 → {xσ(i)}N  i=1 ∈ CN), and  for any j ∈ S and k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}\\S,  λ  � ∂  ∂xj  F  ����  xj=xk  =  ∂  ∂xk  �  F  ���  xj=xk  �  ∈ C[x1, x−1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ˆxj, ˆx−1  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (36)  In particular, DFSym(N, {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, λ) = C[x1, x−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN, x−1  N ]SN , so in this sense DF-symmetry  is a generalization of symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (Our disallowal of Laurent polynomials in the symmetric case was  without loss of generality, as we could shuffle factors of x1 · · · xN between the polynomial and the  rest of the Selberg integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' However, it is useful here to allow general Laurent polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=')  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  9  For each λ and S, DFSym(N, S, λ) is a (unital) C-subalgebra of C[x1, x−1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN, x−1  N ], and  λ−  �  j∈S  xj + λ+  �  k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}\\S  xk ∈ DFSym(N, S, λ),  λ−  �  j∈S  1  xj  + λ+  �  k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}\\S  1  xk  ∈ DFSym(N, S, λ)  (37)  whenever λ−1  − (λ− + λ+) = λ, so DFSym(N, S, λ) contains polynomials of all degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The key  method of constructing DF-symmetric Laurent polynomials is the following:  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any M ∈ N+ and matrix-valued polynomials ϕ, ψ ∈ xCM×M[x] such that  if Aj is the coefficient of xj in ϕ, then Aj is strictly upper-triangular,  if Bj is the coefficient of xj in ψ, then Bj is strictly lower-triangular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Suppose that the A’s all commute with each other and that the B’s all commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (We do not assume that the A’s commute with the B’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=') Then, the matrix elements of  exp  �  λ−  �  j∈S  ψ(x−1  j ) + λ+  �  k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}\\S  ψ(x−1  k )  �  exp  �  λ−  �  j∈S  ϕ(xj) + λ+  �  k∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}\\S  ϕ(xk)  �  (38)  lie in DFSym(N, S, λ), where λ = λ−1  − (λ−+λ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The vertex operators which Dotsenko and Fateev in-  tegrate to define the minimal model 3-point coefficients have this form up to some scalar factors which  are part of the Selberg integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In this example, the coefficients of ϕ are annihilation operations  on some Fock space, and the coefficients of ψ are creation operators (with all operators truncated  to some finite dimensional subspace of the Fock space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' See [DF84][KT86b][KT86a][Fel89][PM97,  Chp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  For a set S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and α±, β±, γ±, γ0 ∈ C, let αDF0,S, βDF0,S ∈ CN be given by  αj =  �  α+  (j ∈ S),  α−  (j /∈ S),  (39)  βj =  �  β+  (j ∈ S),  β−  (j /∈ S),  (40)  and let γDF0,S ∈ CN(N−1)/2 be given by  γj,k =  �  �  �  �  �  �  �  γ+  (j, k ∈ S),  γ0  (j ∈ S, k /∈ S or vice versa),  γ−  (j, k /∈ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (41)  Let  ˙W DF0,S  N  [F] = {(α−, α+, β−, β+, γ−, γ0, γ+) ∈ C7 : (αDF0,S, βDF0,S, γDF0,S) ∈ ˙VN[F]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (42)  Define, for (α−, α+, β−, β+, γ−, γ0, γ+) ∈ ˙W DF0,S  N  [F],  ˙IDF0,S  N  [F](α−, α+, β−, β+, γ−, γ0, γ+) = ˙IN[F](αDF0,S, βDF0,S, γDF0,S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (43)  Now let ˙W DF,S  N  [F] denote the set of (α+, β+, γ+) ∈ C3 such that γ+ ̸= 0 and, setting  γ− = γ−1  + ,  α− = −γ−α+,  β− = −γ−β+  (44)  — cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [DF85a, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2] — it is the case that (α−, α+, β−, β+, γ−, −1, γ+) ∈ ˙W DF0,S  N  [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is an  open and dense subset of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  ˙IDF,S  N  [F](α+, β+, γ+) = ˙IDF0,S  N  [F](−γ−1  + α+, α+, −γ−1  + β+, β+, γ−1  + , −1, γ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (45)  for (α+, β+, γ+) ∈ ˙W DF,S  N  [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Set N+ = |S| and N− = N − N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  10  ETHAN SUSSMAN  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix γ+ ∈ C\\{0, 1} and S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that  F ∈ DFSym(N, S, γ−1  + (1 − γ+)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (46)  There exist an entire function IDF,S  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α+, β+, γ+) : C2  α+,β+ → C such that  ˙IDF,S  N  [F](α+, β+, γ+) =  � �  ±  N±  �  j=1  sin(π(α± + β± + (N± + j − 2)γ±))  sin(π(α± + (j − 1)γ±)) sin(π(β± + (j − 1)γ±))  �  × IDF,S  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α+, β+, γ+)  (47)  when α−, β−, γ− are related to α+, β+, γ+ by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (44) and the left-hand side is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  If desired, it is possible to replace the sines with Γ-functions with appropriate integral shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' When F = 1, Dotsenko and Fateev claim in [DF85a, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='8, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='35]1 that the integral  above is given by  ˙IDF,S  N  [1](α+, β+, γ+) ∝ γ2N−N+  ∓  � N±  �  j=1  e−iπ(j−1)γ± Γ(jγ±) sin(πjγ±)  Γ(γ±) sin(πγ±)  �  ×  � N∓  �  j=1  e−iπ(j−1)γ∓ Γ(jγ∓ − N±) sin(πjγ∓)  Γ(γ∓) sin(πγ∓)  �� N±  �  j=1  Γ(1 + α± + (j − 1)γ±)Γ(1 + β± + (j − 1)γ±)  Γ(2 − 2N∓ + α± + β± + (N± − 2 + j)γ±)  �  ×  � N∓  �  j=1  Γ(1 + α∓ + (j − 1)γ∓ − N±)Γ(1 + β∓ + (j − 1)γ∓ − N±)  Γ(2 − N± + α∓ + β∓ + (N∓ − 2 + j)γ∓)  �  (48)  for each choice of sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Associahedra  We use the term ‘mwc’ to mean manifold-with-corners in the sense of Melrose – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [Mel][HMM97],  these possessing C∞-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In this section, we define the mwcs that will be used to resolve the  singularities of Selberg- and Dotsenko-Fateev-like integrands:  in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, we define the associahedra Kℓ,m,n, used to meromorphically continue the Selberg-like  integrals, and  in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2 we define the mwcs Aℓ,m,n, used to meromorphically continue the DF-like integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Since K0,N,0 is the usual N-dimensional associahedra, we refer to the mwcs defined below as  associahedra as well, hence the title of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If M is a mwc, we use F(M) to denote the set  of faces of M, where by faces we mean only the boundary hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We use “facet” to refer to  the higher codimension boundary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  It is worth comparing Melrose’s notion of mwc to that of polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  A mwc is locally a  polyhedron, but the converse is not true, as the basic requirement of M being locally diffeomorphic  to a relatively open neighborhood of [0, ∞)N means that every facet f ⊊ M is the intersection of  at most N faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' While the (closed) ball, tetrahedron, cube, and dodecahedron are all mwcs, the  octahedron and icosahedron are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It is critical for the argument in §3 that the associahedra  Aℓ,m,n and Kℓ,m,n are not just polyhedra, but rather mwcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, the notion of “mwc” used here  plays a similar role to that of “polyhedra in general position” in [Var95, §10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7], but they are not  equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For the purposes of this paper, we find it more natural (and technically simpler, as it  avoids the need for polyhedral realizations) to use the language of mwcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  1There seem to be a couple typos in [DF85a, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='35], which we have fixed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The first few cases of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (48)  have been numerically checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  11  We keep track of the full C∞-structure of these mwcs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Were it required, we could keep  track of Cω- (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' real analytic) structure, but since this would require going somewhat beyond the  existent literature on mwcs, and since this level of precision is not needed for the rest of the paper,  we will restrict ourselves to the smooth category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  If f is a facet of M, then the blowup [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f] is a mwc, and the blowdown map  bd : [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f] → M  (49)  is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For convenience, we can identify the interior [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f]◦ with M◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (If F is a codimension ≤ 1  facet of M, then we can identify [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' F] with M itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=') Naturally, if f has codimension ≥ 2, then  F([M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f]) = {[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f ∩ F] : F ∈ F(M)} ∪ {ff},  (50)  where ff = bd−1(f) is the front face of the blowup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, given bdfs xF ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+) of the faces  F ∈ F(M), we can choose bdfs x[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='f∩F], xff of the faces of [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f] such that, for each F ∈ F(M),  xF ◦ bd =  �  x[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='f∩F]xff  (f ⊆ F),  x[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='f∩F]  (otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (51)  (We identify polyhomogeneous – in particular, smooth – functions on [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f] with their restrictions  to the interior, so, going forwards, we can drop the “◦ bd.”) Specifically, in addition to defining  x[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='f∩F] = xF if f ̸⊆ F, we can take  xff =  �  F∈F(M),f⊆F  xF,  (52)  and, if f ⊆ F, then  x[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='f∩F] = xF  �  �  F∈F(M),f⊆F  xF  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (53)  This follows from the analogous result for blowing up a facet of [0, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Note that because M is a  mwc and not just a polyhedron, if F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , Fd ∈ F(M) are distinct faces with ∩δFδ ̸= ∅, then the  connected components of ∩δFδ are codimension d facets of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (The 2D lens is an example of a  mwc with two faces whose intersection is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=')  If U is an open subset of a mwc, then U can be considered as a mwc in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We will say  that some function x ∈ C∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [0, ∞)) is a bdf in U of F ∈ F(M) if it is a bdf of the face F ∩ U of U,  assuming that F ∩ U ̸= ∅, in which case it is automatically a face of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let Rt = Rt ∪ {−∞, +∞}  denote the “radial” compactification of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is a (C∞-)manifold-with-boundary, with 1/t serving  as a bdf for {∞} in {t > 0} and −1/t serving as a bdf for {−∞} in {t < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Associahedra Kℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We now define the mwc Kℓ,m,n for ℓ, m, n ∈ N not all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The  blowup procedure below is a generalization of that in [KT86a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We begin with the set  △ℓ,m,n = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) ∈ RN : x1 ≤ · · · ≤ xℓ ≤ 0  ≤ xℓ+1 ≤ · · · ≤ xℓ+m ≤ 1 ≤ xℓ+m+1 ≤ · · · ≤ xN},  (54)  where N = ℓ + m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is a compact sub-mwc of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Naturally,  △ℓ,m,n ∼= △ℓ,0,0 × △0,m,0 × △0,0,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (55)  Also, △ℓ,0,0 ∼= △ℓ, △0,m,0 ∼= △m, and △0,0,n ∼= △n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For example, in the case N = 2, we have six cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' These are △2,0,0, △0,2,0, △0,0,2, each of which  is diffeomorphic to the triangle △2, and △1,1,0, △1,0,1, △0,1,1, each of which is diffeomorphic to the  square □2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  If ℓ, n = 0, in which case m = N, then △ℓ,m,n is just the standard N-simplex △N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  12  ETHAN SUSSMAN  We call a subset I ⊆ Z/(N + 3)Z consecutive if it is of the form {k mod (N + 3), · · · , k +  κ mod (N + 3)Z} for some k ∈ Z/(N + 3)Z and κ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (Thus, the empty set will not be considered  consecutive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=')  We label the facets (of any codimension, possibly zero) of △ℓ,m,n using (unordered) partitions I  of Z/(N + 3)Z into consecutive subsets I, with no two of 0, ℓ + 1, ℓ + m + 2 ∈ Z/(N + 3)Z appearing  together in any element I ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Specifically,  f0,I =  �  (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) ∈ △ℓ,m,n :  �  I ∈ I ⇒  �  j ∈ I ⇒ tj = ±∞  (0 ∈ I)  j, k ∈ I ⇒ tj = tk  (0 /∈ I)  ��  ,  (56)  where  tj = xj for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ,  tℓ+1 = 0,  tℓ+1+j = xℓ+j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , m,  tℓ+m+2 = 1, and  tℓ+m+2+j = xℓ+m+j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The dimension of f0,I is given by  dim f0,I = |I| − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (57)  For notational simplicity, if I0 ⊆ I is I with the singletons removed, then we define fI0 = f0,I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus,  f∅ denotes the “bulk” of △ℓ,m,n, and the faces of △ℓ,m,n are of the form f{I} for I a consecutive  pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Rephrasing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (57),  codim fI =  �  I∈I  (|I| − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (58)  As a bdf of f{I} for I = {k mod Z/(N + 3)Z, k + 1 mod Z/(N + 3)} when k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N + 1}, we  can take  xf{I} = tk+1 − tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (59)  For the remaining two cases of F{0,1} (which only exists if ℓ ≥ 1) and f{N+2,N+3} (which only exists  if n ≥ 1), we can take xf{0,1} = −1/x1 and xf{N+2,N+3} = 1/xN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let Fℓ,m,n = Fℓ,m,n(△) denote the family of facets fI of △ℓ,m,n such that I = {I} for some  consecutive subset I ⊂ Z/(N + 3)Z of size |I| ≥ 2 not containing any two of 0, ℓ + 1, ℓ + m + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In other words, Fℓ,m,n is the set of facets fI for I defining a partition of Z/(N + 3)Z into a single  interval of length at least two (not containing any two of 0, ℓ + 1, ℓ + m + 2) and a number of  singletons which are being omitted from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For each d ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, let Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d denote the set of elements of Fℓ,m,n of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then,  the mwc Kℓ,m,n is defined by the iterated blowup  Kℓ,m,n = [△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n,N] = [· · · [△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0] · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (60)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=', we first blow up the elements of the collection Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 (which may be empty, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' if ℓ, m, n are  all nonzero), and then, proceeding from higher to lower codimension, iteratively blow up the lifts of  the facets in Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d (meaning the closures of the lifts of the interiors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We should check that the blowup eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (60) is well-defined, which concretely means that, for each  d, the blow-ups in the step in which we blow up the lifts of the elements of Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This  can be done via a somewhat tedious inductive argument, which we only sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  When the time has come to blow up the facets f ̸= f′ in the lifted Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d, their intersection is –  if nonempty – either a point (which we denote K0,0,0) or else an associahedron Kℓ∩,m∩,n∩ (which  will not change upon performing further blowups) of dimension < N, and a neighborhood thereof is  diffeomorphic to  [0, 1)N−d  t  × Kℓ∩,m∩,n∩ × [0, 1)N−d  t′  ,  (61)  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  13  with f corresponding to {t = 0} and f′ corresponding to {t′ = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' the blowups of these two faces in  the product above commute, with the result being naturally diffeomorphic to  [[0, 1)N−d  t  , {0}] × Kℓ∩,m∩,n∩ × [[0, 1)N−d  t′  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {0}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (62)  In order to prove the claimed decomposition, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (61), it is first useful to note when f ∩ f′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If  I, I′ satisfy |I| = N − d + 1 = |I′| and I ∩ I′ ̸= ∅, then the corresponding facets  f = cl[△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Fℓ,m,n,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Fℓ,m,n,d−1]f◦  {I}  (63)  f′ = cl[△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Fℓ,m,n,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Fℓ,m,n,d−1]f◦  {I′}  (64)  of [△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n,d−1] satisfy f ∩ f′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed, I ∩ I′ ̸= ∅ implies  f{I} ∩ f{I′} = f{I∪I′} ∈ Fℓ,m,n(△),  (65)  and since this is blown up in an earlier stage of the construction, f and f′ cannot intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  So, if our two facets f, f′ to be blown up have nonempty intersection, then they must be the  lifts of f{I} and f{I′} for I, I′ satisfying I ∩ I′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The intersection f ∩ f′ lies in the preimage of  f{I} ∩ f{I′} = f{I,I′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This facet of △ℓ,m,n is of the form △ℓ∩,m∩,n∩ for ℓ∩ + m∩ + n∩ = 2d − N ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  As seen inductively, the lift of this facet after performing the blow-ups so far is Kℓ∩,m∩,n∩, although  this is not crucial for the proof that the construction is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Since this has dimension  2d − N, a neighborhood of this facet in our partially blown-up manifold automatically has the form  [0, 1)2N−2d × Kℓ∩,m∩,n∩,  (66)  so it just needs to be checked that f, f′ sit inside of this in the expected way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed, the projections  of f, f′ onto the [0, 1)2N−2d factor are necessarily transversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' With the fact that they both have  dimension d, this implies that we can decompose  [0, 1)2N−2d = [0, 1)N−d  t  × [0, 1)N−d  t′  (67)  such that f corresponds to {t = 0} and f′ corresponds to {t′ = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This completes our sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We now discuss the combinatorial structure of Kℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' All of the faces of △ℓ,m,n are in Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='N−1,  so every face of Kℓ,m,n is the front face of one of our blowups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' So, the faces of Kℓ,m,n are in bijection  with the elements of Fℓ,m,n and thus with I as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Such a subset is uniquely specified by its  endpoints j, k ∈ Z/(N + 3)Z, since only two consecutive subsets of Z/(N + 3)Z have the same  endpoints as I, namely I itself and I∁ ∪ {j, k}, and the latter contains two of 0, ℓ + 1, ℓ + m + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let Jℓ,m,n denote the set of unordered pairs {j, k} arising in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For {j, k} ∈ Jℓ,m,n, let  I(j, k) = I(k, j) denote the unique consecutive subset of Z/(N + 3)Z having these endpoints and  containing at most one member of {0, ℓ + 1, ℓ + m + 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For such j, k, let Fj,k = Fk,j denote the  corresponding face of Kℓ,m,n, and let xFj,k = xFk,j denote a bdf of that face constructed inductively  as in the introduction to this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (Note that these bdfs may depend on the particular order in  which the elements of the Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d are blown up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=')  There are 2−1N(N + 3) faces in Kℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consider the case N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, up to essential equivalence, the cases to consider are  K1,1,0 and K0,2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' These are depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The mwc K1,1,0 is identical to A1,1,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2  we introduce notation for labeling the faces of the Aℓ,m,n, and this notation appears in Figure 4  alongside that used for the Kℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We have introduced an additional notation for the faces of  Kℓ,m,n, indicating I in the subscript using the following conventions:  The elements 0, ℓ + 1, ℓ + m + 2 ∈ Z/5Z are depicted using a ‘◦,’ and 0 is omitted if not  included in I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The other elements of Z/5Z are depicted using a ‘•.’  Except for 0, the elements of Z/5Z are depicted in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If 0 is to be depicted, it is listed  either first or last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The elements included in I are enclosed in parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  14  ETHAN SUSSMAN  F3,4 = F•◦(•◦)  F2,3 = F•(◦•)◦  F0,1 = F(◦•)◦•◦  F1,2 = F(•◦)•◦  F1,3 = F(•◦•)◦  x2  w1  F1,2 = F(◦•)•◦  F3,4 = F◦•(•◦)  F2,3 = F◦(••)◦  F1,3 = F(◦••)◦  F2,4 = F◦(••◦)  1 − x2  x1  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The associahedra K1,1,0 (left) and K0,2,0 (right), realized as polyhedra  roughly in accordance with the blowup procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the first figure, the horizontal  axis is roughly w1 = 1/(1 − x1), increasing to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the second figure, it is  just (roughly) x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In both figures, the vertical axis is (roughly) x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  F4,5 = F•◦•(◦•) = F{3},∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F0,1 = F(◦•)◦•◦• = F{1},∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  F1,5 = F◦•)◦•◦(• = F{1},{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  F2,3 = F•(◦•)◦• = F{2},∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  x2  y3  w1  F1,3 = F(•◦•)◦• = F{1},{2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F3,4 = F•◦(•◦)• = F{2},{∅};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F3,5 = F•◦(•◦•) = F{2},{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F1,2 = F(•◦)•◦• = F{1},{∅};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F0,5 = F5,6 = F•◦•◦(•◦) = F∅,{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The mwc K1,1,1, with labeled faces, realized as a polyhedron roughly in  accordance with the blowup procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Here w1 = 1/(1 − x1) and y3 = (x3 − 1)/x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The faces in the line of sight are F1,2 = F(•◦)•◦•, F1,3 = F(•◦•)◦•, F3,4 = F•◦(•◦)•,  F3,5 = F•◦(•◦•), and F0,5 = F•◦•◦(•◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consider the case N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, up to essential equivalence, the cases to consider are  K1,1,1, K1,2,0, and K0,3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' These are depicted in Figure 4, Figure 5, Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The mwc K1,1,1 is  identical to A1,1,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We have modified the “•” notation from the previous example and used it to label the faces in  the figures, alongside the notation used in the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For instance, when considering  K0,3,0, “◦(• • •)◦” denotes {2, 3, 4} ⊂ Z/6Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' When considering K1,2,0, “◦(• ◦ •)• ◦” denotes {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  When considering K1,1,1, “•) ◦ • ◦ (•◦” denotes {0, 1, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  The Kℓ,m,n satisfy the following “universal property:”  For any subsets S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, Q ⊆ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, R ⊆ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} that are not  all empty, let forg : △ℓ,m,n → △|S|,|Q|,|R| denote the forgetful map forgetting the variables xj  for j /∈ S ∪ Q ∪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, forg lifts to a smooth b-map [Mel]  forg : Kℓ,m,n → K|S|,|Q|,|R|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (68)  Given any face F of K|S|,|Q|,|R|, forg  ∗xF vanishes to first order at each face in forg  −1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  This can be proven by inducting on the number of blowups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  15  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The mwc K1,2,0, with  labeled faces, realized as a polyhe-  dron roughly in accordance with  the blowup procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' As above,  w1 = 1/(1 − x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The faces  in the line of site are F1,2 =  F(•◦)••◦, F1,3  = F(•◦•)•◦, and  F4,5 = F•◦•(•◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  F3,4 = F•◦(••)◦  F2,3 = F•(◦•)•◦  F0,1 = F(◦•)◦••◦  F3,5 = F•◦(••◦)  F1,4 = F(•◦••)◦  F2,4 = F•(◦••)◦  x2  x3 − x2  w1  F4,5 = F•◦•(•◦)  F1,2 = F(•◦)••◦  F1,3 = F(•◦•)•◦  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The mwc K0,3,0, with  labeled faces, realized as a polyhe-  dron roughly in accordance with  the blowup procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The faces  in the line of sight are F4,5 =  F◦••(•◦) and F3,5 = F◦•(••◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  [KT86a, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2], where the full  blowup procedure is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  F1,4 = F(◦•••)◦  F3,4 = F◦•(••)◦  F1,3 = F(◦••)•◦  F2,4 = F◦(•••)◦  F1,2 = F(◦•)••◦  F2,3 = F◦(••)•◦  F2,5 = F◦(•••◦)  x2 − x1  x3 − x2  x1  F4,5 = F◦••(•◦)  F3,5 = F◦•(••◦)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that µ ∈ C∞(△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Ω△ℓ,m,n) is a strictly positive smooth density on  △ℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, the lift of µ to Kℓ,m,n has the form  �  �  {j,k}∈Jℓ,m,n  x|j−k|−1  Fj,k  �  µ  (69)  for a strictly positive µ ∈ C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩKℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Here, for j, k ∈ Z/(N + 3)Z, we use the notation  |j − k| = min{|j0 − k0|, |k0 − j0| : j0, k0 ∈ Z : j0 ≡ j mod (N + 3), k0 ≡ k mod (N + 3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  In the product, each unordered pair is counted only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We recall the following lemma:  Suppose that M is a mwc and µ ∈ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩM) is a strictly positive smooth density on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Then, if f is a facet of M of codimension d ∈ N+, the lift of µ to [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f] has the form xd−1  ff  ν  and ν a strictly positive smooth density on [M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Working in local coordinates, this follows from the case of blowing up a facet in [0, ∞)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In this  case, we can use cylindrical coordinates (that is, spherical coordinates if the facet we are blowing up  is the corner).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The result follows from the form of the Lebesgue measure in cylindrical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The proposition follows from an inductive application of the lemma, once we note that |j − k| is  the codimension of Fj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Lebesgue measure on RN, which defines a strictly positive smooth density  on △◦  ℓ,m,n, has the form  �  ℓ  �  j=1  (1 − xj)2��  N  �  j=ℓ+m+1  x2  j  �  µ  (70)  for µ ∈ C∞(△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Ω△ℓ,m,n) a strictly positive smooth density on △ℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  16  ETHAN SUSSMAN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It is the case that the 1-form dxj ∈ Ω1△◦  ℓ,m,n defines an extendable 1-form on △ℓ,m,n if  j ∈ {ℓ + 1, · · · , ℓ + m}, and the extension is nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The same holds for  dwj = (1 − xj)−2 dxj for wj = 1/(1 − xj) if j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and  dyj = x−2  j  dxj for yj = (xj − 1)/xj if j ∈ {ℓ + m + 1, · · · , N},  since △ℓ,m,n is a submanifold of RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The µ in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (70) can therefore be taken to be |dw1 ∧ · · · ∧  dwℓ ∧ dxℓ+1 ∧ · · · ∧ dxℓ+m ∧ dyℓ+m+1 ∧ · · · ∧ dyN|, which lies in C∞(△ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Ω△ℓ,m,n) and is strictly  positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  We now record the results of lifting the factors xi, 1 − xi, and xj − xk comprising the Selberg  integrand to Kℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Beginning with the first two cases:  If i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, then  −xi ∈  �  N+3  �  j=ℓ+m+3  ℓ  �  k=i  x−1  Fj,k  ��  i�  j=1  ℓ+m+1  �  k=ℓ+1  xFj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (71)  (1 − xi) ∈  �  N+3  �  j=ℓ+m+3  ℓ  �  k=i  x−1  Fj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (72)  If i ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, then  xi ∈  � ℓ+1  �  j=1  ℓ+m+1  �  k=i+1  xFj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (73)  (1 − xi) ∈  �  i+1  �  j=ℓ+2  N+2  �  k=ℓ+m+2  xFj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (74)  If i ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, then  xi ∈  �  i+2  �  j=ℓ+m+3  ℓ  �  k=0  x−1  Fj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (75)  −(1 − xi) ∈  �  i+2  �  j=ℓ+m+3  ℓ  �  k=0  x−1  Fj,k  �� ℓ+m+2  �  j=ℓ+2  N+2  �  k=i+2  xFj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (76)  If N = 1, then these are all trivial to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By applying the universal property of the associahedra,  the N ≥ 2 case follows from the N = 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In a similar manner, by working out the case of K0,2,0 in detail and applying the universal  property, we get, for k > j:  If j, k ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, then  (xk − xj) ∈  � ℓ+1  �  j0=1  ℓ+m+1  �  k0=k+1  xFj0,k0  ��  j+1  �  j0=ℓ+2  N+2  �  k0=k+1  xFj0,k0  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (77)  Indeed, in the case of ℓ, n = 0 and m = 2, this says that x2 − x1 ∈ xF1,3xF2,3xF2,4C∞(K0,2,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Indeed, if we construct K0,2,0 by first blowing up F1,3 and then blowing up F2,4, we get  xF1,3 = x2,  xF2,3 =  x2 − x1  2x2 − x2  2 − x1  ,  xF2,4 = 2x2 − x2  2 − x1  x2  ,  (78)  so that xF1,3xF2,3xF2,4 = x2 − x1, on the nose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' On the other hand, if we reverse the order of the  blowups, then we get  xF1,3 = x2 − x2  1  1 − x1  ,  xF2,3,0 = x2 − x1  x2 − x2  1  ,  xF2,4 = 1 − x1,  (79)  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  17  so we still get xF1,3xF2,3xF2,4 = x2 − x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  From this, we can deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  If j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, then, in terms of wi = −xi/(1−xi), (xk−xj) = (1−wj)−1(1−wk)−1(wj −  wk), so,  (xk − xj) ∈  �  ℓ  �  j0=j  N+3  �  k0=ℓ+m+3  x−1  Fj0,k0  ��  j�  j0=1  ℓ+m+1  �  k0=k  xFj0,k0  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (80)  If j, k ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, then, in terms of yi = 1/xi, (xk − xj) = y−1  j y−1  k (yj − yk), so  (xk − xj) ∈  �  j+2  �  j0=ℓ+2  N+2  �  k0=k+2  xFj0,k0  ��  k+2  �  j0=ℓ+m+3  ℓ  �  k0=0  x−1  Fj0,k0  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (81)  The next three follow from the K1,1,0, K1,0,1, and K0,1,1 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We illustrate the K1,1,0 case, and  the others are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  If j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and k ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, then (xk − xj) = (1 − wj)−1(wj + xk − xkwj), so  (xk − xj) ∈  �  ℓ  �  j0=j  N+3  �  k0=ℓ+m+3  x−1  Fj0,k0  ��  j�  j0=1  ℓ+m+1  �  k0=k+1  xFj0,k0  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (82)  In the case ℓ, m = 1, n = 0, this says that (x2 − x1) ∈ x−1  F1,5xF1,3C∞(K1,1,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed,  the bdf xF1,3 of F1,3 in K1,1,0 is defined by  xF1,3 = (1 − w1) + x2 = −  x1  1 − x1  + x2,  (83)  and xF1,5 = xF0,1 = w1 = 1/(1 − x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' So,  x−1  F1,5xF1,3 = x2 − x1 − x1x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (84)  The supposed C∞(K1,1,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+) term above is therefore (x2 − x1)(x2 − x1 − x1x2)−1 =  (1 − x2x1/(x2 − x1))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' One way (besides checking in a system of local coordinate charts)  to see that this is smooth (and positive) on K1,1,0 is the identity  −  x2x1  x2 − x1  =  xF1,2xF1,3xF2,3  xF1,2 + xF2,3xF0,1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (85)  The faces F0,1, F2,3 are disjoint from F1,2 (see Figure 3), so the denominator on the right-hand  side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (85) is nonvanishing, so the quotient is indeed smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Likewise:  If j ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m} and k ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, then (xk − xj) = y−1  k (1 − xjyk), so  (xk − xj) ∈  �  j+1  �  j0=ℓ+2  N+2  �  k0=k+2  xFj0,k0  ��  ℓ  �  j0=0  k+2  �  k0=ℓ+m+3  x−1  Fj0,k0  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (86)  If j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and k ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, then (xk − xj) = y−1  k (1 − wj)−1(1 − wj + wjyk),  so  (xk − xj) ∈  �  ℓ  �  j0=j  N+3  �  k0=k+3  x−1  Fj0,k0  ��  ℓ  �  j0=0  k+2  �  k0=ℓ+m+3  x−1  Fj0,k0  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (87)  We associate to each face F• ∈ F(Kℓ,m,n) an affine functional  ρ• : C2N+N(N−1)/2 ∋ (α, β, γ) �→ ρ•(α, β, γ) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (88)  Suppose that we are given some α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If one of  (I) j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}  18  ETHAN SUSSMAN  (II) j, k ∈ {ℓ + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m + 1},  (III) j, k ∈ {ℓ + m + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N + 2}  holds, then, letting k denote the larger of {j, k},  ρj,k = k − j + 2  �  j′≤j0<k0≤k′  γj0,k0,  (89)  where, for each i ∈ {j, k}, i′ = i if i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, i′ = i − 1 if i ∈ {ℓ + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m + 1}, and i′ = i − 2  if i ∈ {ℓ + m + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N + 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The other cases are:  If j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + 1} and k ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m + 1} and j ̸= k, then  ρj,k = k − j − 1 +  ℓ  �  i=j  αi +  k  �  i=ℓ+2  αi−1 + 2  �  j≤j0<k0≤k−1  γj0,k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (90)  If j ∈ {ℓ + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m + 2} and k ∈ {ℓ + m + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N + 2} and j ̸= k, then  ρj,k = k − j − 1 +  ℓ+m+1  �  i=j  βi−1 +  k  �  i=ℓ+m+3  βi−2 + 2  �  j−1≤j0<k0≤k−2  γj0,k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (91)  If j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and k ∈ {ℓ + m + 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N + 3} and at least one of j ̸= 0, k ̸= N + 3 holds,  then  ρj,k = k − j − N − 4 −  j  �  i=1  αi −  j  �  i=1  βi −  N+2  �  i=k  αi−2 −  N+2  �  i=k  βi−2 − 2  j  �  j′=1  N  �  i=1,i̸=j′  γi,j′  − 2  N  �  k′=k−2  N  �  i=1,i̸=k′−2  γi,k′ + 2  �  1≤j′<k′≤j  γj′,k′ + 2  �  k−2≤j′<k′≤N  γj′,k′ + 2  j  �  j′=1  N  �  k′=k−2  γj′,k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (92)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given any α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2, the Selberg-  like integrand  N  �  i=1  |xi|αi|1 − xi|βi  �  1≤j<k≤N  (xk − xj)2γj,k|dx1 · · · dxN| ∈ C∞(△◦  ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Ω△◦  ℓ,m,n)  (93)  lifts, via the blowdown map bd : Kℓ,m,n → △ℓ,m,n, to an extendable density of the form  �  �  {j,k}∈Jℓ,m,n  xρj,k  Fj,k  �  µℓ,m,n(α, β, γ),  (94)  for some strictly positive smooth density µℓ,m,n(α, β, γ) ∈ C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩKℓ,m,n), depending entirely  on α, β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Each ρj,k is an affine function of α, β, γ, so it suffices to check 2N + N(N − 1)/2 + 1 cases,  the case when all three of α, β, γ are zero and 2N + N(N − 1)/2 cases where the triple (α, β, γ)  ranges over a basis of C2N+N(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Write  ρj,k(α, β, γ) = ρ(0)  j,k + ρ(1)  j,k(α, β, γ),  (95)  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  19  where ρ(0)  j,k = ρj,k(0, 0, 0) and ρ(1)  j,k(α, β, γ) = ρj,k(α, β, γ) − ρ(0)  j,k is the linear part of ρj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, we  want to show that, upon lifting to Kℓ,m,n,  |dx1 · · · dxN| ∈  �  �  {j,k}∈Jℓ,m,n  x  ρ(0)  j,k  Fj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩKℓ,m,n),  (96)  N  �  i=1  |xi|αi|1 − xi|βi  �  1≤j<k≤N  (xk − xj)2γj,k ∈  �  �  {j,k}∈Jℓ,m,n  x  ρ(1)  j,k(α,β,γ)  Fj,k  �  C∞(Kℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (97)  with it sufficing to check eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (97) on a basis of C2N+N(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Equation (96) is simply a restatement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For a basis of C2N+N(N−1)/2, we look at (α, β, γ) such that all of the components α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , αN,  β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , βN, γ1,2, · · · of α, β, γ are all 0 except for one, which we set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The result then  follows, via a bit of algebra, from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (71) through eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (87).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Let T(ℓ, m, n) denote the collection of maximal families I of consecutive subsets I ⊊ Z/(N + 3)Z  such that  2 ≤ |I| ≤ N + 1 for all I ∈ I,  no two of 0, ℓ + 1, ℓ + m + 2 are in any I ∈ I together, and  if I, I′ ∈ I satisfy I ∩ I′ ̸= ∅, then either I ⊆ I′ or I′ ⊆ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The elements of T(ℓ, m, n) can be thought of as specifying valid ways of adding parentheses to group  together the elements of Z/(N + 3)Z without grouping any of 0, ℓ + 1, ℓ + m + 2 together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The  minimal facets of Kℓ,m,n are in bijective correspondence with the elements of T(ℓ, m, n), with  fI =  �  I(j,k)∈I  Fj,k  (98)  the facet corresponding to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Associahedra Aℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We now define the mwc Aℓ,m,n for ℓ, m, n ∈ N not all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We  begin with the N = ℓ + m + n cube □N = [0, 1]N  t , which we identify with  [−∞, 0]ℓ  x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',xℓ × [0, 1]m  xℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',xℓ+m × [1, ∞]n  xℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',xN  (99)  via the coordinate changes ti = 1/(1 − xi) for xi ∈ [−∞, 0] and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and ti = (xi − 1)/xi  for xi ∈ [1, ∞] and i ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The facets of □N we label by sextuples (S, Q, S′, Q′, S′′, Q′′) consisting of (possibly empty)  subsets S, Q ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, S′, Q′ ⊆ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, and S′′, Q′′ ⊆ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} such that  S ∩ Q = S′ ∩ Q′ = S′′ ∩ Q′′ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  FS,Q,S′,Q′,S′′,Q′′ =  �  (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , tN) ∈ □N :  s ∈ S ∪ S′ ∪ S′′ ⇒ ts = 0  q ∈ Q ∪ Q′ ∪ Q′′ ⇒ tq = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (100)  For instance, □N = F∅,∅,∅,∅,∅,∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Now let Fℓ,m,n = Fℓ,m,n(□) denote the family of facets defined by  Fℓ,m,n = ({FS,∅,∅,∅,∅,Q′′}S,Q′′ ∪ {F∅,Q,S′,∅,∅,∅}Q,S′ ∪ {F∅,∅,∅,Q′,S′′,∅}Q′,S′′)\\{□N}  (101)  where S, S′, S′Q, Q′, Q′′ range over all subsets as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For each d ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N − 1}, let Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d  denote the set of elements of Fℓ,m,n of dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, Aℓ,m,n is defined by the iterated blowup  Aℓ,m,n = [□N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n] = [□N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='N−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (102)  20  ETHAN SUSSMAN  t2  t3  t1  A1,1,1  A1,2,0  A0,3,0  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The 3-cube □3 and the three blowups A1,1,1 = K1,1,1, A1,2,0, A0,3,0  thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The □ℓ,m,n(S, S′, S′′) are eight subcubes corresponding to the eight vertices  of □3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' One such cube is depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  As in the previous section, we should check that, for each d = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N, having already blown up  Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d0 for d0 < d, the blowups of the closures of the lifts of the interiors of all of the F ∈ Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d  all commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' One way to see this is to split  □N =  �  S,S′,S′′  □ℓ,m,n(S, S′, S′′),  (103)  where S varies over all subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, S′ varies over all subsets of {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, and S′′  varies over all subsets of {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, and  □ℓ,m,n(S, S′, S′′) =  �  (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , tN) ∈ □N : i ∈ S ∪ S′ ∪ S′′ ⇒ ti ∈ [0, 2/3)  i /∈ S ∪ S′ ∪ S′′ ⇒ ti ∈ (1/3, 1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (104)  Once we have established that blowing up Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0, · · · , Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d−1 is fine, then  [□N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d−1] =  �  S,S′,S′′  [□ℓ,m,n(S, S′, S′′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d−1]  (105)  naturally, with the left-hand side being well-defined if the right-hand side is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, it suffices to  check that the blowups [□ℓ,m,n(S, S′, S′′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' · · · ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='d] are all well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' To see this,  identify  □ℓ,m,n(S, S′, S′′) =  ��  0, 2  3  �S  {ti}i∈S  ×  �1  3, 1  �(S′′)∁  {ti}i∈(S′′)∁  �  ×  ��  0, 2  3  �S′  {ti}i∈S′ ×  �1  3, 1  �S∁  {ti}i∈S∁  �  ×  ��  0, 2  3  �S′′  {ti}i∈S′′ ×  �1  3, 1  �(S′)∁  {ti}i∈(S′)∁  �  (106)  and note that the blowup prescription is just that of performing the total boundary blowup [HMM97]  on each of the three factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (Note that this is not the same as the total boundary blowup of the  product of the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=') Here,  S∁ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}\\S,  (S′)∁ = {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}\\S′,  and (S′′)∁ = {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}\\S′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Thus,  Aℓ,m,n =  �  S,S′,S′′  �  ��  0, 2  3  �S  ×  �1  3, 1  �(S′′)∁�  tb ×  ��  0, 2  3  �S′  ×  �1  3, 1  �S∁�  tb ×  ��  0, 2  3  �S′′  ×  �1  3, 1  �(S′)∁�  tb  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (107)  The faces of Aℓ,m,n are in bijection with the elements of Fℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We label the faces of Aℓ,m,n as  follows:  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  21  F∅,{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  F∅,{2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{2,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{2,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F{1},{2,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F{1},{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F{1},{2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{1,2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{2,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{1,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{1,2,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  F∅,{3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{1,2,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{1,2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{1,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  F∅,{2,3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The eleven faces of A1,2,0 and the fourteen faces of A0,3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  for S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and Q ⊆ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, the face corresponding to FS,∅,∅,∅,∅,Q is  labeled as FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞ = FQ,S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞,  for Q ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and S ⊆ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, the face corresponding to F∅,Q,S,∅,∅,∅ is  labeled as as FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 = FQ,S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0, and  for S ⊆ {ℓ+m+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and Q ⊆ {ℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ+m}, the face corresponding to F∅,∅,∅,Q,S,∅  is labeled as FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 = FQ,S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Here, S, Q are not allowed to both be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For any subsets S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, Q ⊆ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, R ⊆ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} that are not  all empty, let forg : □ℓ,m,n → □|S|,|Q|,|R| denote the forgetful map forgetting the coordinates xi for  i /∈ S ∪ Q ∪ R, Then, forg lifts to a smooth b-map  forg : Aℓ,m,n → A|S|,|Q|,|R|,  (108)  and given any face F of A|S|,|Q|,|R|, the pullback forg  ∗xF vanishes to first order at each face F0  satisfying  F0 ⊆ forg  −1(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (109)  This is the “universal property” of the Aℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Via the decomposition in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (107), it follows from  the corresponding universal property of the total boundary blowup of a product, which is essentially  given by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that µ is a strictly positive smooth density on □ℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, the lift of  µ to Aℓ,m,n has the form  �  �  S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  Q⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  x|S∪Q|−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  ��  �  S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  Q⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  x|S∪Q|−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  ��  �  S⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  Q⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  x|S∪Q|−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  �  µ  (110)  for a strictly positive smooth density µ ∈ C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩAℓ,m,n) on Aℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  As a notational convenience, we are setting xF∅,∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0 = 1 for each x0 ∈ {0, 1, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Follows via induction on the number of blowups, as in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Lebesgue measure on RN, which defines a strictly positive smooth density  on □◦  ℓ,m,n, has the form  �  ℓ  �  j=1  (1 − xj)2��  N  �  j=ℓ+m+1  x2  j  �  µ  (111)  22  ETHAN SUSSMAN  for some strictly positive smooth density µ ∈ C∞(□ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Ω□ℓ,m,n) on □ℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Follows from the same computation as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For each pair of distinct i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} such that either i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ},  i, j ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, or i, j ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, the set Hj,k = clAℓ,m,n{p ∈ □◦  ℓ,m,n : xj = xk}  is a p-submanifold of Aℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  See [MS08, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2] for the definition of “p-submanifold.”  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consider the neighborhood bd−1(□ℓ,m,n(S, S′, S′′)) ⊆ Aℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If one of j, k is in S ∪ S′ ∪ S′′  and the other is not, then the intersection of Hj,k with bd−1(□ℓ,m,n(S, S′, S′′)) is a submanifold  disjoint from the boundary and therefore a p-submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It therefore suffices to consider the case  when j, k ∈ S ∪ S′ ∪ S′′ (and the case when neither are in S ∪ S′ ∪ S′′ is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For notational  simplicity, we only consider the case when j, k ∈ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then,  Hj,k ∩ bd−1(□ℓ,m,n(S, S′, S′′)) =  ��  0, 2  3  �S  ×  �1  3, 1  �(S′′)∁�  tb ×  � ˜Hj,k ∩  ��  0, 2  3  �S′  ×  �1  3, 1  �S∁�  tb  �  ×  ��  0, 2  3  �S′′  ×  �1  3, 1  �(S′)∁�  tb,  (112)  where ˜Hj,k is the closure of {xj = xk} in ([0, 2/3)S′ × (1/3, 1]S∁)tb, which is a p-submanifold [MS08]  (this also follows from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, Hj,k ∩ bd−1(□ℓ,m,n(S, S′, S′′)) is a p-submanifold of  bd−1(□ℓ,m,n(S, S′, S′′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' As the neighborhoods bd−1(□ℓ,m,n(S, S′, S′′)) cover Aℓ,m,n, the conclusion  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  This result is illustrated in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We now record the results of lifting xi and 1−xi to Aℓ,m,n, these being derivable via the universal  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  If i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, then  −xi ∈  �  �  i∈Q⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  S⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  ��  �  i∈S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  Q⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  �  C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (113)  (1 − xi) ∈  �  �  i∈Q⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  S⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �  C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (114)  If i ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, then  xi ∈  �  �  S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  i∈Q⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  �  C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (115)  (1 − xi) ∈  �  �  i∈S⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  Q⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  �  C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (116)  If i ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, then  xi ∈  �  �  i∈S⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  Q⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �  C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (117)  −(1 − xi) ∈  �  �  S⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  i∈Q⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  ��  �  i∈S⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  Q⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �  C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (118)  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  23  Let I1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, I2 = {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, and I3 = {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For j, k ∈ I• for the  same • ∈ {1, 2, 3}, let yj,k denote a defining function of Hj,k, with the sign chosen so as to have the  same sign as xj − xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, for all distinct j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N},  (xj − xk) ∈ Yj,kXj,kC∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+),  (119)  where Xj,k = Xk,j is given by  Xj,k =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  � �  S⊆I3  j∈Q⊆I1 or k∈Q⊆I1  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �� �  j,k∈S⊆I1  Q⊆I2  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  �  (j, k ∈ I1),  � �  S⊆I1  j,k∈Q⊆I2  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  �� �  j,k∈S⊆I2  Q⊆I3  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  �  (j, k ∈ I2),  � �  S⊆I2  j,k∈Q⊆I3  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  �� �  j∈S⊆I3 or k∈S⊆I3  Q⊆I1  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �  (j, k ∈ I3),  � �  j∈S⊆I1  Q⊆I3  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �� �  j∈S⊆I1  k∈Q⊆I2  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  �  (j ∈ I1, k ∈ I2),  � �  S⊆I1  k∈Q⊆I3  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �� �  j∈S⊆I2  k∈Q⊆I3  xFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  �  (j ∈ I2, k ∈ I3),  � �  S⊆I1,Q⊆I3  {j,k}∩S∪Q̸=∅  x−1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  �  (j ∈ I3, k ∈ I1),  (120)  and Yj,k = yj,k if, for some • ∈ {1, 2, 3}, we have j, k ∈ I•, and Yj,k = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We associate to each face F• ∈ F(Aℓ,m,n) an affine functional  ϱ• : C2N+N(N−1)/2 ∋ (α, β, γ) �→ ϱF•(α, β, γ) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (121)  Suppose that we are given some α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then,  ϱ•(α, β, γ) is defined as follows:  For S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ} and Q ⊆ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m},  ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 = |S| + |Q| − 1 +  �  j∈S∪Q  αj + 2  �  j,k∈S∪Q  j>k  γj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (122)  For S ⊆ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m} and Q ⊆ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N},  ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 = |S| + |Q| − 1 +  �  j∈S∪Q  βj + 2  �  j,k∈S∪Q  j>k  γj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (123)  For S ⊆ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and Q ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ},  ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞ = −|S| − |Q| − 1 −  �  j∈S∪Q  αj −  �  j∈S∪Q  βj − 2  �  j>k  j∈S∪Q or k∈S∪Q  γj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (124)  Then, letting ∆ ⊂ □ℓ,m,n be defined by ∆ = ∪3  =1 ∪j̸=k,j,k∈I• {xj = xk}:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given any α, β ∈ CN and γ = {γj,k = γk,j}1≤j<k≤N ∈ CN(N−1)/2,  N  �  i=1  |xi|αi|1−xi|βi  �  1≤j<k≤N  (xk −xj +i0)2γj,k|dx1 · · · dxN| ∈ C∞(□◦  ℓ,m,n\\∆;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C⊗Ω(□◦  ℓ,m,n\\∆)) (125)  lifts, via the blowdown map bd : Aℓ,m,n → □ℓ,m,n, to  �  �  S⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  Q⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  xϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  ��  �  S⊆{ℓ+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ+m}  Q⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  xϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  ��  �  S⊆{ℓ+m+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  Q⊆{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',ℓ}  xϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞  ��  �  1≤j<k≤ℓ  (yk,j + i0)2γj,k  �  �  �  ℓ+1≤j<k≤ℓ+m  (yk,j + i0)2γj,k  ��  �  ℓ+m+1≤j<k≤N  (yk,j + i0)2γj,k  �  µℓ,m,n  (126)  24  ETHAN SUSSMAN  A1,2,0  A0,3,0  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The sets Hj,k ∈ P in A1,2,0 and A0,3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Pictured are H1,2, H1,3, and H2,3,  where the axes are oriented as in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In A0,3,0, the intersection H1,2∩H1,3∩H2,3  has been indicated with an extra dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  for some strictly positive smooth density µℓ,m,n ∈ C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩAℓ,m,n) on Aℓ,m,n, depending entirely  on α, β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Follows from the preceding computations, along with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  If M is an orientable mwc, we say that a collection P of interior p-submanifolds each of codimension  one is consistently orientable if we can choose an orientation on each such that, for any p ∈ M, the  subset  �  P∈P,p∈P  ++N∗  p P ⊂ T ∗  p M  (127)  does not contain zero, where ++N∗P ⊂ +N∗P ⊂ T ∗M is the induced positively oriented conormal  bundle, sans the zero section, and T ∗M is the extendable cotangent bundle of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Whether or  not this holds does not depend on the choice of orientation of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Choosing defining functions  {yP }P∈P ⊂ C∞(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R) for the P ∈ P such that  dyP (p) ∈ +N∗  p P  (128)  for each p ∈ P, we say that the {yP }P∈P are consistently oriented defining functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In □◦  0,3,0 = (0, 1)3, consider P = {H◦  1,2, H◦  2,3, H◦  3,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The functions x2 −x1, x3 −x2, x1 −x3  are not consistently oriented defining functions, as  0 = d(x2 − x1) + d(x3 − x2) + d(x1 − x3),  (129)  but x2 − x1, x3 − x2, and x3 − x1 are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Let  P = {Hj,k}j,k∈I1,j̸=k ∪ {Hj,k}j,k∈I2,j̸=k ∪ {Hj,k}j,k∈I3,j̸=k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (130)  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The collection P defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (130) is consistently orientable, and {yk,j}j<k is  a set of consistently oriented defining functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We will show that, for any p ∈ Aℓ,m,n and {λj,k}p∈Hj,k∈P ∈ [0, ∞), if the 1-form  �  Hj,k∈P s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' p∈Hj,k  λj,kdyk,j ∈ Ω1(Aℓ,m,n)  (131)  vanishes at p, then λj,k = 0 for all Hj,k ∈ P such that p ∈ Hj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Put differently, we want to  show that if P is any partition of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} into nonempty subsets S ⊂ I1, I2, I3, then, given any  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  25  {λj,k}j,k∈S∈P,j<k ⊂ R≥0 not all zero, then  �  j,k∈S∈P  j<k  λj,k dyk,j  (132)  is nonvanishing on ∩j,k∈S∈P,j<kHj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If P consists only of singletons, then this is vacuously true, so  it suffices to consider the case when at least one member of P has cardinality > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  This is certainly true for p ∈ □◦  ℓ,m,n, as dyk,j ∝ dxk − dxj on □◦  ℓ,m,n ∩ Hj,k, where the coefficient  of proportionality is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed, by the results above,  xk − xj = fj,kyk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='j  (133)  for some fj,k ∈ C∞(Aℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R≥0) that is nonvanishing in the interior, so  dyk,j = f−1  j,k (dxk − dxj) − f−1  j,k yk,j dfj,k  (134)  in □◦  ℓ,m,n, which is equal to f−1  j,k (dxk − dxj) on Hj,k ∩ □◦  ℓ,m,n, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This argument does not  work for p ∈ ∂Aℓ,m,n, as fj,k may vanish there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  A homogeneity argument can be used to show that, for any p ∈ ∂Aℓ,m,n, there exists a tubular  neighborhood T : U → U0 of a neighborhood U0 ⊂ F0 of p in F0, where F0 is the smallest facet  containing p, such that the intersections U ∩ P of this neighborhood with the P ∈ P are all vertical  subsets, meaning of the form T −1(B) for some B ⊂ U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This implies that if the 1-form above  vanishes at p, then it also vanishes on the fiber of the tubular neighborhood over p and hence  somewhere in □◦  ℓ,m,n ∩Hj,k∋p Hj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  We illustrate the preceding argument with an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consider the case when the only one  of ℓ, m, n that is nonzero is m, and consider p ∈ ∩Hj,k∈PHj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The set ∩Hj,k∈PHj,k ⊂ A0,N,0 (the  “small diagonal”) is a p-submanifold located away from all but the very first two blowups involved  in the construction of A0,N,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Near this p-submanifold, A0,N,0 is canonically diffeomorphic to  [□0,N,0, {0}, {1}], the result of blowing up two opposite corners of the N-cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We consider the  situation near the blowup of  {0} = F∅,∅,{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N},∅,∅,∅,  (135)  and the situation near the opposite corner is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the interior of the front face of that blowup,  we can use ϱ = x1 as a bdf and coordinates ˆxj = xj/x1 for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N as parametrizing the face  itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In terms of these coordinates,  ∩Hj,k∈P Hj,k = {ˆx2, · · · , ˆxN = 1}  (136)  locally, and, for 1 ≤ j < k ≤ N, we can write yk,j = ˜yk,jC∞(A0,N,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' R+) for ˜yk,j given locally by  ˜yk,j = ϱ−1(xk − xj) = ˆxk − ˆxj, where ˆx1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This satisfies  d˜yk,j =  �  dˆxk  (j = 1),  dˆxk − dˆxj  (j ̸= 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (137)  So, if λk,j ≥ 0, then �  1≤j<k≤N λj,kd˜yk,j = 0 ⇒ λj,k = 0 for all k, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Since the yk,j differ from the  ˜yk,j by a (smooth) positive factor, the yk,j have the same property on ∩Hj,k∈PHj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  There is a more direct argument using the coordinates in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 (with the decomposition  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (107)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Namely, using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (107), the result follows from the analogous result for [0, 1)N  tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given  any σ ∈ SN, consider the coordinates ϱ, ˆxσ(2), · · · , ˆxσ(N) defined in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, these giving a  C∞-atlas as σ varies over all permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In these coordinate systems, the relevant p-submanifolds  are, locally,  Hj,k = {ˆxj+1 · · · ˆxk = 1} ⊂ [0, 1)N  tb,  (138)  26  ETHAN SUSSMAN  so have defining functions yk,j = −1 + ˆxj+1 · · · ˆxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This satisfies  dyk,j =  k  �  i=j+1  dˆxi  ˆxi  (139)  on Hj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The 1-forms, ωj,k = �k  i=j+1 ˆx−1  i  dˆxi, defined by the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (139) satisfy  {λj,k}1≤j<k≤N and j,k∈S∈P ⊂ [0, ∞) and  �  1≤j<k≤N and j,k∈S∈P  λj,kωj,k = 0  ⇒ λj,k = 0 for all j < k in S ∈ P,  (140)  from which the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let ΣT(ℓ, m, n) denote the collection of maximal families I of pairs (x0, S) of x0 ∈ {0, 1, ∞} and  nonempty S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} such that  if (x0, S), (x0, Q) ∈ I, either S ⊆ Q or Q ⊆ S, (x0, S) ∈ I ⇒  �  �  �  �  �  �  �  S ∩ I3 = ∅  (x0 = 0),  S ∩ I1 = ∅  (x0 = 1),  S ∩ I2 = ∅  (x0 = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (141)  The minimal facets of Aℓ,m,n are in bijective correspondence with the elements of ΣT(ℓ, m, n), with  fI =  �  (x0,S)∈I, S∪Q⊆S  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0  (142)  the facet corresponding to I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Meromorphic continuation  We now turn to the analytic extension of Selberg-like integrals to dense, open subsets of the space  of possible exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' As discussed in the introduction, the results in this section are apparently  sharp for generic Selberg-like integrals, but for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' symmetric Selberg-like integrals they are only  preliminary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Nevertheless, the results we prove here will be useful in establishing the sharp results  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For our discussion of the symmetric and DF-symmetric cases, it is useful to consider somewhat  more general integrals than eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let ℓ, m, n ∈ N satisfy ℓ + m + n = N ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix a finite  collection D of indexed sets  {dF}F∈F(Kℓ,m,n) ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (143)  Define  Sℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n[F](α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ) =  �  △ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n  � N  �  i=1  |xi|αi|1 − xi|βi  ��  �  1≤j<k≤N  (xk − xj)2γj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='k  �  F dx1 · · · dxN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (144)  for (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ) ∈ Ωℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n[D],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' where  Ωℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n[D] denotes the set of (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ) ∈ CN  α × CN  β × CN(N−1)/2  γ  such that  � N  �  i=1  |xi|αi|1 − xi|βi  ��  �  1≤j<k≤N  (xk − xj)2γj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='k  ��  �  F∈F(Kℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n)  xdF  F  �  ∈ L1(△ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' dx1 · · · dxN)  (145)  for all {dF}F∈F(Kℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n) ∈ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' and  F has the form  F =  �  {dF}F∈F(Kℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n)  �  �  F∈F(Kℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n)  xdF  F  �  F{dF}F∈F(Kℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n)  (146)  for some F{dF}F∈F(Kℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n) ∈ C∞(Kℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  27  We denote the set of such F by AD(Kℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Observe that Ωℓ,m,n[D] is a nonempty, open, and  connected subset of C2N+N(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the case N = 1, we consider Ωℓ,m,n[D] as a subset of C2  α,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We write Ω[F] to denote Ωℓ,m,n[D] for arbitrary D such that F ∈ AD(Kℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  Ωℓ,m,n = Ωℓ,m,n[{{0}F∈F(Kℓ,m,n)}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (147)  If ϕ ∈ C∞(△ℓ,m,n), then we can consider ϕ as an element of C∞(Kℓ,m,n), so Sℓ,m,n[ϕ] : Ωℓ,m,n → C  is well-defined, and Ωℓ,m,n[ϕ] ⊇ Ωℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If f ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN], then the lift of fϕ is also a classical  symbol on Kℓ,m,n (it is smooth if ℓ, n = 0, but not necessarily otherwise), so  Sℓ,m,n[fϕ] : Ωℓ,m,n[f] → C  (148)  is well-defined, except now we may have Ωℓ,m,n[fϕ] ̸⊆ Ωℓ,m,n if ℓ ̸= 0 or n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In the special case when ℓ, n = 0 and m = N, we use the abbreviations Ω0,N,0 = ΩN, Ω0,N,0[•] =  ΩN[•], and  S0,N,0[F](α, β, γ) = SN[F](α, β, γ),  (149)  this being consistent with our earlier notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  As in the introduction, when α, β, γ are constant, we just write ‘α’ in place of ‘α,’ ‘β’ in place of  ‘β,’ and ‘γ’ in place of ‘γ.’ Let Uℓ,m,n[•] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ Ωℓ,m,n[•]  holds when α = α, β = β, and γ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Similar abbreviations will be used throughout the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In addition to the general Selberg-like integral above, we have the following general integral of  Dotsenko–Fateev type:  Iℓ,m,n[F](α, β, γ) =  �  □ℓ,m,n  � N  �  i=1  |xi|αi|1 − xi|βi  ��  �  1≤j<k≤N  (xk − xj + i0)2γj,k  �  F dx1 · · · dxN  (150)  for (α, β, γ) ∈ Vℓ,m,n[D], where now D denotes a finite collection of indexed sets {dF}F∈F(Aℓ,m,n) ⊆ C,  Vℓ,m,n[D] denotes the set of (α, β, γ) ∈ C2N+N(N−1)/2 for which the integrand in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (150)  lies in L1(□ℓ,m,n, dx1 · · · dxN) – that is the set of (α, β, γ) such that  � N  �  i=1  |xi|αi|1 − xi|βi  ��  �  1≤j<k≤N  |xk − xj|2γj,k  ��  �  F∈F(Kℓ,m,n)  xdF  F  �  ∈ L1(□ℓ,m,n, dx1 · · · dxN)  (151)  for all {dF}F∈F(Aℓ,m,n) ∈ D, and  F has the form eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (146) for F{dF}F∈F(Aℓ,m,n) ∈ C∞(Aℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (150), xγ = eπiγeγ log |x| if x < 0 and xγ = eγ log x if x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We apply abbreviations for  Dotsenko–Fateev-like integrals that are analogous to those used for Selberg-like integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let Wℓ,m,n[•] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ Vℓ,m,n[•] holds when α = α,  β = β, and γ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let W DF0  ℓ,m,n[F] denote the set of (α−, α+, β−, β+, γ−, γ0, γ+) ∈ C7 such that  (αDF0, βDF0, γDF0) ∈ Vℓ,m,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  IDF0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n [F](α−, α+, β−, β+, γ−, γ0, γ+) = Iℓ,m,n[F](αDF0, βDF0, γDF0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (152)  This section is split into many short subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The general analytic framework in which the  extension is performed is discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, and the specific application to Selberg-like integrals  is contained in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We prove a family of identities relating Iℓ,m,n, Iℓ,n,m, In,ℓ,m, · · · in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' As  preparation for our discussion of singularity removal in the DF-symmetric case, we discuss in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4 an  alternative regularization procedure suggested by Dotsenko–Fateev that works for some suboptimal  range of parameters (in particular allowing γ0 = −1, but not allowing the real parts of α−, α+, β−, β+  to be too negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Iℓ,m,n are related to the Selberg-like integrals Sℓ,m,n in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A key lemma  used in the removal of singularities is in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This lemma is a generalization of a result proven by  Aomoto [Aom87] and discussed heuristically by Dotsenko–Fateev [DF85a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For completeness and  28  ETHAN SUSSMAN  later convenience, we record in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7 the symmetric and DF-symmetric cases of the results in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2  regarding the Dotsenko–Fateev integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let Sℓ,m,n = Sℓ × Sm × Sn, which we consider as the subgroup of SN leaving each of I1, I2, I3  invariant, where I1, I2, I3 are as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given a permutation σ ∈ Sℓ,m,n, let  Iℓ,m,n[F](α, β, γ)σ =  �  □ℓ,m,n  � N  �  i=1  |xi|αi|1−xi|βi  ��  �  1≤j<k≤N  (xσ(k)−xσ(j)+i0)2γσ(j),σ(k)  �  F dx1 · · · dxN,  (153)  defined for (α, β, γ) ∈ Vℓ,m,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If we define ασ, βσ, γσ by ασ  j = ασ(j), βσ  j = βσ(j), and γσ  j,k =  γσ(j),σ(k), and  F σ(y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , yN) = F(yσ−1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , yσ−1(N)),  (154)  then Iℓ,m,n[F](α, β, γ)σ = Iℓ,m,n[F σ](ασ, βσ, γσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This relation will be very useful below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' More  generally, for any σ ∈ SN, let  Iℓ,m,n[F](α, β, γ)σ = Iℓ,m,n[F σ](ασ, βσ, γσ)  (155)  Sℓ,m,n[F](α, β, γ)σ = Sℓ,m,n[F σ](ασ, βσ, γσ),  (156)  defined for (α, β, γ) ∈ Vℓ,m,n[F] in the former case or for  (α, β, γ) ∈ Ωℓ,m,n[F]σ = {(α, β, γ) ∈ C2N+N(N−1)/2 : (ασ, βσ, γσ) ∈ Ωℓ,m,n[F σ]}  (157)  in the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We will use similar notation for other subsets of C2N+N(N−1)/2 below, as well as  for the meromorphic extensions of Sℓ,m,n[F] and Iℓ,m,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Some Generalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let N ∈ N be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For a Fréchet space X, let O(CN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' X) denote the  Fréchet space of entire X-valued functions on CN, where the topology is that of uniform convergence  in compact subsets, as measured with respect to each Fréchet seminorm on X, and similarly for X  an LF-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let E′(RN) denote the LCTVS of compactly supported distributions on RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By the  Schwartz representation theorem,  E′(RN) = ∪m∈RHm,s  c  (RN),  (158)  where Hm  c (RN) is the set of compactly supported elements of Hm(RN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let N ∈ N+, k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, and κ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any  ψ ∈ C∞  c (Rk  t1,··· ,tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,··· ,tN )) =  �  m,s∈R  C∞  c (Rk  t1,··· ,tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Hm,s  sc,c (RN−k))  (159)  let, for ρ = (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ρk),  IN,k,κ[ψ](ρ) =  � ∞  0  · ·  � ∞  0  tρ1  1 · · · tρk  k ⟨1, ψ(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , tk, −)⟩ dt1 · · · dtk,  (160)  which we abbreviate as  IN,k,κ[ψ](ρ) =  �  RN  k  tρ1  1 · · · tρk  k ψ(t) dNt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (161)  Here, RN  k = [0, ∞)k  t1,··· ,tk × Rn−k  tk+1,··· ,tN, and IN,k,κ[ψ](ρ) is defined initially for ℜρ1, · · · , ℜρk > −1,  for which the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (160) is a well-defined integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let  O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk  t1,··· ,tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,··· ,tN ))) =  �  m,s∈R  O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk  t1,··· ,tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Hm,s  sc,c (RN−k))), (162)  endowed with the strongest topology such that the inclusions  O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk  t1,··· ,tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Hm,s  sc,c (RN−k))) �→ O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk  t1,··· ,tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,··· ,tN )))  (163)  are all continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  29  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that, for each ρ ∈ Ck and δ ∈ Cκ, we are given some ψ(−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ρ, δ) as in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (159), depending entirely on ρ, δ in the sense that the map  Ck × Cκ ∋ (ρ, δ) �→ ψ ∈ C∞  c (Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k))  (164)  is entire, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' lies in O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk  t1,··· ,tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,··· ,tN ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Define  IN,k,κ[ψ](ρ, δ) = IN,k,κ[ψ(ρ, δ)](ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (165)  Then, the function JN,k,κ[ψ] defined by  IN,k,κ[ψ](ρ, δ) =  �  k  �  j=1  Γ(ρj + 1)  �  JN,k,κ[ψ](ρ, δ)  (166)  extends to an entire function on Ck  ρ × Cκ  δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Moreover, the function  JN,k,κ[−] : O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk  t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tN ))) ∋ ψ �→ JN,k,κ[ψ] ∈ O(Ck × Cκ)  (167)  is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [GS64][Var95, Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The k = 0 case is essentially tautologous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We now proceed inductively on k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let k ≥ 1, and assume that we have proven the result for  smaller k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Expanding ψ in Taylor series around t1 = 0, there exist  ψ(j) ∈ O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk−1  t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tN )))  (168)  E(j) ∈ O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞(Rt1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk−1  t2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tN ))))  (169)  such that  ψ(t1, · · · , tN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ρ, δ) =  J  �  j=0  tj  1ψ(j)(t2, · · · , tN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ρ, δ) + tJ+1  1  E(J+1)(t1, · · · , tN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ρ, δ)  (170)  for all J ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let K ⊂ Ck+κ be an arbitrary nonempty compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There exists some T > 0  such that supp ψ(−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ρ, δ) ⊆ {−T ≤ t1 ≤ T} for all (ρ, δ) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, if ℜρ1, · · · , ℜρk > −1 and  (ρ, δ) ∈ K,  IN,k,κ[ψ](ρ, δ) =  J  �  j=0  IN−1,k−1,κ[ψ(j)]( ˆρ, δ)  ρ1 + j + 1  T ρ1+j+1 +  � T  0  tρ1+J+1  1  IN−1,k−1[E(J+1)(t1, −)]( ˆρ, δ) dt1,  (171)  where ˆρ = (ρ2, · · · , ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We now define JN,k,κ[ψ](ρ, δ) : {ℜρ1 > −2 − J} × Cκ  δ → C by  JN,k,κ[ψ](ρ, δ) =  1  Γ(ρ1 + 1)  J  �  j=0  JN−1,k−1,κ[ψ(j)]( ˆρ, δ)  ρ1 + j + 1  T ρ1+j+1  +  1  Γ(ρ1 + 1)  � T  0  tρ1+J+1  1  JN−1,k−1[E(J+1)(t1, −)]( ˆρ, δ) dt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (172)  By construction, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (166) holds when ℜρ1, · · · , ℜρk > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  By the continuity clause of the  inductive hypothesis, the integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (171) is a well-defined Bochner integral, for each individual  (ρ, δ) ∈ {ℜρ1 > −2 − J} × Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Moreover, the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (172) depends analytically on  (ρ, δ) ∈ {ℜρ1 > −1 − J} × Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By the inductive hypothesis, this is true for the sum on the first line  (multiplied by Γ(ρ1 + 1)−1), as the simple poles due to the factors of 1/(ρ1 + j + 1) cancel with those  30  ETHAN SUSSMAN  of Γ(ρ1 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' So, in order to show that the whole right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (172) depends analytically  on (ρ, δ) in this domain, we can show it for  � T  0  tρ1+J+1  1  JN−1,k−1[E(J+1)(t1, −)]( ˆρ, δ) dt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (173)  Justifying differentiation under the integral sign, this is a C1-function of (ℜρ1, ℑρ1) ∈ {(u, v) ∈  R2, u > −1 − J}, and it satisfies the Cauchy-Riemann equations, so it follows that the integral in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (173) is analytic as a function of ρ1 ∈ {ℜρ1 > −1 − J}, for each fixed ˆρ ∈ Ck−1 and δ ∈ Cκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Adding ˆρ, δ-dependence does not change the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  So, the formula eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (171) yields an analytic extension of IN,k,κ, and we can take a union over all  J ∈ N, the various partial extensions agreeing with each other via analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The continuity clause  is evident from the formula eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (172) and the inductive hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Consequently, IN,k,κ[ψ] admits an analytic continuation ˙IN,k,κ[ψ] : Ω → C to the set Ω =  (Ck  ρ\\ �  j∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',k}{ρj ∈ Z≤−1}) × Cκ  δ, and the map  ˙IN,k,κ[−] : O(Ck × Cκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' C∞  c (Rk  t1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' E′(RN−k  tk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',tN ))) ∋ ψ �→ ˙IN,k,κ[ψ] ∈ O(Ω)  (174)  is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  If P is a consistently orientable collection of codimension-1 interior p-submanifolds on a mwc M,  then, letting xF for F ∈ F(M) denote a bdf of the face F, it is the case that, for any δ ∈ CP and  ρ ∈ CF(M), the product  ω(ρ, δ) =  �  F∈F(M)  xρF  F  �  P∈P  (yP + i0)δP : ˙C∞  c (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩM) ∋ µ �→ lim  ε→0+  �  M  �  F∈F(M)  �  P∈P  xρF  F (yP + iε)δP µ  (175)  is a well-defined classical distribution on M, where {yP }P∈P are consistently oriented defining  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (Here, ˙C∞  c (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩM) is the set of compactly supported smooth densities on M that are  Schwartz at each boundary hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=') That is, ω is an extendable distribution on M and defines,  for small ϵ > 0, an element of C∞([0, ϵ)xF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' D′(F)) for each face F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We write the right-hand side of  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (175) as  �  M ω(ρ, δ)µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' More generally, if µ ∈ C∞  c (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩM), then  lim  ε→0+  �  M  �  F∈F(M)  �  P∈P  xρF  F (yP + iε)δP µ =  �  M  ω(ρ, δ)µ  (176)  exists whenever ρF > −1 for all F ∈ F(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let κ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that we are given some entire family  µ : CF(M) × CP × Cκ → C∞  c (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩM)  (177)  of compactly supported smooth densities µ(ρ, δ, λ) ∈ C∞  c (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' ΩM) on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consider the function  I[M, µ](ρ, δ, λ) : {(ρ, δ, λ) ∈ CF(M) × CP × Cκ : ρF > −1 for all F ∈ F(M)} → C  (178)  defined by  I[M, µ](ρ, δ, λ) =  �  M  ω(ρ, δ)µ(ρ, δ, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (179)  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that, for some N0 ∈ N+, we are given an affine map L = (L1, L2, L3) :  CN0  ϱ  → CF(M)  ρ  × CP  δ × Cκ  λ such that, for each F ∈ F(M), the affine functional  (L•)F : CN0 ∋ ϱ �→ (L1ϱ)F ∈ C  (180)  is nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, there exist entire functions Ireg,f[M, µ](L•) : CN0  ϱ  → C associated to the  minimal facets f of M such that  I[M, µ](Lϱ) =  �  f  �  �  F∈F(M),F⊇f  Γ(1 + (Lϱ)F)  �  Ireg,f[M, µ](Lϱ)  (181)  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  31  for all ϱ ∈ CN0 for which the left-hand side is defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (179).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Pass to a partition of unity subordinate to a system of coordinate charts on M and apply  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Then, letting L = {(L•)F : F ∈ F(M)},  � �  Λ∈L  1  Γ(1 + Λ(ϱ))#Λ  �  I[M, µ](Lϱ)  (182)  extends to an entire function CN0  ϱ  → C, where #Λ ∈ N+ is the maximum size of any set S ⊆ F(M)  of faces such that ∩F∈SF ̸= ∅ and (L•)F = Λ for all F ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed, this follows from the proposition  above since, for each facet f,  � �  Λ∈L  1  Γ(1 + Λ(ϱ))#Λ  �  �  F∈F(M),F⊇f  Γ(1 + (Lϱ)F)  (183)  is entire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Specialization to Generic Selberg- and DF-like Integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We now apply the results of  the previous section to the specific case of the integrals eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (144) and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (150).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix ℓ, m, n ∈ N  satisfying ℓ + m + n = N, N ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Selberg case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix F ∈ AD(Kℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let ρj,k = ρj,k(α, β, γ) be defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (89), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (90),  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (91), and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Recalling the definition of T(ℓ, m, n) given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There exist entire functions  Sℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I,{dF}F∈F(Kℓ,m,n)[F] : C2N+N(N−1)/2  α,β,γ  → C,  (184)  associated to pairs of minimal facets f of Kℓ,m,n and collections {dF}F∈F(ℓ,m,n) ∈ D of weights such  that  Sℓ,m,n[F](α, β, γ) =  �  I∈T(ℓ,m,n)  �  {dF}F∈F(Kℓ,m,n)∈D  �  �  I(j,k)∈I  Γ(1 + ρj,k + dFj,k)  �  × Sℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I,{dF}F∈F(Kℓ,m,n)[F](α, β, γ)  (185)  for all (α, β, γ) ∈ Ωℓ,m,n[D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is a corollary of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2, using the fact that the minimal  facets of Kℓ,m,n are in correspondence with the elements of T(ℓ, m, n) via eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Consequently, there exists an analytic extension ˙Sℓ,m,n[F] :  ˙Ωℓ,m,n[D] → C of Sℓ,m,n[F] :  Ωℓ,m,n[D] → C, where  ˙Ωℓ,m,n[D] = C2N+N(N−1)/2  α,β,γ  ��  �  {dF}F∈F(Kℓ,m,n)∈D  �  �  {j,k}∈Jℓ,m,n  {ρj,k + dFj,k ∈ Z≤−1}  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (186)  This is an open and connected subset of full measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' namely, it is the complement of a locally finite  collection of complex (affine) hyperplanes in C2N+N(N−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the case m = N, this agrees with  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  As a corollary of the previous proposition, there exists an entire function  Sℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F] : C2N+N(N−1)/2  α,β,γ  → C  (187)  such that  Sℓ,m,n[F](α, β, γ) =  �  �  {j,k}∈Jℓ,m,n  Γ(1 + ρj,k + dmin  Fj,k)  �  Sℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F](α, β, γ)  (188)  32  ETHAN SUSSMAN  holds for all (α, β, γ) ∈ Ωℓ,m,n[D], where dmin  F  = min{dF : {dF0}F0∈F(Kℓ,m,n) ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The case of the proposition above where m = N gives Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed, if F ∈ C∞(△N), F  lifts to an element of C∞(K0,N,0), and the orders of vanishing of F at the relevant facets of △N  imply the same order of vanishing at the lift in K0,N,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Dotsenko–Fateev case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix F ∈ AD(Aℓ,m,n), where D is now a collection of orders for  the faces of Aℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Recalling the definition of ΣT(ℓ, m, n) given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There exist entire functions  Iℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I,{dF}F∈F(Aℓ,m,n)[F] : C2N+N(N−1)/2  α,β,γ  → C  (189)  associated to the I ∈ ΣT(ℓ, m, n) such that  Iℓ,m,n[F](α, β, γ) =  �  I∈ΣT(ℓ,m,n)  �  {dF}F∈F(Aℓ,m,n)∈D  ��  �  (x0,S)∈I  Γ(1 + ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0 + dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0)  �  × Iℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg,I{dF}F∈F(Aℓ,m,n)[F](α, β, γ)  �  (190)  for all (α, β, γ) ∈ Vℓ,m,n[D], where we have abbreviated I1 ∩ S, I2 ∩ S, and I3 ∩ S as S or Q as  appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Consequently, Iℓ,m,n[F] : Vℓ,m,n[D] → C admits an analytic continuation ˙Iℓ,m,n[F] : ˙Vℓ,m,n[D] → C,  where  ˙Vℓ,m,n[D] = C2N+N(N−1)/2  α,β,γ  �  �  {dF}F∈F(Aℓ,m,n)  �  x0∈{0,1,∞}  �  S,Q  {ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0 + dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0 ∈ Z≤−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (191)  Note that ˙Vℓ,m,n[F] ⊇ ∩σ∈Sℓ,m,n ˙Ωℓ,m,n[F]σ, as every functional (α, β, γ) �→ ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0(α, β, γ) has the  form ρj,k(ασ, βσ, γσ) for some σ ∈ Sℓ,m,n and {j, k} ∈ Jℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  As a corollary of the previous proposition, there exists a function  Iℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F] : C2N+N(N−1)/2  α,β,γ  → C  (192)  such that, for all (α, β, γ) ∈ Vℓ,m,n[D],  Iℓ,m,n[F](α, β, γ) =  �  �  x0∈{0,1,∞}  �  S,Q  Γ(1 + ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0 + dmin  FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='x0)  �  Iℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='reg[F](α, β, γ),  (193)  where S, Q vary over subsets of I1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, I2 = {ℓ+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ+m}, and I3 = {ℓ+m+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N},  depending on x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The m = N case of the previous proposition is Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A simple identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For each permutation σ of {0, 1, ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  (ℓ′, m′, n′) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (ℓ, m, n)  (σ = 1),  (n, m, ℓ)  (σ = (0 1)),  (ℓ, n, m)  (σ = (0 ∞)),  (m, ℓ, n)  (σ = (1 ∞)),  (n, ℓ, m)  (σ = (0 1 ∞)),  (m, n, ℓ)  (σ = (1 0 ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (194)  In other words, if the elements of {0, 1, ∞} label the vertices of a triangle and the edges are labeled  accordingly – that is, ‘ℓ’ labels the edge between 0 and ∞, ‘m’ labels the edge between 0 and 1, and  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  33  ‘n’ labels the edge between 1 and ∞ – then (ℓ′, m′, n′) is the permutation of (ℓ, m, n) resulting from  applying σ to the triangle and reading off the new labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let Tσ : CP 1 → CP 1 denote the unique automorphism acting on {0, 1, ∞} via σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' These are  T1(z) = z,  T(0 1)(z) = 1 − z,  T(0 ∞)(z) = 1  z ,  T(1 ∞)(z) = −  z  1 − z ,  (195)  T(0 1 ∞)(z) =  1  1 − z ,  T(0 ∞ 1)(z) = z − 1  z  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (196)  Let σparam : C2N+N(N−1)/2 → C2N+N(N−1)/2 denote the affine map  σparam(α, β, γ) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (α, β, γ)  (σ = 1),  (β, α, γ)  (σ = (0 1)),  (−2 − α − β − 2γ⌟1, β, γ)  (σ = (0 ∞)),  (α, −2 − α − β − 2γ⌟1, γ)  (σ = (1 ∞)),  (−2 − α − β − 2γ⌟1, α, γ)  (σ = (0 1 ∞)),  (β, −2 − α − β − 2γ⌟1, γ)  (σ = (1 0 ∞)),  (197)  where γ⌟1 ∈ CN has jth component �  k̸=j γj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let rev ∈ Sℓ′,m′,n′ denote the permutation  that reverses the order of the elements in each of the sets {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ′}, {ℓ′ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ′ + m′}, and  {ℓ′ + m′ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let |σ| denote the order of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If (α, β, γ) ∈ ˙Vℓ,m,n, then σparam(α, β, γ) ∈ ˙Vℓ′,m′,n′, and if (α, β, γ) ∈ ˙Ωℓ,m,n,  then σparam(α, β, γ) ∈ ˙Ωrev|σ|  ℓ′,m′,n′, and  ˙Iℓ,m,n[1](α, β, γ) = ˙Iℓ′,m′,n′[1](σparam(α, β, γ))rev|σ|,  ˙Sℓ,m,n[1](α, β, γ) = ˙Sℓ′,m′,n′[1](σparam(α, β, γ))rev|σ|  (198)  for all (α, β, γ) ∈ ˙Ωℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It can be checked case-by-case that  {ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='• ◦ σparam : • ∈ {0, 1, ∞}, S, Q as above} = {ϱS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='• : • ∈ {0, 1, ∞}, S, Q as above},  (199)  where on the left-hand side (S, Q) varies over appropriate pairs of subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ′}, {ℓ′ +  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ′ + m′}, and {ℓ′ + m′ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and on the right-hand side (S, Q) varies over appropriate  pairs of subsets {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, and {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, depending on •.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It can be  seen from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (199) that  ˙Vℓ,m,n = (σparam)−1( ˙Vℓ′,m′,n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (200)  The case of ˙Ωℓ,m,n is similar but more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Equation (198) can be proven for (α, β, γ) ∈ Ωℓ,m,n by way of a change-of-variables by substituting  x = Tσ−1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The full result follows via analytic continuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' An imperfect alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For I ∈ {(−∞, 0], [0, 1], [1, ∞)} and r > 0, let ΓI,±,r : (0, 1) → C  be defined by  Γ[0,1],±,r(t) =  �  �  �  �  �  �  �  t ± irt  (t ∈ (0, 1/3)),  t ± ir/3  (t ∈ [1/3, 2/3]),  t ± ir/3 ∓ ir(t − 2/3)  (t ∈ (2/3, 1)),  (201)  Γ[1,∞),±,r(t) = Γ[0,1],∓,r(1 − t)−1, and Γ(−∞,0],±,r(t) = 1 − Γ[1,∞),∓,r(1 − t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Note that the images of  these contours are permuted amongst themselves by the transformations Tσ above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  34  ETHAN SUSSMAN  ℑz  ℜz  Γ[0,1],+,1  Γ[0,1],+,4  0  1  Γ[1,∞),+,1  Γ(−∞,0],+,1  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The contours Γ(−∞,0],+,1, Γ[0,1],+,1, Γ[0,1],+,4, Γ[1,∞),+,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [DF85a,  Figure 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (For our purposes, the contours drawn by Dotsenko & Fateev approach  ±∞ with imaginary part too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is why our ΓI,±,r look different for I ̸= [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Suppose that F ∈ C[x1, x−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN, x−1  N ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any compact K ⋐ C with nonempty interior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' let  O = O[F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' K] denote the set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' which depends on ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' n ∈ N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' though we suppress this dependence  notationally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' of (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' β) ∈ C2N such that  �  Γ(−∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  · ·  �  Γ(−∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='ℓ−1  � �  Γ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  · ·  �  Γ[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m−1  � �  Γ[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0  · ·  �  Γ[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n−1  � N  �  j=1  zαj  j (1 − zj)βj  �  �  1≤j<k≤N  (zk − zj)2γj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='kF0 dzN · · · dzℓ+m+1  �  dzℓ+m · · · dzℓ+1  �  dzℓ · · · dz1  (202)  is an absolutely convergent Lebesgue integral whenever γj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='k ∈ K for all j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' k ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} with j < k,  for every monomial F0 in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the definition of the integral above we are defining the integrand  such that the branch cuts are not encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For such (α, β, γ),  (α, β, γ) ∈ ˙Vℓ,m,n[F],  (203)  and the integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (202) is equal to ˙Iℓ,m,n(α, β, γ)[F], assuming that we choose our branches  appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The latter part of this statement can be proven by checking that the integral defined  above depends analytically on its parameters and agrees with Iℓ,m,n(α, β, γ)[F] for (α, β, γ) ∈  Vℓ,m,n[F], which in turn is proven via a contour deformation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The set O is nonempty, open, and contains an affine cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If  αj has sufficiently large real part for j ∈ I1 ∪ I2 and sufficiently negative real part for j ∈ I3,  and  βj has sufficiently large real part for j ∈ I2 ∪ I3 and sufficiently negative real part for j ∈ I1,  then (α, β) ∈ O[F, K], where what “sufficiently large” means depends on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consequently, given  any subset S ⊆ Sℓ × Sm × Sn, the set OS∩ defined by  OS∩ = {(α, β) ∈ C2N : (ασ, βσ) ∈ O[F σ, Kσ] for all σ ∈ S}  (204)  is open and nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If K contains e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' −1, then O[F, K] contains some (α, β) such that (α, β, γ) /∈  Vℓ,m,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' So, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (202) gives us an alternative definition of ˙Iℓ,m,n(α, β, γ)[F] for some range of  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  35  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Consider • ∈ {1, 2, 3} and j, k ∈ I• with j < k and |j − k| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that  γj,k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let τ ∈ Sℓ,m,n denote the transposition swapping j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then,  Iℓ,m,n[F](α, β, γ) − Iℓ,m,n[F](α, β, γ)τ =  �  Γ(−∞,0],+,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1  · ·  �  Γ(−∞,0],+,ℓ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='ℓ  �  �  Γ[0,1],+,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='ℓ+1  · ·  �  Γ[0,1],+,m−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='ℓ+m  � �  Γ[1,∞),+,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='ℓ+m+1  · ·  �  Γ[1,∞),+,n−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='N  �  N  �  j0=1  z  αj0  j0 (1 − zj0)βj0  �  ×  �  �  1≤j0<k0≤N  (zk0 − zj0)2γj0,k0  �  F dzN · · · dzℓ+m+1  �  dzℓ+m · · · dzℓ+1  �  dzℓ · · · dz1,  (205)  whenever (α, β, γ) ∈ O∩{1,τ}, where ΓI,+,r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='i = ΓI,+,r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='i unless i = j, in which case ΓI,+,r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='i =  ΓI,+,r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='i({zi0}i0̸=j) is a small counterclockwise circle around zk not winding around any of the other  z’s or 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It suffices to consider the case F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Indeed, if F is a monomial, then we can simply absorb  it into a redefinition of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The set O∩{1,τ} is decreasing with the set of monomials in F, so once the  result has been proven for monomials, it follows for all Laurent polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For • = 2, the proposition follows via a straightforward countour deformation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The  case • ∈ {1, 3} can be reduced to • = 3 via Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Symmetrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let F ∈ AD(Aℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any (α, β, γ) ∈ ∩σ∈Sℓ,m,n ˙Ωℓ,m,n[F]σ,  ˙Iℓ,m,n[F](α, β, γ) =  �  σ∈Sℓ,m,n  eiΘ(σ−1) ˙Sℓ,m,n[F](α, β, γ)σ,  (206)  where Θ(σ) = 2π �  1≤j<k≤N 1σ(j)>σ(k)γj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By analyticity, it suffices to prove the result when the quantities above are well-defined  Lebesgue integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Decomposing □ℓ,m,n into ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' copies of △ℓ,m,n,  Iℓ,m,n[F](α, β, γ) =  �  σ∈Sℓ,m,n  �  △ℓ,m,n  N  �  j=1  |xj|ασ(j)|1 − xj|βσ(j)  �  1≤j<k≤N  (xσ−1(k) − xσ−1(j) + i0)2γj,k  F(xσ−1(1), · · · , xσ−1(N)) dNx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (207)  The right-hand side is  �  σ∈Sℓ,m,n  eiΘ(σ−1)  �  △ℓ,m,n  N  �  j=1  |xj|ασ(j)|1 − xj|βσ(j)  �  1≤j<k≤N  (xk − xj)2γσ(j),σ(k)(F σ) dNx,  (208)  which is the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (206).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that α, β, γ are invariant under all σ ∈ Sℓ,m,n, and suppose now that  F ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN]Sℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, for all (α, β, γ) ∈ ˙Ωℓ,m,n[F],  ˙Iℓ,m,n[F](α, β, γ) =  �  ℓ  �  k=1  1 − e2πikγ1  1 − e2πiγ1  �� m  �  k=1  1 − e2πikγ2  1 − e2πiγ2  ��  n  �  k=1  1 − e2πikγ3  1 − e2πiγ3  � ˙Sℓ,m,n[F](α, β, γ),  (209)  where, for each • ∈ {1, 2, 3}, γ• = γj,k for all distinct j, k ∈ I•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Here, we are treating (1 − e2πiγ)−1(1 − e2πikγ) as an entire function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  36  ETHAN SUSSMAN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Applying the previous proposition,  Iℓ,m,n[F](α, β, γ) =  � �  σ∈Sℓ  eπi o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (σ)γ�� �  σ∈Sm  eπi o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (σ)γ�� �  σ∈Sn  eπi o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (σ)γ�  Sℓ,m,n[F](α, β, γ),  (210)  where o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (σ) is the number of out-of-order pairs in σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We appeal to the algebraic identity [Bón12]  Z[ζ] ∋  �  σ∈SN  ζo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (σ) =  N−1  �  n=0  n  �  m=0  ζm =  N  �  n=1  1 − ζn  1 − ζ ,  (211)  which holds for all N ∈ N and encodes the bijection between SN and the set of possible runs of the  bubble sort algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Plugging in ζ = eπiγ, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (210) becomes eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (209).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Aomoto-Dotsenko–Fateev relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix N ∈ N+ and F ∈ C[x1, x−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN, x−1  N ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, let σj ∈ SN be the permutation that takes 1 and inserts it in the jth  position while maintaining the relative order of the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' That is, σj = (1 j j − 1 · · · 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For any ℓ ∈ N+ and m, n ∈ N with ℓ + m + n = N, let  ˙Λℓ,m,n[F] = ˙Vℓ,m,n[F] ∩ ˙Vℓ−1,m+1,n[F]σℓ ∩ ˙Vℓ−1,m,n+1[F]σℓ+m  = ˙Vℓ,m,n[F] ∩ ˙Vℓ−1,m+1,n[F]σℓ+m ∩ ˙Vℓ−1,m,n+1[F]σN ,  (212)  ˙℧ℓ,m,n[F] = (∩ℓ  j=1 ˙Ωℓ,m,n[F]σj) ∩ (∩ℓ+m  j=ℓ ˙Ωℓ−1,m+1,n[F]σj) ∩ (∩N  j=ℓ+m ˙Ωℓ−1,m,n+1[F]σj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (213)  Note that ˙℧ℓ,m,n[F], ˙Λℓ,m,n[F] are open, dense, and connected subsets of C2N+N(N−1)/2, being the  complements of locally finite unions of complex affine hyperplanes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any (α, β, γ) ∈ ˙℧ℓ,m,n[F],  0 =  ℓ  �  j=1  e±iθj ˙Sℓ,m,n[F](α, β, γ)σj +  ℓ+m  �  j=ℓ  e±iϑj ˙Sℓ−1,m+1,n[F](α, β, γ)σj  +  N  �  j=ℓ+m  e±iϕj ˙Sℓ−1,m,n+1[F](α, β, γ)σj  (214)  holds for each choice of sign, where θj = 2π �  2≤j0≤j γ1,j0, ϑj = πα + 2π �  2≤j0≤j γ1,j0, and ϕj =  πα + πβ + 2π �  2≤j0≤j γ1,j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Without loss of generality, we may assume F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  ℧ℓ,m,n =  �  �  �  �  �  �  �  (α, β, γ) ∈ C2N+N(N−1)/2 : (ασj, βσj, γσj) ∈  �  �  �  �  �  �  �  Ωℓ,m,n  (j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ})  Ωℓ−1,m+1,n  (j ∈ {ℓ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m})  Ωℓ−1,m,n+1  (j ∈ {ℓ + m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N})  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (215)  (This is to be read such that, whenever (α, β, γ) ∈ ℧ℓ,m,n, (ασℓ, βσℓ, γσℓ) ∈ Ωℓ,m,n ∩ Ωℓ−1,m+1,n and  (ασℓ+m, βσℓ+m, γσℓ+m) ∈ Ωℓ−1,m+1,n ∩ Ωℓ−1,m,n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=')  Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For each ϝ1, ϝ2, ϝ3 > 0 and γ, γ ∈ (−(N −1)−1, 0) with γ < γ, let ℧0,ϝ,γ,γ (suppressing  the ℓ, m, n dependence for brevity) denote the set of (α, β, γ) ∈ C2N+N(N−1)/2 such that  γ < ℜγj,k < γ for all j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} with j ̸= k,  ϝ1 < ℜαj < ϝ2 for each j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ}, ℜαj > ϝ1 for each j ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, and  ℜαj < −ϝ3 for each j ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N},  ϝ1 < ℜβj < ϝ2 for each j ∈ {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, ℜβj > ϝ1 for each j ∈ {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m},  and ℜβj < −ϝ3 for j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ},  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  37  where ϝ = (ϝ1, ϝ2, ϝ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The set ℧0,ϝ,γ,γ is open and nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (7) and the analogue of  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (7) for the m < N case, there exist ϝ00, ϝ0, ϝ01 > 0 (depending on ℓ, m, n, γ, γ) such that  ℧ϝ,γ,γ  def  = {(α, β, γ) ∈ ℧0,ϝ,γ,γ and (α1, β1) ∈ Ω1,0,0 ∩ Ω0,1,0 ∩ Ω0,0,1} ⊂ ℧ℓ,m,n  (216)  whenever ϝ2 > ϝ1 > ϝ0 and ϝ3 > ϝ01ϝ2 + ϝ00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Observe that Ω1,0,0 ∩ Ω0,1,0 ∩ Ω0,0,1 is the subset  of C2  α,β defined by the inequalities −1 < ℜα, ℜβ and ℜα + ℜβ < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The set  {(r1, r2) ∈ R2 : −1 < r1, r2 and r1 + r2 < −1}  (217)  is a nonempty triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' So, ℧ϝ,γ,γ is an open and nonempty subset of C2N+N(N−1)/2 and moreover  of ˙℧ℓ,m,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For such ϝ and (α, β, γ) ∈ ℧ϝ,γ,γ, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (214) (with F = 1) just reads  0 =  ℓ  �  j=1  e±iθjSℓ,m,n[1](α, β, γ)σj +  ℓ+m  �  j=ℓ  e±iϑjSℓ−1,m+1,n[1](α, β, γ)σj  +  N  �  j=ℓ+m  e±iϕjSℓ−1,m,n+1[1](α, β, γ)σj  (218)  (note the absence of the dots over the S’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By the analyticity of all of the functions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (214) on  ˙℧ℓ,m,n, it suffices to prove that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (218) holds for such (α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  By Fubini’s theorem, the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (218) is  �  △ℓ−1,m,n  ω(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN)  � � +∞  −∞  (−x1 ± i0)α1(1 − x1 ± i0)β1� N  �  j=2  (xj − x ± i0)2γ1,j  �  dx  �  dx2 · · · dxN,  (219)  where ω(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) = [�N  j=2 |xj|αj|1 − xj|βj] �  2≤j<k≤N(xk − xj)2γj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The claim then follows from  0 =  � +∞  −∞  (−x ± i0)α(1 − x ± i0)β� N  �  j=2  (xj − x ± i0)2γj  �  dx,  (220)  which holds for every (x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) ∈ (R\\{0, 1})N−1 such that x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN are pairwise distinct and  all α, β, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , γN ∈ C for which  the integrand of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (220) lies in L1(R) and  ℜγj ∈ (−1, 0) for all j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Denote the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (220) by I± = I±(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' α, β, γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , γN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For R >  max{|x1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , |xN−1|},  0 =  �  Γ∓(R)  (−z ± i0)α(1 − z ± i0)β  N  �  j=2  (xj − z ± i0)2γj dz,  (221)  where Γ±(R) = Γ±(R)(x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) ⊂ C is the semicircular contour (with N + 1 semicircular insets  placed so that the contour avoids x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN) connecting −R and +R, with the arc and semicircular  insets in the half-plane {z ∈ C : ±ℑz ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' See Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (221), the integrand is defined  taking the branch cut along the negative real axis, so  (x − z ± i0)2γj =  �  exp(2γj(log |x − z| + i arg(x − z)))  (+ case, ℑz ≤ 0),  exp(2γj(log |x − z| − 2πi + i arg(x − z)))  (− case, ℑz ≥ 0),  (222)  for any x ∈ R, where arg(x − z) ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We orient Γ+ counter-clockwise and Γ− clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  38  ETHAN SUSSMAN  ℑz  ℜz  0  1  x2  x1  x3  −R  R  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The contour Γ+(R) in the case ℓ = 2, m = 1, n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let Γ++(R) denote the large arc of Γ+(R) and Γ+0(R) denote the rest, and likewise let Γ−−(R)  denote the large arc of Γ−(R) and Γ−0(R) denote the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then,  I± = lim  R→∞  �  Γ∓0(R)  (−z ± i0)α(1 − z ± i0)β  N  �  j=2  (xj − z ± i0)2γj dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (223)  On the other hand, for R sufficiently large,  ���  �  Γ∓∓(R)  (−z ± i0)α(1 − z ± i0)β  N  �  j=2  (xj − z ± i0)2γj dx  ��� ≤ π(2R)1+ℜα+ℜβ = O(R−ε)  (224)  for some ε > 0 depending on (α, β) ∈ Ω1,0,0 ∩ Ω0,1,0 ∩ Ω0,0,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Combining eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (221), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (223), and  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (224), we get I± = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any F ∈ C[x1, x−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN, x−1  N ],  0 = ˙Iℓ,m,n[F](α, β, γ) + e+πi(α+2�ℓ  j=2 γ1,j) ˙Iℓ−1,m+1,n[F](α, β, γ)σℓ  + e+πi(α+β+2�ℓ+m  j=2 γ1,j) ˙Iℓ−1,m,n+1[F](α, β, γ)σℓ+m  (225)  0 = ˙Iℓ,m,n[F](α, β, γ)σℓ + e−πi(α+2�ℓ  j=2 γ1,j) ˙Iℓ−1,m+1,n[F](α, β, γ)σℓ+m  + e−πi(α+β+2�ℓ+m  j=2 γ1,j) ˙Iℓ−1,m,n+1[F](α, β, γ)σN  (226)  both hold, for all (α, β, γ) ∈ ˙Λℓ,m,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Via analyticity, it suffices to prove this for all (α, β, γ) ∈ ∩σ∈Sℓ,m,n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' σ(1)=1 ˙℧ℓ,m,n[F]σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For  such (α, β, γ), we can cite the previous proposition to get  0 =  �  σ∈Sℓ,m,n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' σ(1)=1  eπiΘ(σ−1)�  ℓ  �  j=1  e±iθσ  j ˙Sℓ,m,n[F](α, β, γ)σjσ  +  ℓ+m  �  j=ℓ  e±iϑσ  j ˙Sℓ−1,m+1,n[F](α, β, γ)σjσ +  N  �  j=ℓ+m  e±iϕσ  j ˙Sℓ−1,m,n+1[F](α, β, γ)σjσ�  ,  (227)  where θσ  j = 2π �  2≤j0≤j γ1,σ(j0), ϑσ  j = πα+2π �  2≤j0≤j γ1,σ(j0), and ϕσ  j = πα+πβ+2π �  2≤j0≤j γ1,σ(j0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The order of multiplication is such that σjσ is a permutation satisfying (σjσ)(1) = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (227),  Θ is defined as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  39  Every σ0 ∈ Sℓ,m,n has the form σ0 = σjσ for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and σ ∈ Sℓ,m,n satisfying  σ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It can be seen that  Θ(σ−1  0 ) = Θ(σ−1) + θσ  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (228)  Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7, we check that the two cases of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (227) yield the two results, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (225) and  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (226).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For instance,  �  σ∈Sℓ,m,n s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' σ(1)=1  eπiΘ(σ−1)  ℓ  �  j=1  e+iθσ  j ˙Sℓ,m,n[F](α, β, γ)σjσ =  �  σ∈Sℓ,m,n  eπiΘ(σ−1) ˙Sℓ,m,n[F](α, β, γ)σ  = ˙Iℓ,m,n[F](α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (229)  Similar statements apply to the other two sums in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (227) in the ‘+’ case, thus yielding eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (225).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Similar computations apply to the ‘−’ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The symmetric and DF-symmetric cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix F ∈ AD(Aℓ,m,n), not necessarily symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We assume that dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='• ∈ Z for all FS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='• ∈ F(Aℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Let  δk = min{dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 : S ⊆ I1, Q ⊆ I2, |S ∪ Q| = k}  (230)  for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m},  δ  k = min{dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 : S ⊆ I2, Q ⊆ I3, |S ∪ Q| = k}  (231)  for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , m + n}, and  dk = − min{dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞ : S ⊆ I3, Q ⊆ I1, |S ∪ Q| = k}  (232)  for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Here, we are ranging over all {dF}F∈F(Aℓ,m,n) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let ˙Wℓ,m,n[D] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ ˙Vℓ,m,n[D] whenever α, β, γ  have components given by αj = α and βj = β for all indices j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and γj,k = γ for all  j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} with j < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There exists an entire function Iℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] : C3 → C such that  ˙Iℓ,m,n[F](α, β, γ) =  � ℓ+m  �  k=1  Γ(δk + k(1 + α + (k − 1)γ))  �� m+n  �  k=1  Γ(  δ  k + k(1 + β + (k − 1)γ))  �  ×  � ℓ+n  �  k=1  Γ(−dk − k(1 + α + β + (2N − k − 1)γ))  �  Iℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ)  (233)  for all (α, β, γ) ∈ ˙Wℓ,m,n[D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  For later reference, consider the special case F ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN]SN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Referring to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (14), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (15),  and eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (28), set dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 = δj[F], dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 =  δ  j[F], and dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞ = degj[F], for S, Q ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} as  usual, where, for each S and Q, j = |S ∪ Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, as follows straightforwardly from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (71),  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (73), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (75),  F ∈  �  F∈F(Aℓ,m,n)  xdF  F C∞(Aℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (234)  Thus, letting D denote the collection of the integers above, F ∈ AD(Aℓ,m,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We can therefore apply  the results above, with δj = δj[F],  δ  j =  δ  j[F], and dj = − degj[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We now turn to the “DF0-symmetric” case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, let  ˙W DF0,S  ℓ,m,n [F] = {(α−, α+, β−, β+, γ−, γ0, γ+) ∈ C7 : (αDF0, βDF0, γDF0) ∈ ˙Vℓ,m,n[F]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (235)  40  ETHAN SUSSMAN  This is a dense, open, and connected subset of C7 and depends on S only through the numbers  |S ∩ Ij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Actually, we need a slightly refined version of this later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' let  ˙W DF1,S  ℓ,m,n [F] = {(α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ−, γ0, γ+) ∈ C9  : (αDF1, βDF1, γDF0) ∈ ˙Vℓ,m,n[F]},  (236)  where αDF1, βDF1 are defined as their DF0-counterparts, but defining the jth component using  α+,ν in place of α+ and β+,ν in place of β+ for ν ∈ Iν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For (α−, α+, β−, β+, γ−, γ0, γ+) ∈ ˙W DF0,S  ℓ,m,n [F], let  ˙IDF0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n [F](α−, α+, β−, β+, γ−, γ0, γ+) = ˙Iℓ,m,n[F](αDF0, βDF0, γDF0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (237)  Let ℓ+ = S ∩ I1, ℓ− = ℓ − ℓ+, m+ = S ∩ I2, m− = m − m+, n+ = S ∩ I3, and n− = n − n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Set  N+ = |S| and N− = N − N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Suppose now that F ∈ AD(Aℓ,m,n) is symmetric in the variables {xi}i∈S and {xi}i/∈S separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let  δj−,j+ = min{dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 : S ⊆ I1, Q ⊆ I2, |(S ∪ Q)\\S| = j−, |(S ∪ Q) ∩ S| = j+}  (238)  for j− ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ− + m−} and j+ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ+ + m+},  δ  j−,j+ = min{dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 : S ⊆ I2, Q ⊆ I3, |(S ∪ Q)\\S| = j−, |(S ∪ Q) ∩ S| = j+}  (239)  for j− ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , m− + n−} and j+ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , m+ + n+}, and  dj−,j+ = − min{dFS,Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞ : S ⊆ I3, Q ⊆ I1, |(S ∪ Q)\\S| = j−, |(S ∪ Q) ∩ S| = j+}  (240)  for j− ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ− + n−} and j+ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ+ + n+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A similar argument to that above yields:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' There exists an entire function IDF0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] : C7 → C such that  ˙IDF0  ℓ,m,n[F](α−, α+, β−, β+, γ−, γ0, γ+) = IDF0  ℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α−, α+, β−, β+, γ−, γ0, γ+)  ×  � ℓ−+m−  �  j−=1  ℓ++m+  �  j+=1  Γ(δj−,j+ + j−(1 + α− + (j− − 1)γ−) + j+(1 + α+ + (j+ − 1)γ+) + 2γ0j−j+)  �  ×  � m−+n−  �  j−=1  m++n+  �  j+=1  Γ(  δ  j−,j+ + j−(1 + β− + (j− − 1)γ−) + j+(1 + β+ + (j+ − 1)γ+) + 2γ0j−j+)  �  ×  � ℓ−+n−  �  j−=1  ℓ++n+  �  j+=1  Γ(−dj−,j+ − j−(1 + α− + β− + (2N− − j− − 1)γ−)  − j+(1 + α+ + β+ + (2N+ − j+ − 1)γ+) − 2γ0j−j+)  �  (241)  holds whenever (α−, α+, β−, β+, γ−, γ0, γ+) ∈ ˙W DF0  ℓ,m,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■□  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Removing singularities  As in previous sections, fix ℓ, m, n ∈ N not all zero, and let N = ℓ + m + n and I1 = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ},  I2 = {ℓ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , ℓ + m}, and I3 = {ℓ + m + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For k ∈ N, let  ϝk : Cγ\\{kγ ∈ Z≤−1 and γ /∈ Z} → C  (242)  denote the analytic function given by ϝk(γ) = Γ(1 + γ)−1Γ(1 + kγ) for kγ /∈ Z≤−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We can consider  ϝ−1  k  as an entire function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  41  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix F ∈ C[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , xN]SN , and let δj,  δ  j, dj ∈ N be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let ˙Uℓ,m,n[F] denote the set of (α, β, γ) ∈ C3 such that (α, β, γ) ∈ ˙Ωℓ,m,n[F] whenever α, β, γ  have components given by αj = α and βj = β for all indices j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and γj,k = γ for all  j < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, we can define  ˙Sℓ,m,n[F](α, β, γ) = ˙Sℓ,m,n[F](α, β, γ)  (243)  for any (α, β, γ) ∈ ˙Uℓ,m,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The function Sreg  ℓ,m,n[F] : ˙Uℓ,m,n[F] → C defined by  Sreg  ℓ,m,n[F](α, β, γ) =  � ℓ+m  �  k=1  Γ(δk + k(1 + α + (k − 1)γ))  �−1  � m+n  �  k=1  Γ(  δ  k + k(1 + β + (k − 1)γ))  �−1� ℓ+n  �  k=1  Γ(−dk − k(1 + α + β + (N + k − 2)γ))  �−1  �  ℓ  �  k=1  1  ϝk(γ)  �� m  �  k=1  1  ϝk(γ)  ��  n  �  k=1  1  ϝk(γ)  � ˙Sℓ,m,n[F](α, β, γ)  (244)  extends to an entire function C3  α,β,γ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Since the prefactor on the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (244) consisting of all of the Γ-  function reciprocals is entire, Sreg  ℓ,m,n[F] extends to an analytic function on ˙Uℓ,m,n[F], the  domain of ˙Sℓ,m,n[F](α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For all (α, β, γ) ∈ ˙Uℓ,m,n[F], we have  �  ℓ  �  k=1  1 − e2πikγ  1 − e2πiγ ϝk(γ)  �� m  �  k=1  1 − e2πikγ  1 − e2πiγ ϝk(γ)  ��  n  �  k=1  1 − e2πikγ  1 − e2πiγ ϝk(γ)  �  Sreg  ℓ,m,n[F](α, β, γ)  = Iℓ,m,n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ)  (245)  by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='11, this extends to an entire function C3  α,β,γ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The product ϝk(γ)(1 − e2πikγ)(1 − e2πiγ)−1, with its removable singularities removed,  vanishes if and only if kγ ∈ N and γ /∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, Sreg  ℓ,m,n[F] extends to an analytic function on  C3  α,β,γ\\ ∪M  k=2 {kγ ∈ N, γ /∈ N},  (246)  where M = max{ℓ, m, n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Combining these two observations, Sreg  ℓ,m,n[F] extends to an analytic function on ˙Uℓ,m,n[F] ∪  (C3  α,β,γ\\ ∪M  k=2 {kγ ∈ N, γ /∈ N}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The set ∪M  k=2{kγ ∈ N, γ /∈ N} is a union of hyperplanes, and it is disjoint from  N  �  k=1  {k(k + 1)γ ∈ Z≤−k},  (247)  so ˙Uℓ,m,n[F]∪(C3  α,β,γ\\∪M  k=2 {kγ ∈ N, γ /∈ N}) is the complement in C3  α,β,γ of a locally finite collection  of complex codimension-2 affine subspaces of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The result therefore follows from Hartog’s extension  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  For any ℓ ∈ N+ and m, n ∈ N,  {(α, β, γ) ∈ C3 : (α, β, γ) ∈ ˙℧ℓ,m,n[F]} = ˙Uℓ,m,n[F] ∩ ˙Uℓ−1,m+1,n[F] ∩ ˙Uℓ−1,m,n+1[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (248)  42  ETHAN SUSSMAN  The symmetric case of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='9 reads, after multiplying through by 1 − e±2iγ,  0 = (1 − e±2πiℓγ) ˙Sℓ,m,n[F](α, β, γ) + e±πi(α+2(ℓ−1)γ)(1 − e±2πi(m+1)γ) ˙Sℓ−1,m+1,n[F](α, β, γ)  + e±πi(α+β+2(ℓ−1+m)γ)(1 − e±2πi(n+1)γ) ˙Sℓ−1,m,n+1[F](α, β, γ)  (249)  for all (α, β, γ) in the set defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (248).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Define  ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 = {(α, β, γ) ∈ C3 : α + mγ /∈ Z for any j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N − 1}},  (250)  ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 = {(α, β, γ) ∈ C3 : β + mγ /∈ Z for any j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N − 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (251)  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2 (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [DF85a][Aom87][FW08]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  For all (α, β, γ) ∈ ˙UN,0,0[F] ∩ ˙U0,N,0[F] ∩ ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1,  ˙S0,N,0[F](α, β, γ) = (−1)N� N−1  �  m=0  sin(π(α + β + (N + m − 1)γ))  sin(π(β + mγ))  � ˙SN,0,0[F](α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (252)  For all (α, β, γ) ∈ ˙U0,N,0[F] ∩ ˙U0,0,N[F] ∩ ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0,  ˙S0,N,0[F](α, β, γ) = (−1)N� N−1  �  m=0  sin(π(α + β + (N + m − 1)γ))  sin(π(α + mγ))  � ˙S0,0,N[F](α, β, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (253)  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We prove the second claim, and the proof of the first is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that  (α, β, γ) ∈  N  �  n=0  ˙U0,N−n,n[F] ∩  N−1  �  n=0  ˙U1,N−1−n,n[F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (254)  We can apply eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (249) for ℓ = 1 and all pairs of m, n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N − 1} such that m + n = N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Combining the plus and minus cases of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (249) to eliminate the ˙S1,N−n−1,n[F] term,  1  2i  �  e+πiα 1 − e+2πi(m+1)γ  1 − e+2πiγ  − e−πiα 1 − e−2πi(m+1)γ  1 − e−2πiγ  � ˙S0,N−n,n[F](α, β, γ)  = − 1  2i  �  e+πi(α+β+2(N−n−1)γ) 1 − e+2πi(n+1)γ  1 − e+2πiγ  − e−πi(α+β+2(N−n−1)γ) 1 − e−2πi(n+1)γ  1 − e−2πiγ  �  × ˙S0,N−n−1,n+1[F](α, β, γ)  (255)  if γ /∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We calculate:  1  2i  �  e+πiα 1 − e+2πi(N−n)γ  1 − e+2πiγ  −e−πiα 1 − e−2πi(N−n)γ  1 − e−2πiγ  �  = 2s(γ)s(α + (N − n − 1)γ)s((N − n)γ)  1 − cos(2πγ)  (256)  and  1  2i  �  e+πi(α+β+2(N−n−1)γ) 1 − e+2πi(n+1)γ  1 − e+2πiγ  − e−πi(α+β+2(N−n−1)γ) 1 − e−2πi(n+1)γ  1 − e−2πiγ  �  =  2s(γ)  1 − cos(2πγ)s(α + β + (2N − n − 2)γ)s((n + 1)γ),  (257)  where s(t) = sin(πt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' So, for (α, β, γ) as above such that none of the trigonometric factors on the  right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (256) vanish,  ˙S0,N−n,n[F](α, β, γ) = −s(α + β + (2N − n − 2)γ)s((n + 1)γ)  s(α + (N − n − 1)γ)s((N − n)γ)  ˙S0,N−1−n,n+1[F](α, β, γ)  (258)  Applying this recursively for n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N − 1, we end up with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (253).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  43  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The sets in S1, S2, S3 in R3  α,β,γ ∩ {β = 1/5} in the case N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In summary, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (253) holds for a nonempty, open subset of (α, β, γ) ∈ ˙U0,N,0[F]∩ ˙U0,0,N[F]∩ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  By analyticity, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The function SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ) defined by  SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ) =  � N  �  j=1  Γ(2 + ¯dj + α + β + (N + j − 2)γ)  Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯  δ  j + β + (j − 1)γ)ϝj(γ)  �  SN[F](α, β, γ)  (259)  extends to an entire function SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] : C3  α,β,γ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We begin by defining the following open (and dense) subsets of C3:  QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 = {(α, β, γ) ∈ C3 : ¯δj + α + (j − 1)γ /∈ N for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}},  QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 = {(α, β, γ) ∈ C3 : ¯  δ  j + β + (j − 1)γ /∈ N for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}},  QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞ = {(α, β, γ) ∈ C3 : ¯dj + α + β + (N + j − 2)γ /∈ Z≤−2 for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}},  UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 = {(α, β, γ) ∈ C3 : δj + j(α + (j − 1)γ) /∈ Z≤−j for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}},  UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 = {(α, β, γ) ∈ C3 :  δ  j + j(β + (j − 1)γ) /∈ Z≤−j for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}},  UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞ = {(α, β, γ) ∈ C3 : −dj − j(1 + α + β + (N + j − 2)γ) /∈ Z≤0 for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}}  = {(α, β, γ) ∈ C3 : dj + j(1 + α + β + (N + j − 2)γ) /∈ N for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (260)  We write  SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ) = Υ0(α, β, γ)Υ1(α, β, γ)  ×  � N  �  j=1  Γ(δj + j(1 + α + (j − 1)γ))Γ(  δ  j + j(1 + β + (j − 1)γ))ϝj(γ)  �−1  SN[F](α, β, γ)  (261)  for  Υ0(α, β, γ) =  � N  �  j=1  Γ(δj + j(1 + α + (j − 1)γ))Γ(  δ  j + j(1 + β + (j − 1)γ))  Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯  δ  j + β + (j − 1)γ)  �  ,  (262)  Υ1(α, β, γ) =  � N  �  j=1  Γ(2 + ¯dj + α + β + (N + j − 2)γ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (263)  S1  4  2  0  4  2  0  2  4  QS2  4  2  0  2  4  4  2  0  2  4S3  4  2  0  2  4  4  2  0  2  444  ETHAN SUSSMAN  By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, the second line on the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (261) defines an entire function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Since Υ0 extends to an analytic function on UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 and Υ1 extends to an analytic function on  QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞, SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] extends to an analytic function on UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 ∩ QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  In ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ ˙U0,N,0 ∩ ˙U0,0,N, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2 gives  SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ) = (−1)NΥ2(α, β, γ)Υ3(α, β, γ)  ×  � N  �  j=1  Γ(−dj − j(1 + α + β + (N + j − 2)γ))Γ(  δ  j + j(1 + β + (j − 1)γ))ϝj(γ)  �−1 ˙S0,0,N[F](α, β, γ),  (264)  where  Υ2 =  � N  �  j=1  Γ(  δ  j + j(1 + β + (j − 1)γ))  s(α + (j − 1)γ)Γ(1 + ¯δj + α + (j − 1)γ)Γ(1 + ¯  δ  j + β + (j − 1)γ)  �  (265)  Υ3 =  � N  �  j=1  s(α + β + (N + j − 2)γ)Γ(2 + ¯dj + α + β + (N + j − 2)γ)  × Γ(−dj − j(1 + α + β + (N + j − 2)γ))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (266)  By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, the function on the second line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (264) extends to an entire function of  α, β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' On the other hand, Υ2 extends to an analytic function on QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, and Υ3 extends to  an analytic function on UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Combining these observations, SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] analytically continues to  QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Likewise, SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] extends analytically to UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞, using ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 in place of ON;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 and  the other part of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  So, SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α, β, γ) analytically continues to  U = (UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 ∩ QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞) ∪ (UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞) ∪ (QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 ∩ UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (267)  This is  U = C3���  �  H1∈S1,H2∈S2,H3∈S3  H1 ∩ H2 ∩ H3)  �  ,  (268)  where  S1 is the set of hyperplanes that are contained in the complement of one of UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0, UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞,  S2 is the set of hyperplanes that are contained in the complement of one of UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0, QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞,  and  S3 is the set of hyperplanes that are contained in the complement of one of QN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0, UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, UN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let  H = {H1 ∩ H2 ∩ H3 ̸= ∅ : H1 ∈ S1, H2 ∈ S2, H3 ∈ S3},  (269)  so that SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] defines an analytic function on U = C3\\∪H∈HH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Observe that every H ∈ H is  an affine subspace of C3 of complex codimension two or three (since S1 ∩ S2 ∩ S3 = ∅), and the  collection H is locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Hartog’s theorem therefore implies that SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] analytically continues to the entirety of C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  45  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The DF-symmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given γ+ ∈ C\\{0, 1} and α+, β+ ∈ C, let γ− = γ−1  + , α− = −γ−α+,  and β− = −γ−β+ as in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Given γ+ ̸= 0, 1 and F ∈ DFSym(N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' S, λ) for λ = γ−1  + (γ+ − 1), let ˙W DF,S  ℓ,m,n[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] denote the set  of (α+, β+) ∈ C2 such that  (α−, α+, β−, β+, γ−, −1, γ+) ∈ ˙W DF0,S  ℓ,m,n [F].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (270)  For (α+, β+) ∈ ˙W DF,S  ℓ,m,n[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], let  ˙IDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n[F](α+, β+, γ+) = IDF0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n [F](α−, α+, β−, β+, γ−, −1, γ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (271)  Then, as adumbrated by Dotsenko and Fateev:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any σ ∈ Sℓ,m,n,  ˙IDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n[F](α+, β+, γ+) = ˙IDF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n[F](α+, β+, γ+)σ  (272)  for all (α+, β+) ∈ ˙W DF,S  ℓ,m,n[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Since ˙W DF,S  ℓ,m,n[F] depends only on S through |S ∩ I1|, |S ∩ I2|, |S ∩ I3|,  ˙W DF,S  ℓ,m,n[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] = ˙W DF,S  ℓ,m,n[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+]σ,  (273)  so the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (272) is defined for any (α+, β+) ∈ ˙W DF,S  ℓ,m,n[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Since Sℓ,m,n is generated by transpositions τ of adjacent elements of I1, I2, I3, it suffices to  consider the case when σ is such a transposition, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For notational simplicity, we consider the case  when τ is a transposition of some j, j + 1 ∈ I2 and j ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The other cases are similar but involve  some notational changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Let ˙W DF,1,S  ℓ,m,n [F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ⊆ C6 denote the set of (α1,+, α2,+, α3,+, β1,+, β2,+, β3,+) ∈ C6 such that  (α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ−, −1, γ+) ∈ ˙W DF1,S  ℓ,m,n [F],  (274)  where α−,ν = −γ−α+,ν and β−,ν = −γ−β+,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' It suffices to prove that, for any σ ∈ Sℓ,m,n,  ˙IDF,1,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n [F](α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ+)  = ˙IDF,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n [F](α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ+)σ  (275)  for all (α1,+, α2,+, α3,+, β1,+, β2,+, β3,+) ∈ ˙W DF,1,S  ℓ,m,n [F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], where  ˙IDF,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='S  ℓ,m,n [F](α−,1, α−,2, α−,3, α+,1, α+,2, α+,3, β−,1, β−,2, β−,3, β+,1, β+,2, β+,3, γ+) =  ˙Iℓ,m,n[F](αDF,1, βDF,1, γDF),  (276)  where αDF,1, βDF,1 are defined as αDF1, βDF1, using α−,ν = −γ−α+,ν and β−,ν = −γ−β+,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  First observe that there exists a nonempty, open subset  O ⊂ ˙W DF,1,S  ℓ,m,n [F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+]  (277)  (containing an affine cone) such that (αDF,1, βDF,1, γDF) ∈ O{1,τ} whenever (α+,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , β+,3) ∈ O,  where O{1,τ} is defined as in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We can choose O such that ℜα±,2, ℜβ±,2 > 0 everywhere in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Since ˙W DF,S  ℓ,m,n[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] is connected, it suffices via analyticity to prove the result for  (α1,+, α2,+, α3,+, β1,+, β2,+, β3,+) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (278)  We write α± in place of α±,2 and β± in place of β±,2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  46  ETHAN SUSSMAN  We can apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6 for (α+,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , β+,3) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6, it suffices to check  that, whenever all of the zk’s besides zj and zj+1 are somewhere in the interior of the corresponding  contour in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (205),  �  Γ[0,1],+,j−ℓ  �  zα+  j  (1 − zj)β+zα−  j+1(1 − zj+1)β−�  �  1≤j0<k0≤N  {j0,k0}∩{j,j+1}̸=∅  (zk0 − zj0)2γj0,k0  �  F dzj dzj+1 = 0,  (279)  where the inner integral is taken over a small circle around zj+1, for each zj+1 ∈ Γ[0,1],+,j−ℓ\\{0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Since the integrand is holomorphic in zj in a punctured neighborhood of zj+1, we apply the  Cauchy residue theorem to deduce that the left-hand side is proportional to  �  Γ[0,1],+,j−ℓ  G0  ∂G  ∂zj  ���  zj=zj+1  dzj+1,  (280)  where  G0(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , zN) = zα−  j+1(1 − zj+1)β−�  �  j0∈([N]\\S)\\{j+1}  (zj+1 − zj0)2γ−��  �  j0∈S\\{j}  (zj+1 − zj0)−2�  , (281)  G(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , zN) = zα+  j  (1 − zj)β+�  �  j0∈([N]\\S)\\{j+1}  (zj − zj0)−2��  �  j0∈S\\{j}  (zj − zj0)2γ+�  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (282)  We are choosing branch cuts such that we do not encounter any as zj, zj+1 are integrated along  Γ[0,1],+,j−ℓ (except at the endpoints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Other than that, it is not important what the precise choice  of branch cuts are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The integrand in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (280) is computed to be  G0  ∂G  ∂zj  ���  zj=zj+1  =  � α+  zj+1  −  β+  1 − zj+1  + 2γ+  �  j0∈S\\{j}  1  zj+1 − zj0  − 2  �  j0∈([N]\\S)\\{j+1}  1  zj+1 − zj0  + ∂zjF  F  ���  zj=zj+1  �  H,  (283)  where H = G0G|zj=zj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' On the other hand,  ∂H  ∂zj+1  =  �α− + α+  zj+1  − β− + β+  1 − zj+1  +(2γ+−2)  �  j0∈S\\{j}  1  zj+1 − zj0  +(2γ−−2)  �  j0∈([N]\\S)\\{j+1}  1  zj+1 − zj0  + ∂zj+1(F|zj=zj+1)  F|zj=zj+1  �  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (284)  Since 1 − γ− = α−1  + (α− + α+) = β−1  + (β− + β+) = γ−1  + (γ+ − 1) = λ, and since F ∈ DFSym(N, S, λ),  G0  ∂G  ∂zj  ���  zj=zj+1  = 1  λ  ∂H  ∂zj+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (285)  Consequently,  �  Γ[0,1],+,j−ℓ  G0  ∂G  ∂zj  ���  zj=zj+1  dzj+1 = 1  λ  �  Γ[0,1],+,j−ℓ  ∂H  ∂zj+1  dzj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (286)  The right-hand side is proportional to  �  p  ∂H  ∂zj+1  dzj+1  (287)  if α+, β+ /∈ Z, where p is a Pochhammer contour in C2\\{0, 1} staying sufficiently close to Γ[0,1],+,j−ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Lifting to a cover of a neighborhood of Γ[0,1],+,j−ℓ on which H lifts to a single-valued analytic  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  47  function, we can conclude (using analyticity) that the integral in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (287) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' By analyticity,  we can remove the nonintegrality constraint on α+, β+ to conclude that  �  Γ[0,1],+,j−ℓ  ∂H  ∂zj+1  dzj+1 = 0  (288)  for all (α+,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , β+,3) ∈ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='5 (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' [DF85a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given the setup above, for arbitrary S:  For all (α+, β+) ∈ ˙W DF,S  N,0,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ∩ ˙W DF,S  0,N,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] such that β± + m±γ± /∈ Z for any m± ∈  {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N± − 1}  ˙IDF,S  0,N,0[F](α+, β+, γ+) = (−1)N ˙IDF,S  N,0,0[F](α+, β+, γ+)  � N+−1  �  m+=0  sin(π(α+ + β+ + (N+ + m+ − 1)γ+))  sin(π(β+ + m+γ+))  �� N−−1  �  m−=0  sin(π(α− + β− + (N− + m− − 1)γ−))  sin(π(β− + m−γ−))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (289)  For all (α+, β+) ∈ ˙W DF,S  0,N,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ∩ ˙W DF,S  0,0,N[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] such that α± + m±γ± /∈ Z for any m± ∈  {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N± − 1},  ˙IDF,S  0,N,0[F](α+, β+, γ+) = (−1)N ˙IDF,S  0,0,N[F](α+, β+, γ+)  � N+−1  �  m+=0  sin(π(α+ + β+ + (N+ + m+ − 1)γ+))  sin(π(α+ + m+γ+))  �� N−−1  �  m−=0  sin(π(α− + β− + (N− + m− − 1)γ−))  sin(π(α− + m−γ−))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (290)  For S = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N+}, we also have:  For all (α+, β+) ∈  ˙W DF,S  N+,N−,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ∩ ˙W DF,S  0,N,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] such that β+ + m+γ+ /∈ Z for any  m+ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N+ − 1}  ˙IDF,S  0,N,0[F](α+, β+, γ+) = (−1)N+ ˙IDF,S  N+,N−,0[F](α+, β+, γ+)  � N+−1  �  m+=0  sin(π(α+ + β+ + (N+ + m+ − 1)γ+))  sin(π(β+ + m+γ+))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (291)  For all (α+, β+) ∈  ˙W DF,S  0,N,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ∩ ˙W DF,S  0,N+,N−[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] such that α− + m−γ− /∈ Z for any  m− ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N− − 1},  ˙IDF,S  0,N,0[F](α+, β+, γ+) = (−1)N− ˙IDF,S  0,N+,N−[F](α+, β+, γ+)  � N−−1  �  m−=0  sin(π(α− + β− + (N− + m− − 1)γ−))  sin(π(α− + m−γ−))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (292)  Similarly, for S = {N − N+ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, we have:  For all (α+, β+) ∈  ˙W DF,S  N−,N+,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ∩ ˙W DF,S  0,N,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] such that β− + m−γ− /∈ Z for any  m− ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N− − 1}  ˙IDF,S  0,N,0[F](α+, β+, γ+) = (−1)N− ˙IDF,S  N−,N+,0[F](α+, β+, γ+)  � N−−1  �  m−=0  sin(π(α− + β− + (N− + m− − 1)γ−))  sin(π(β− + m−γ−))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (293)  48  ETHAN SUSSMAN  For all (α+, β+) ∈  ˙W DF,S  0,N,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ∩ ˙W DF,S  0,N−,N+[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] such that α+ + m+γ+ /∈ Z for any  m+ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N+ − 1},  ˙IDF,S  0,N,0[F](α+, β+, γ+) = (−1)N+ ˙IDF,S  0,N−,N+[F](α+, β+, γ+)  � N+−1  �  m+=0  sin(π(α+ + β+ + (N+ + m+ − 1)γ+))  sin(π(α+ + m+γ+))  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (294)  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Follows from a repeated application of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='10, as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The only difference with the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2 is that we appeal to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4 to show  (instead of it being an automatic consequence of symmetry) that  ˙Iℓ,m,n[F](αDF,S, βDF,S, γDF,S) = ˙Iℓ,m,n[F](αDF,S, βDF,S, γDF,S)σℓ  (295)  ˙Iℓ−1,m+1,n[F](αDF,S, βDF,S, γDF,S)σℓ = ˙Iℓ−1,m+1,n[F](αDF,S, βDF,S, γDF,S)σℓ+m  (296)  ˙Iℓ−1,m,n+1[F](αDF,S, βDF,S, γDF,S)σℓ+m = ˙Iℓ−1,m,n+1[F](αDF,S, βDF,S, γDF,S)σN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (297)  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For γ+ ∈ C\\{0, 1}, the functions IDF,S  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α+, β+, γ+) defined by  ˙IDF,S  N  [F](α+, β+, γ+) =  � �  ±  N±  �  j=1  sin(π(α± + β± + (N± + j − 2)γ±))  sin(π(α± + (j − 1)γ±)) sin(π(β± + (j − 1)γ±))  �  × IDF,S  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F](α+, β+, γ+)  (298)  extend to entire functions IDF,S  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[F] : C2  α+,β+ → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■□  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The proof is very similar to that used to prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Using the previous proposition  with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4, it suffices to note that the union of all nine of the sets  ˙W DF,S−  N,0,0 [F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], ˙W DF,S−  0,N,0 [F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], ˙W DF,S−  0,0,N [F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], ˙W DF,S−  N−,N+,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], ˙W DF,S−  N−,0,N+[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+],  ˙W DF,S−  0,N−,N+[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], ˙W DF,S+  N+,N−,0[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], ˙W DF,S+  N+,0,N−[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+], ˙W DF,S+  0,N+,N−[F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' γ+] ⊂ C2  α+,β+,  (299)  where S+ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N+} and S− = S∁, is the complement of locally finite set of points, and then the  result follows via Hartog’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The N = 2 case  We now consider the N = 2 case in some detail, beginning with the formula  S2(α, β, γ) = Γ(1 + α1)Γ(1 + β2)Γ(2 + 2γ1,2 + α1 + α2)Γ(1 + 2γ1,2)  Γ(2 + α1 + 2γ1,2)Γ(3 + α1 + α2 + β2 + 2γ1,2)  3F2(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1),  (300)  a = (a1, a2, a3) = (1 + α1, −β1, 2 + 2γ1,2 + α1 + α2) and b = (b1, b2) = (2 + α1 + 2γ1,2, 3 + α1 + α2 +  β2 + 2γ1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is asymmetric in the role of the α’s and β’s, but there is an analogous formula  with the α’s and β’s on the right-hand side switched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Some of the singularities of S2 are manifest in  this formula, but others are hidden in the 3F2 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Consider now the Dotsenko–Fateev-like integral  IDF0  2  (α1, α2, β1, β2, γ) =  � 1  0  � 1  0  xα1  1 xα2  2 (1 − x1)β1(1 − x2)β2(x2 − x1 + i0)2γ dx1 dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (301)  By the previous proposition:  THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  49  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ˙IDF0  2  (α1, α2, β1, β2, γ) = Γ(2 + 2γ + α1 + α2)Γ(1 + 2γ)  �  Γ(1 + α1)Γ(1 + β2)3F2(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1)  Γ(2 + α1 + 2γ)Γ(3 + α1 + α2 + β2 + 2γ)  + e2πiγ  Γ(1 + α2)Γ(1 + β1)3F2(a′, b′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1)  Γ(2 + α2 + 2γ)Γ(3 + α1 + α2 + β1 + 2γ)  �  ,  (302)  where a′ = (a′  1, a′  2, a′  3) = (1 + α2, −β2, 2 + 2γ + α1 + α2) and b′ = (b′  1, b′  2) = (2 + α2 + 2γ, 3 + α1 +  α2 + β1 + 2γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■□  The formula eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (302) is not suitable for analytic continuation to γ = −1, for which we instead  use the method described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' That yields  ˙I2(α1, α2, β1, β2, γ) = Γ(1 + α1)Γ(1 + β1)  Γ(2 + α1 + β1)  �  Γ  �  zα2+2γ(1 − z)β2  × 2F1  �  − 2γ, 1 + α1, 2 + α1 + β1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1  z  ��  dz,  (303)  where the à is a trapezoidal contour in the upper-half of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This formula can be  used to numerically compute ˙IDF0  2  (α1, α2, β1, β2, γ) for γ with large negative real part, as long as  α1, α2, β1, β2 have sufficiently large positive real part relative to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We illustrate the method of proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 with the computation of the residues associated  with α− + α+ + 2γ ∈ Z−2−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Introducing coordinates ϱ = x2 and λ = x1/x2,  S2(α1, α2, β1, β2, γ) =  � 1  0  � x2  0  xα1  1 xα2  2 (1 − x1)β1(1 − x2)β2(x2 − x1)2γ dx1 dx2  =  � 1  0  � 1  0  ϱ1+α1+α2+2γλα1(1 − λϱ)β1(1 − ϱ)β2(1 − λ)2γ dλ dϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (304)  Expanding (1 − λϱ)β1(1 − ϱ)β2 in Taylor series around ϱ = 0, we have  (1 − λϱ)β1(1 − ϱ)β2 =  ∞  �  k=0  k  �  κ,κ,κ′=0  ak,κ,κ,κ′βκ  1 βκ′  2 λκϱk  (305)  for some ak,κ,κ,κ′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then, computing the outer integral term by term and using the formula for  the β-function for the inner integral,  S2(α1, α2, β1, β2, γ) ∼  ∞  �  k=0  1  2 + α1 + α2 + 2γ + k  k  �  κ,κ,κ′=0  ak,κ,κ,κ′βκ  1 βκ′  2  Γ(1 + α1 + κ)Γ(1 + 2γ)  Γ(2 + α1 + 2γ + κ)  (306)  where the ‘∼’ means modulo an error which is not singular at (all but a positive codimension subset  of) the hyperplane under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The right-hand side of this has an apparent pole whenever  α1 + α2 + 2γ ∈ Z≤−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  We now examine some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Fix d ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' First consider  S2[xd](α, β, γ) = S2(α, α+d, β, β, γ) =  � 1  0  � x2  0  xα  1 xd+α  2  (1−x1)β(1−x2)β(x2−x1)2γ dx1 dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (307)  By Equation (300),  S2[xd](α, β, γ) = Γ(1 + α)Γ(1 + β)Γ(1 + 2γ)Γ(2 + 2α + 2γ + d)  Γ(2 + α + 2γ)Γ(3 + 2α + β + 2γ + d)  3F2(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1),  (308)  where now a = (a1, a2, a3) = (1+α, −β, 2+2α+2γ+d) and b = (b1, b2) = (2+α+2γ, 3+2α+β+2γ+d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  A numerically generated plot of the absolute value of the right-hand side is given in the case d = 2  50  ETHAN SUSSMAN  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The absolute values of the right-hand sides of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (308) (left) and  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (311) (right) plotted against α, for β = 1/2 and γ = 1/3 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The singularities  predicted in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (309), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (312) have been drawn as dotted vertical lines, those  associated with {α ∈ Z} in blue and those associated with {α + γ ∈ 2−1Z} in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  It appears that all of the poles that could be present are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The apparent  zeroes of S2[F](α, 1/2, 1/3) in the depicted range of α have been marked with dotted  black lines and numerically computed to be ≈ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='48503 for F = x2 and ≈ −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='06833,  −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='57013, and −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='08562 for F = y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Applying Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, in which α1,∗ = α, α2,∗ = d+2α +2γ, β1,∗ = β, β2,∗ = 2β +2γ,  and γ1,2,∗ = 2γ, we deduce that S2[xd](α, β, γ) extends to an analytic function on  C3  α,β,γ  ��  {α ∈ Z≤−1}∪{α+γ ∈ 2−1Z≤−2−d}∪{β ∈ Z≤−1}∪{β +γ ∈ 2−1Z≤−2}∪{γ ∈ 2−1Z≤−1}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (309)  In the d = 2 case, this can be seen in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  On the other hand, consider  S2[yd](α, β, γ) = S2(α+d, α, β, β, γ) =  � 1  0  � x2  0  xd+α  1  xα  2 (1−x1)β(1−x2)β(x2 −x1)2γ dx1 dx2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (310)  By Equation (20),  S2[yd](α, β, γ) = Γ(1 + α + d)Γ(1 + β)Γ(1 + 2γ)Γ(2 + 2α + 2γ + d)  Γ(2 + α + 2γ + d)Γ(3 + 2α + β + 2γ + d)  3F2(a′, b′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1),  (311)  where a′ = (a′  1, a2, a3) = (1 + d + α, −β, 2 + d + 2α + 2γ) and b′ = (b′  1, b2) = (2 + d + α + 2γ, 3 + d +  2α + β + 2γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We again apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, but now α1,∗ = d + α, α2,∗ = d + 2α + 2γ, in order to  deduce that S2[yd](α, β, γ) extends analytically to  C3  α,β,γ  ��  {α ∈ Z≤−1−d}∪{α+γ ∈ 2−1Z≤−2−d}∪{β ∈ Z≤−1}∪{β+γ ∈ 2−1Z≤−2}∪{γ ∈ 2−1Z≤−1}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (312)  See Figure 13 for a numerically generated plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  If we instead pick F(x, y) = xd + yd, which is in some sense the symmetrization of the previous  two examples, the situation looks very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Combining the formulas above yields  S2[xd + yd](α, β, γ) = (S2[xd] + S2[yd])(α, β, γ) = Γ(1 + α)Γ(1 + β)Γ(1 + 2γ)Γ(2 + 2α + 2γ + d)  Γ(2 + α + 2γ)Γ(3 + 2α + β + 2γ + d)  ×  �  3F2(a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1) + Γ(1 + α + d)Γ(2 + α + 2γ)  Γ(1 + α)Γ(2 + α + 2γ + d) · 3F2(a′, b′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (313)  [S2[α-](α, 1/2, 1/3)]  VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Qα  40  30 20  10  4  0S2[y21(α, 1/2, 1/3)  VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' α  40  30  20  10  2  4  αTHE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  51  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The absolute value of the right side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (313), plotted against  α ∈ (0, −6) (left) and α ∈ (−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1, −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='75) (right), for β = 1/2 and γ = 1/3 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Singularities associated with {α ∈ Z} are marked with blue lines and those with  {α + γ ∈ Z} with red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The zeroes associated with {α + β + γ ∈ Z} are marked in  green and those with {α + β + 2γ ∈ Z} in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The location of the second plot is  marked (roughly) as an inset on the left plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  On the other hand, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2, we know that S2[xd + yd] extends analytically to  C3  α,β,γ  ��  {α ∈ Z≤−1−¯δ1} ∪ {α + γ ∈ Z≤−2−¯δ2} ∪ {β ∈ Z≤−1−¯  δ  1} ∪ {β + γ ∈ Z≤−1−¯  δ  2}  ∪ {γ ∈ 2−1Z≤−1, γ /∈ Z}  �  ,  (314)  where ¯δ1, ¯  δ  1, ¯δ2, ¯  δ  2 are as in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In this example, ¯δ1, ¯  δ  1, ¯  δ  2 = 0, ¯δ2 = ⌈d/2⌉, ¯d2 = d, and  ¯d1 = ⌊d/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' See Figure 14, in which d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In the sum eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (313), the poles of the individual  summands at such 2α + 2γ ∈ Z≤−2−d ∩ (2Z + 1) (which we can see from Figure 13 exist) must  cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  By eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (306), the residue of S2[xd + yd](α, β, γ) at 2α + 2γ = −2 − d − k for k ∈ N is proportional  to  k  �  κ,κ,κ′=0  ak,κ,κ,κ′βκ+κ′�  Γ(1 + α + κ)  Γ(2 + α + 2γ + κ) +  Γ(1 + α + κ + d)  Γ(2 + α + 2γ + κ + d)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (315)  Since this has to vanish whenever −2 − d − k is odd and (α, γ) ∈ {2α + 2γ = −2 − d − k} ⊂ C2  α,γ is  such that the function in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (315) is well-defined, we conclude that, for such k and (α, γ),  k  �  κ=0  ��  �  κ′+κ′′=κ  ak,κ,κ′,κ′′  ��  Γ(1 + α + κ)  Γ(2 + α + 2γ + κ) +  Γ(1 + α + κ + d)  Γ(2 + α + 2γ + κ + d)  ��  = 0  (316)  for each κ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  A direct algebraic proof of this fact is not entirely trivial, but it is  straightforward to check case-by-case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The function S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[xd + yd](α, β, γ) is plotted as a function of α ∈ C in Figure 15, still in the  case d = 2 – for fixed β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' As expected, it appears to have no singularities, in accordance with  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Unlike in the case F = 1, where Selberg’s formula shows that S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[1](α, β, γ) is  constant, S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[xd + yd] is nonconstant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  [S2[α² +y²](α, 1/2, 1/3)  VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' α  12  10  8  6  2  (Inset)  2  0S2α- + y-(α, 1/2, 1/3)/ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' α (Inset)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='00  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='6 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='0 -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='852  ETHAN SUSSMAN  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The function S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[x2 + y2](α, 1/2, 1/3) defined by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (29), plotted as  a function of α ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Consider now ˙IDF  2  (α+, β+) = ˙IDF0  2  (α+, −α+, β+, −β+, 1), which is a DF-symmetric integral with  γ± = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This is given concretely by  ˙IDF  2  (α+, β+) = Γ(1 + α+)Γ(1 + β+)  Γ(2 + α+ + β+)  �  Γ  �  zα++2γ(1 − z)β+  × 2F1  �  − 2γ, 1 + α+, 2 + α+ + β+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 1  z  ��  dz  (317)  when the real parts of α+, β+ are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The Dotsenko–Fateev claim, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (48), is, up to  a sign, that  ˙IDF  2  (α+, β+) = −Γ(1 + α+)Γ(1 + β+)Γ(α+)Γ(β+)  2Γ(1 + α+ + β+)Γ(α+ + β+)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  (318)  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Explicit coordinates on [0, 1)N  tb  In this appendix, we discuss the total boundary blowup [0, 1)N  tb, the mwc constructed by blowing  up all of the facets of [0, 1)N, starting with those of the lowest dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For each nonempty subset  S ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}, let FS denote the face of [0, 1)N  tb corresponding to the facet {j ∈ S ⇒ xj = 0}  of [0, 1)N  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Tracing through the construction of the total boundary blowup, we have the following  explicit choice of boundary defining functions (which are different, if N ≥ 3, from the recursively  defined boundary defining functions discussed in the introduction to §2) of the various faces of  [0, 1)N  tb:  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The function  xFS = xFS,N =  �  S0⊇S  � �  j∈S0  xj  �(−1)|S|−|S0|  (319)  serves as a bdf of FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Suppose that I is a (possibly empty) set of nested nonempty subsets of  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Then,  fI = {xFS = 0 for all S ∈ I}  (320)  is a codimension |I| facet of [0, 1)N  tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' This defines a bijective correspondence between the set of nested  nonempty subsets of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N} and the set of facets of [0, 1)N  tb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' If p lies in the interior of fI, then,  letting σ ∈ SN denote any permutation consistent with I,  ϱ = xσ(1), ˆxσ(2) = xσ(2)/xσ(1), · · · , ˆxσ(N) = xσ(N)/xσ(N−1)  (321)  give a local set of coordinates near p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■□  [S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='Reg[α?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' +y°](α, 1/2, 1/3)]  VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Qα  15  10  2  5  1  0  6  0  4  1  2  2  0THE SINGULARITIES OF SELBERG- AND DOTSENKO–FATEEV-LIKE INTEGRALS  53  Here, we say that σ ∈ SN is consistent with I if, whenever j < k, σ(j) ∈ S ∈ I ⇒ σ(k) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We  can cover [0, 1)N  tb with the N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' coordinate charts whose restrictions to the interior are of the form  {0 < xσ(N) < 2xσ(N−1) < · · · < 2Nxσ(1) < 2N} ∩ (0, 1)N  → [0, 1)xσ(1) × [0, 2)xσ(2)/xσ(1) × · · · × [0, 2)xσ(N)/xσ(N−1),  (322)  for σ ∈ SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The preceding proposition is used to prove:  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' For any M ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , N − 1} and nonempty Q ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' , M},  xFQ,M =  �  Q0⊆{M+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N}  xFQ∪Q0,N  (323)  in (0, 1)N  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  ■  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A factor of �  j∈Q∪Q0 xj appears on the right-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (323) to the power  �  Q1⊆Q0  (−1)|Q0|,  (324)  which is, by the binomial theorem, +1 if Q0 = ∅ and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, �  Q0⊆{M+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=',N} xFQ∪Q0,N =  �  j∈Q xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  □  The full proof of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1 is somewhat incidental to the rest of the paper, so we merely  illustrate the argument (which generalizes to the N ≥ 3 case, and applies in an even simpler form  to the N = 2 case) in the case N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The total boundary blowup [0, 1)N  tb is defined as  [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, z = 0}, {x, y = 0}],  (325)  where the first blowup is that of {x, y, z = 0} and must be performed first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The other three blowups  can be performed in any order, and each order yields a canonically diffeomorphic mwc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The first  blowup, yielding [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z = 0}], is  y  z  x  y/(x + y + z)  z/(x + y + z)  x/(x + y + z)  x + y + z  where we are marking the faces with boundary defining functions (using the Cartesian coordinates  x, y, z in place of x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The choice of bdfs on the blowup are in accordance with the prescription  in the introduction of §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  The next blowup, yielding  [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {y, z = 0}],  (326)  is  54  ETHAN SUSSMAN  y/(x + y + z)  z/(x + y + z)  x/(x + y + z)  x + y + z  y/(y + z)  z/(y + z)  x/(x + y + z)  x + y + z  (y + z)/(x + y + z)  Again, the choices of bdfs are in accordance with §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Next, we blow up the facet of [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {y, z = 0}] corresponding to the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Because the previous blowup was located away from the facet being blown up now, we can use the  sum  x  x + y + z +  z  x + y + z =  x + z  x + y + z  (327)  of the bdfs of the adjacent faces in [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z = 0}] as a bdf of the front face of the current  blowup rather than  x  x + y + z +  z  y + z = xy + 2xz + yz + z2  (y + z)(x + y + z) ,  (328)  which would be the prescription in the introduction §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' The choices in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (327), eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (328) are  equivalent, in the sense that their quotient is a smooth, nonvanishing function on [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z =  0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, z = 0}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Given that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (327) serves as a bdf of the front face, the quotient  x/(x + y + z)  (x + z)/(x + y + z) =  x  x + z  (329)  serves as a bdf in [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, z = 0}] for the lift of the yz-plane, and  z/(y + z)  (x + z)/(x + y + z) = z(x + y + z)  (x + z)(y + z)  (330)  serves as a bdf for the lift of the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' In summary, the third blowup is  y/(y + z)  z/(y + z)  x/(x + y + z)  x + y + z  (y + z)/(x + y + z)  y/(y + z)  z(x + y + z)(x + z)−1(y + z)−1  x/(x + z)  x + y + z  (y + z)/(x + y + z)  (x + z)/(x + y + z)  The final blowup, yielding [0, 1)3  tb = [[0, 1)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {y, z = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' {x, z = 0}, {x, y = 0}], is  similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' We use (x + y)/(x + y + z) as a bdf of the blowup of the face corresponding to the z-axis,  and we can then use  x/(x + z)  (x + y)/(x + y + z) = x(x + y + z)  (x + y)(x + z)  (331)  y/(y + z)  (x + y)/(x + y + z) = y(x + y + z)  (x + y)(y + z)  (332)  REFERENCES  55  as bdfs of the faces corresponding to the yz- and xz-planes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Thus, we up with  y(x + y + z)(x + y)−1(y + z)−1  z(x + y + z)(x + z)−1(y + z)−1  x(x + y + z)(x + y)−1(x + z)−1  x + y + z  (y + z)/(x + y + z)  (x + z)/(x + y + z)  (x + y)/(x + y + z)  as our final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  References  [AK11]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Aomoto and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Kita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Theory of hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Springer Monographs in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' With  an appendix by Toshitake Kohno, Translated from the Japanese by Kenji Iohara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Springer-Verlag, Tokyo,  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1007/978-4-431-53938-4 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  [Aom87]  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Aomoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' “On the complex Selberg integral”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Oxford Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (2) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='152 (1987), 385–399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1093/qmath/38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='385 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Japan Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Springer Theses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content='  [MS08]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Scattering configuration spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' arXiv: 0808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='2022 [DG] (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content='  [MSS02]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Shnider, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Stasheff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Operads in algebra, topology and physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Mathematical Surveys  and Monographs 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' American Mathematical Society, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Intersection numbers of twisted cycles and the correlation functions of the  conformal field theory II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content='  [MY03]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' “Intersection numbers of twisted cycles and the correlation functions of the conformal field theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Graduate Texts in Contemporary  Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' “Permutohedra, associahedra, and beyond”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' IMRN 6 (2009), 1026–1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content='1093/imrn/rnn153 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content='  [Sel44]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' “Remarks on a multiple integral”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Norsk Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Tidsskr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 26 (1944), 71–78 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content='  [Sta63]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+page_content='  ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 108 (1963), 293–312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1090/s0002-9947-1963-0158400-5 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  [Tam62]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Tamari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' “The algebra of bracketings and their enumeration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Nieuw Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Wisk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' (3) 10 (1962), 131–146  (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  [Var95]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Varchenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Multidimensional hypergeometric functions and representation theory of Lie algebras and  quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Advanced Series in Mathematical Physics 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' World Scientific Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=', 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1142/2467 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' on pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 2, 5, 10, 29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  [Yos03]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Yoshida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' “A geometric interpretation of the Selberg integral”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' Modern Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' A 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='24  (2003), 4343–4359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='1142/S0217751X03015192 (cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='  Email address: ethanws@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
+page_content='edu  Department of Mathematics, Massachusetts Institute of Technology, Massachusetts 02139-4307,  USA' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE2T4oBgHgl3EQfOgbi/content/2301.03750v1.pdf'}
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+Transport Model for the Propagation of Partially-Coherent,
+Polarization-Gradient Vector Beams
+J. M. Nichols and D. V. Nickel
+Naval Research Laboratory
+4555 Overlook Ave.
+SW.
+Washington D.C. 20375
+F. Bucholtz
+Jacobs Technology, Inc.
+2551 Dulles View Drive
+Herndon, VA 20171
+G. Rohde
+University of Virginia
+Dept.
+of Electrical and Computer Engineering
+Dept.
+of Biomedical Engineering
+415 Lane Rd.
+Charlottesville, VA
+In a recent work [20], we predicted and experimentally validated a new physical mechanism for
+altering the propagation path of a monochromatic beam. Specifically, we showed that by properly
+tailoring the spatial distribution of the linear state of polarization transverse to the direction of
+propagation, the beam followed a curved trajectory in free space. Here we extend the model to the
+partially coherent, polychromatic case by redefining the beam amplitude, phase, and polarization
+angle as appropriate statistical quantities.
+In particular, we propose an entirely new definition
+of linear polarization gradient as an average over the third generalized Stokes parameter in the
+spatial frequency domain. In the new model, the beam curvature matches that of our previous work
+in the fully coherent case, but is predicted to gradually vanish as the beam loses coherence and
+becomes depolarized. The model also clearly predicts, however, that there does not exist a natural
+mechanism in free space or due to atmospheric turbulence that will cause significant depolarization.
+Simulated beam trajectories are shown for varying levels of initial partial coherence and for different
+polarization profiles. A new class of non-diffracting beams is also suggested by way of example.
+Lastly, as a byproduct of the derivation we generalize a previously known result concerning the
+free-space propagation of generalized Stokes parameters [13] by showing the result also holds for
+propagation in inhomogeneous media.
+I.
+INTRODUCTION
+Recently we demonstrated that through proper choice of spatial distribution of linear polarization
+angle, a coherent, monochromatic beam follows a curved trajectory in free space [20]. The effect is similar
+in magnitude to that of Airy beam bending (both scale as k−2
+0
+where k0 is the free-space wavenumber).
+However, in our approach it is the centroid of the beam that experiences a transverse acceleration
+as opposed to the Airy beam case where the main intensity lobe transversely accelerates, but where
+the centroid of the beam does not.
+The physics of the vector beam bending were predicted via a
+transport model, whereby the parameters defining the optical field (intensity and phase) were shown to
+be governed by the transport of intensity equation (TIE) and an eikonal equation, expressing conservation
+of transverse linear momentum. The model was re-formulated in Lagrangian coordinates in order to solve
+for the transverse beam location which, under appropriate choice of polarization gradient, was shown to
+bend in the transverse plane. These predictions were subsequently verified in experiment [20].
+In this work we extend the model to the more general case of partially coherent light, showing that with
+appropriate definitions of optical phase, amplitude, and polarization gradient the model still predicts a
+bending effect. For perfectly coherent light, the governing equations reduce to the previously derived
+coherent case, while for incoherent light, the effect disappears. The resulting model also predicts that the
+state of linear polarization (including the degree of polarization) will remain unchanged on propagation.
+Collectively, these model predictions represent the main contribution of this work.
+A second, but related contribution is to provide a new definition of polarization angle gradient that
+is appropriate for modeling partially coherent vector beams. Our definition is based on the normalized
+integral of the third generalized Stokes parameter and incorporates the degree of polarization directly.
+While this definition falls naturally from the vector transport equations, it is also consistent with recent
+arXiv:2301.03994v1  [physics.optics]  10 Jan 2023
+
+work on scalar, partially coherent beams where a similar definition of generalized phase was proposed
+[35]. In what follows, we show that this scalar definition is also easily extended to the vector case and is
+also a natural consequence of the transport model.
+Finally, in the process of deriving this result, we generalize a previously known result from Korotkova
+[13]. In that work, the author showed that the transverse integral of the generalized Stokes parameters
+were conserved under free-space propagation. We confirm this result using an entirely different approach
+and extend it to the case of propagation in an inhomogeneous medium.
+Our approach is as follows.
+1. We begin with the wave equation in the general case of a weakly-inhomogenous medium.
+We
+make the paraxial assumption via the slowly-varying envelope approximation to obtain Helmholtz
+equations for each field component. In the instance where the medium warrants a probabilistic
+description, we additionally invoke the Markov Approximation [30].
+2. To accommodate the probabilistic nature of the partially coherent problem, the fields are
+represented in the coherency matrix formalism where each matrix element is an expectation of
+a product of finite-time Fourier transforms [24] of field components at two different transverse
+spatial locations. Importantly, we retain the dependence on spatially-separated points throughout
+the calculation in the form of Wigner Transforms with respect to the transverse spatial coordinate.
+This information is vital in our model as the state of polarization is not uniform in the transverse
+plane.
+3. We obtain a vector transport equation for the generalized Stokes parameters which is the partially
+coherent, vector counterpart to the transport of intensity equations obtained in our earlier paper
+[20]. It is entirely reasonable to expect that, in the case of partial polarization, individual com-
+ponents of the Stokes 4-vector must obey continuity equations while, in the fully-polarized case,
+continuity of the intensity is sufficient to characterize beam propagation.
+4. We extend the method of moments developed for the scalar problem [7, 18] to the vector case,
+leading to the coupled transport equations describing the evolution of expected amplitude, phase,
+and polarization angle. Importantly, the model reduces to our monochromatic, coherent model in
+those respective limits.
+The resulting model extends our prior coherent model variables (phase, polarization angle, and intensity)
+to the partially coherent case by taking appropriate averages over the Wigner spatial frequency and with
+these new definitions, predict the expected beam path.
+II.
+TRANSPORT MODELING OF PARTIALLY COHERENT, POLARIZED LIGHT
+A.
+Spectral Coherency Representation
+We start with the vector wave equation for electric-field vector E(x, t)
+∇2E(x, t) − ϵ(x)
+c2
+0
+∂2E(x, t)
+∂t2
+= 0
+(1)
+which presumes an inhomogeneous, isotropic medium characterized by dimensionless dielectric constant
+ϵ(x) > 1, where x denotes position in three-dimensional Cartesian space, where c0 = 1/√ϵ0µ0 is the
+speed of light in vacuum, and where ϵ0 and µ0 are the permittivity and permeability, respectively, of free
+space. We assume the inhomogeneity is weak and define
+ϵ1(x) = ϵ(x) − ⟨ϵ⟩
+⟨ϵ⟩
+<< 1
+(2)
+where ⟨ϵ⟩ is the average value of ϵ(x) taken over the entire region of interest.
+We assume the wave propagates in the +z-direction and we model the electric field vector as a sta-
+tionary random process, existing for times −T ≤ t ≤ T. Since (1) is linear in E(x, t), we may consider a
+superposition of monochromatic, transverse-wave solutions
+E(x, t) = 1
+2π
+� ∞
+−∞
+{EX(x, ω)T , EY (x, ω)T , 0} e−ik0zeiωtdω,
+−T ≤ t ≤ T
+(3)
+2
+
+represented here by the Fourier integral. Phase accumulation in the direction of propagation is captured
+by the e−ik0z term with wavenumber k0 = (ω/c0) ⟨ϵ⟩1/2 = (2π/λ) ⟨ϵ⟩1/2 at free-space wavelength λ. The
+quantities {EX(x, ω)T , EY (x, ω)T , 0}e−ik0z are therefore the associated Fourier amplitudes at frequency
+ω and possess units of electric field per Hz. Given the above construction, these are given by
+e−ik0zEX,Y (x, ω)T =
+� T
+−T
+EX,Y (x, t)e−iωtdt
+(4)
+as discussed in [24] (sec. 4.7). The explicit dependence of the Fourier amplitudes on T will be retained
+for the moment. The electric field and its Fourier amplitudes are distinguished by their arguments (t
+and ω resp.).
+Substitute (3) into (1) and make the slowly-varying envelope approximation in which it is assumed
+that changes in the amplitude of the field over z−distances of a wavelength are small compared to the
+amplitude itself.
+Then |∂zz(·)| ≪ |k0∂z(·)| and the net result is that spatial gradients in x become
+gradients only in the transverse plane leading to the parabolic equations
+�
+−i2k0∂z + ∇2
+X + k2
+0ϵ1(⃗x, z)
+�
+{EX(⃗x, z, ω)T , EY (⃗x, z, ω)T } = 0
+�
+i2k0∂z + ∇2
+X + k2
+0ϵ1(⃗x, z)
+�
+{E∗
+X(⃗x, z, ω)T , E∗
+Y (⃗x, z, ω)T } = 0
+(5)
+which must be satisfied by each vector component of Fourier amplitudes and their complex conjugates
+(denoted with ∗). Here we have denoted the position in the transverse plane as the vector ⃗x ≡ {x, y} to
+distinguish it from the propagation direction z (x ≡ {⃗x, z}), and have denoted the transverse gradient
+operator ∇X(·) ≡ {∂x(·), ∂y(·)}.
+Note that (5) comprises four separate equations in, respectively, EX, EY , E∗
+X, E∗
+Y which we will call (a-
+d). Now perform the following operations. Multiply (a) and (b) evaluated at ⃗x1 by E∗
+X(⃗x2, z, ω). Multiply
+(a) and (b) evaluated at ⃗x1 by E∗
+Y (⃗x2, z, ω).
+Multiply (c) and (d) evaluated at ⃗x2 by EX(⃗x1, z, ω).
+Multiply (c) and (d) evaluated at ⃗x2 by EY (⃗x1, z, ω). Note that in each of the above operations all four
+multiplications are performed from the right. The results, in order, are eight equations which we denote
+(i)-(viii).
+If we then subtract (v) from (i), (vii) from (ii), (vi) from (iii), and (viii) from (iv) we find (taking into
+account the chain rule for the derivatives in z):
+�
+i2k0∂z +
+�
+∇2
+X2 − ∇2
+X1
+�
++ k2
+0 (ϵ1(⃗x2, z) − ϵ1(⃗x1, z))
+�
+Ei(⃗x1, z, ω)T E∗
+j (⃗x2, z, ω)T =0, i, j ∈ {X, Y }
+(6)
+where the notation ∇2
+Xl indicates that the Laplacian operation is to be performed as a function of the
+transverse variable designated ⃗xl for l = 1, 2. Taking the expected value of (6) over realizations of the
+electric fields and performing the limiting operation
+�
+Ei(⃗x, z, ω)E∗
+j (⃗x, z, ω)
+�
+≡ lim
+T →∞
+�
+E
+�Ei(⃗x, z, ω)T E∗
+j (⃗x, z, ω)T
+2T
+��
+, i, j ∈ {X, Y }
+(7)
+then yields
+i2k∂zWij(⃗x1, ⃗x2, z, ω) +
+�
+∇2
+X2 − ∇2
+X1
+�
+Wij(⃗x1, ⃗x2, z, ω)
++ k2
+0 [ϵ1(⃗x2, z) − ϵ1(⃗x1, z)] Wij(⃗x1, ⃗x2, z, ω) = 0, i, j ∈ {X, Y }
+(8)
+where each Wij(⃗x1, ⃗x2, z, ω) is an element of the spectral density matrix
+Wij ∈ W(⃗x1, ⃗x2, z, ω) =
+� ⟨EX(⃗x1, z, ω)E∗
+X(⃗x2, z, ω)⟩ ⟨EX(⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩
+⟨EY (⃗x1, z, ω)E∗
+X(⃗x2, z, ω)⟩ ⟨EY (⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩
+�
+(9)
+and possesses units of electric field squared per Hz. Note that multiplying (9) by ϵ0/2 gives W(⃗x1, ⃗x2, z, ω)
+in units of power spectral density in Watts/Hz.
+Equation (8) treats the medium properties as deterministic, such as propagation through a medium
+of known refractive index profile, in which case it is common to set ⟨ϵ⟩ = 1 and ϵ1(⃗x, z) = n2(⃗x, z) − 1
+where n(⃗x, z) is the refractive index.
+Alternatively, we can assume the medium is described by its
+statistical properties. Treating ϵ1(⃗x, z) as a zero-mean, Gaussian random variable (e.g., as in a tur-
+bulent atmosphere) requires averaging over both the field amplitudes and the medium properties in
+taking the expectation of Eq.
+(6).
+Define these properties via the covariance ⟨ϵ1(⃗x2, z)ϵ1(⃗x1, z′)⟩ =
+�
+R3 SNN(⃗ξ, κ)ei⃗ξ·(⃗x2−⃗x1)+iκ(z−z′)d⃗ξdκ , that is, as the inverse Fourier Transform of the three-dimensional
+3
+
+spectral density associated with fluctuations in the dielectric constant.
+We then invoke the well-
+established Markov Approximation (MA) of Tatarskii [30], [9] (see extension to the vector electric field
+case in [5]) which assumes the medium is “delta” correlated in the direction of propagation so that
+⟨ϵ1(⃗x2, z)ϵ1(⃗x1, z′)⟩ ≈ δ(z − z′)A(⃗x2 − ⃗x1, z′). Under the MA, Tatarskii [30] showed that the required
+average in (6) can be written
+�
+[ϵ1(⃗x2, z) − ϵ1(⃗x1, z)] Ei(⃗x1, z, ω)T E∗
+j (⃗x2, z, ω)T
+�
+≈ ik0
+2 [A(0, z) − A(⃗x2 − ⃗x1, z)] Wij(⃗x1, ⃗x2, z, ω)
+(10)
+so that Eq. (8) becomes
+i2k∂zWij(⃗x1, ⃗x2, z, ω) +
+�
+∇2
+X2 − ∇2
+X1
+�
+Wij(⃗x1, ⃗x2, z, ω)
++ ik3
+0
+2 [A(0, z) − A(⃗x2 − ⃗x1, z)] Wij(⃗x1, ⃗x2, z, ω) = 0
+(11)
+As we will show, the choice of either a deterministic (8) or a stochastic (11) medium can be easily
+accommodated in the transport model. Also note that the last term in both (8) and (11) possess units
+of [m] so that kA(·) is dimensionless, as is ϵ(·). Note that in [5], the approximation ϵ1 ≈ 2η is invoked
+which would have led to the pre-factor in Eq. (10) being written as 2ik0.
+Equation (8) or (11) therefore contain four separate equations, each governing a different element of
+the spectral density matrix (see also Wolf et al. e.g. [25]). Note that at the spatial location ⃗x1 = ⃗x2 ≡ ⃗x
+the expressions ⟨Ei(⃗x1, z, ω)E∗
+j (⃗x2, z, ω)⟩ reduce to the more familiar power spectral density functions
+Sij(⃗x, z, ω) =
+�
+Ei(⃗x, z, ω)E∗
+j (⃗x, z, ω)
+�
+, i, j ∈ {X, Y }
+(12)
+that is, Sij(⃗x, z, ω) = Wij(⃗x, ⃗x, z, ω).
+B.
+Generalize Stokes Parameter Representation
+It will be convenient to describe propagation in terms of generalized Stokes parameters [15]. Define
+s0(⃗x1, ⃗x2, z, ω) = ⟨EX(⃗x1, z, ω)E∗
+X(⃗x2, z, ω)⟩ + ⟨EY (⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩
+= WXX(⃗x1, ⃗x2, z, ω) + WY Y (⃗x1, ⃗x2, z, ω)
+s1(⃗x1, ⃗x2, z, ω) = ⟨EX(⃗x1, z, ω)E∗
+X(⃗x2, z, ω)⟩ − ⟨EY (⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩
+= WXX(⃗x1, ⃗x2, z, ω) − WY Y (⃗x1, ⃗x2, z, ω)
+s2(⃗x1, ⃗x2, z, ω) = 2Re{⟨EX(⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩} = ⟨EX(⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩ + ⟨EY (⃗x1, z, ω)E∗
+X(⃗x2, z, ω)⟩
+= WXY (⃗x1, ⃗x2, z, ω) + WY X(⃗x1, ⃗x2, z, ω)
+s3(⃗x1, ⃗x2, z, ω) = −2Im{⟨EX(⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩} = i (⟨EX(⃗x1, z, ω)E∗
+Y (⃗x2, z, ω)⟩ − ⟨EY (⃗x1, z, ω)E∗
+X(⃗x2, z, ω)⟩)
+= i (WXY (⃗x1, ⃗x2, z, ω) − WY X(⃗x1, ⃗x2, z, ω)) .
+(13)
+Unlike conventional Stokes parameters which are evaluated at a single point in space, these generalized
+parameters depend on the field relationships at two points ⃗x1 and ⃗x2. As ⃗x2 → ⃗x1 ≡ ⃗x we recover the
+standard definitions. If the above holds, we can also write
+W(⃗x1, ⃗x2, z, ω) = 1
+2
+� s0(⃗x1, ⃗x2, z, ω) + s1(⃗x1, ⃗x2, z, ω) s2(⃗x1, ⃗x2, z, ω) − is3(⃗x1, ⃗x2, z, ω)
+s2(⃗x1, ⃗x2, z, ω) + is3(⃗x1, ⃗x2, z, ω) s0(⃗x1, ⃗x2, z, ω) − s1(⃗x1, ⃗x2, z, ω)
+�
+(14)
+which matches the result of [22] (Eq.
+3.54) and [26] (Eq.
+1.5), the only difference being the sign
+convention chosen for exp (iωt).
+C.
+Wigner Representation
+Because (8,11) are linear in Wij(⃗x1, ⃗x2, z, ω) we can consider still other forms. For example, (8) is also
+seen to govern the generalized Stokes parameters,
+�
+i2k0∂z +
+�
+∇2
+X2 − ∇2
+X1
+�
++ k2
+0 (ϵ1(⃗x2) − ϵ1(⃗x1))
+�
+sν = 0,
+ν = 0, · · · , 3
+(15)
+4
+
+where sν = sν (⃗x1, ⃗x2, z, ω) and where we have omitted the arguments for brevity. Applying the coordi-
+nate transformation ⃗x1,2 → ⃗x ∓ ⃗x ′/2 and noting that, in this new system the operator (∇2
+X2 − ∇2
+X1) →
+2∇X · ∇X′ [23], we have
+�
+i2k0∂z + 2(∇X · ∇X′) + 2k2
+0Φ(⃗x, ⃗x ′, z)
+�
+s′
+ν = 0,
+ν = 0, · · · , 3
+(16)
+where we again use the shorthand notation s′
+ν = s′
+ν(⃗x, ⃗x ′, z, ω) and where
+Φ(⃗x, ⃗x ′, z) = 1
+2
+�
+ϵ1(⃗x + ⃗x ′
+2 , z) − ϵ1(⃗x − ⃗x ′
+2 , z)
+�
+(17)
+will be referred to as the Wigner potential function. In free space Φ(⃗x, ⃗x′, z) = 0 while in the case of a
+stochastic medium, the Wigner potential of Eq. (16) becomes
+Φ(⃗x, ⃗x′, z) = ik0
+4 [A(0, z) − A(⃗x′, z)] .
+(18)
+In these differential coordinates the spectral coherency matrix then becomes
+Wij(⃗x, ⃗x ′, z, ω) ≡
+�
+Ei
+�
+⃗x − ⃗x ′
+2 , z, ω
+�
+E∗
+j
+�
+(⃗x + ⃗x ′
+2 , z, ω
+��
+, i, j = X, Y
+(19)
+Equation (16) matches that of Charnotskii [5] (Equation 22 in the cited work) and states that the
+generalized Stokes parameters obey a parabolic equation of the same basic structure as the complex
+electric field amplitude in the deterministic, coherent case (see also, [8]).
+Now consider the spatial Fourier Transform of (19) with respect to the variable ⃗x ′ and the resulting
+Fourier Transform pair
+�
+Wij(⃗x, ⃗ξ, z, ω) =
+�
+R2 Wij(⃗x, ⃗x ′, z, ω)e−i⃗ξ·⃗x′d⃗x ′
+Wij(⃗x, ⃗x ′, z, ω) =
+� 1
+2π
+�2 �
+R2
+�
+Wij(⃗x, ⃗ξ, z, ω)ei⃗ξ·⃗x ′d⃗ξ
+(20)
+where the hat �
+W denotes the spatial Fourier transform, the operator
+�
+Rn denotes the n−dimensional
+integral over the space of real numbers (e.g.,
+�
+R2 ≡
+� ∞
+−∞
+� ∞
+−∞), and where the differential element in
+transverse real space and reciprocal 2-space are denoted by, respectively, d⃗x ′ ≡ dx′dy′ and d⃗ξ ≡ dξxdξy.
+Note that, unlike in Eqn. (3), we do not need to restrict the limits of integration in defining the Fourier
+Transform as the beam spatial correlations are assumed to be naturally limited in transverse extent, that
+is |Wij(⃗x, ±∞, z, ω)| → 0. The quantity �
+Wij(⃗x, ⃗ξ, z, ω) is thus the Wigner transform [32] associated with
+complex arguments Ei(⃗x, z, ω), Ej(⃗x, z, ω), i, j ∈ X, Y .
+In the expressions (20), Wij(⃗x, ⃗x′, z, ω) is the spatial covariance between the two field components i, j
+as a function of separation ⃗x ′ in the transverse plane, at transverse position ⃗x and downrange position
+z, and within a small optical spectral range {ω, ω + dω}, while �
+Wij(⃗x, ⃗ξ, z, ω) is the covariance between
+Fourier amplitudes of electric field components i, j associated with transverse spatial frequency ⃗ξ, at
+downrange position z and within a small optical spectral range {ω, ω + dω}. Importantly, this latter
+quantity is proportional to the average phase difference between field components i, j with transverse
+separation ⃗x′ and corresponding spatial frequency ⃗ξ (see Priestly section 9.1 [24]). This point will be
+re-visited in section (III C).
+We replace the spectral density matrix in coordinates ⃗x, ⃗x ′ with its Wigner representation to obtain
+i2k0
+� 1
+2π
+�2
+∂z
+�
+R2 �sν
+′ei⃗ξ·⃗x′d⃗ξ + 2(∇X · ∇X′)
+� 1
+2π
+�2 �
+R2 �sν
+′ei⃗ξ·⃗x′d⃗ξ + 2k2
+0Φ(⃗x, ⃗x′, z)sν
+′ = 0
+� 1
+2π
+�2 �
+R2
+�
+∂z�sν
+′ + 1
+k0
+(⃗ξ · ∇X)�sν
+′ −
+�
+R2 ik0Φ(⃗x, ⃗x′, z)sν
+′e−i⃗ξ·⃗x′d⃗x′
+�
+ei⃗ξ·⃗x′d⃗ξ = 0, ν = 0, · · · , 3
+(21)
+where now the generalized Stokes parameters sν ′ have been replaced by their respective spatial Fourier
+transforms �sν ′ with respect to the differential coordinate ⃗x ′ and, hence, are written in terms of the
+5
+
+Wigner distribution functions
+�s0
+′(⃗x, ⃗ξ, z, ω) = �
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�s1
+′(⃗x, ⃗ξ, z, ω) = �
+WXX(⃗x, ⃗ξ, z, ω) − �
+WY Y (⃗x, ⃗ξ, z, ω)
+�s2
+′(⃗x, ⃗ξ, z, ω) = �
+WXY (⃗x, ⃗ξ, z, ω) + �
+WY X(⃗x, ⃗ξ, z, ω)
+�s3
+′(⃗x, ⃗ξ, z, ω) = i
+�
+�
+WXY (⃗x, ⃗ξ, z, ω) − �
+WY X(⃗x, ⃗ξ, z, ω)
+�
+(22)
+which comprise the same result as Luis’ [16] Eq. (19). Recognizing that the term in square brackets
+must be zero in order to satisfy (21), we obtain the vector transport equations,
+∂z �
+S(⃗x, ⃗ξ, z, ω) + 1
+k0
+(⃗ξ · ∇X) �
+S(⃗x, ⃗ξ, z, ω) =
+�
+R2 ik0Φ(⃗x, ⃗x′, z)S(⃗x, ⃗x′, z, ω)e−i⃗ξ·⃗x′d⃗x′
+= − ik0
+(2π)2
+�
+R4 Φ(⃗x, ⃗x′, z, ω) �
+S(⃗x, ⃗ξ′, z, ω)ei(⃗ξ′−⃗ξ)·⃗x′d⃗ξ′d⃗x′
+= − ik0
+(2π)2
+�
+R2 Φ(⃗x, ⃗ξ − ⃗ξ′, z, ω) �
+S(⃗x, ⃗ξ′, z, ω)d⃗ξ′
+(23)
+where the 4-vector �
+S ≡ {�s0 ′, �s1 ′, �s2 ′, �s3 ′}T and S is the associated inverse FT with respect to wavevector
+⃗ξ. Equation (23) thus denotes four separate transport equations, each governing one component �s ′
+ν, ν =
+0 · · · 3. The term involving the vector potential can be written in several useful forms; the second line
+and matches the results of [33] (Eqn. 2.4) and [12] (Eqn. 1.1), albeit with a different convention for the
+Fourier Transform, while the final convolution form is found in [4] (Eqn. 2).
+Expression (23) is the fundamental vector transport equation that governs the evolution of the gener-
+alized Stokes parameters in the direction of propagation, z. In the scalar case (e.g., analysis of the 1D
+Schr¨odinger equation [33] or scalar optical wave propagation [10], [23]), Eq. (23) is often taken as the
+starting point in predicting the evolution of �
+WXX(⃗x, ⃗ξ, z, ω). Here we have extended these prior works
+to the full vector electric field in order to account for spatially dependent polarization. Related works on
+the full vector transport equations for the Wigner distribution include [26] (Eq. 1.6), and [17]. However
+we believe this work to be the first extension of such analyses to partially polarized optical fields.
+III.
+PARTIALLY-COHERENT, VECTOR TRANSPORT OF INTENSITY
+In this section we extend the transport-of-intensity equations (TIEs) from the partially coherent scalar
+case, as described in [23, 35], to the partially-coherent vector case. To arrive at such a model, we follow
+an approach used in quantum mechanics for obtaining the so-called “hydrodynamic” model, so named
+for the resemblance of the model to the continuity and momentum equations commonly found in fluid
+mechanics. Specifically, we take “moments” of Eq. (23) with respect to the wavevector. The first two
+such moments are obtained by multiplying (23) by unity and ⃗ξ respectively, and then integrating over the
+wavevector. The resulting expressions govern the conservation of mass (intensity), momentum (phase),
+and energy of the system. This approach has been used in analysis of the 1-D Schr¨odinger equation (see
+[4, 7, 12]) and has only very recently been extended to the 2-D Pauli equation [17]. While the dominant
+application area has been quantum mechanics, Marcuvitz [18] has suggested the approach as a general
+technique for studying wave propagation in the scalar case.
+In this work, we extend this “method of moments” to the vector transport equation (23) for partially
+coherent propagation problems in optics. The first such conservation equation is obtained by directly
+integrating (23) (that is, taking the “zeroth” moment) which yields
+∂z
+�
+R2
+�
+S(⃗x, ⃗ξ, z, ω)d⃗ξ + ∇X · 1
+k0
+�
+R2
+⃗ξ �
+S(⃗x, ⃗ξ, z, ω)d⃗ξ = 0.
+(24)
+To see that the Wigner potential term integrates to zero, replace the generalized Stokes parameters by
+their inverse spatial Fourier transforms (in frequency variable ⃗ξ′), in which case integrating the right
+6
+
+hand side of (23) over wavevector ⃗ξ can be carried out as
+=
+�
+R6
+ik0
+(2π)2 �
+S(⃗x, ⃗ξ ′, z, ω)ei⃗ξ ′·⃗x′Φ(⃗x, ⃗x ′, z)e−i⃗ξ·⃗x ′d⃗x ′d⃗ξ ′d⃗ξ
+=
+ik0
+(2π)2
+�
+R4
+�
+S(⃗x, ⃗ξ ′, z, ω)ei⃗ξ ′·⃗x ′d⃗ξ ′
+�
+R2 e−i⃗ξ·⃗x ′d⃗ξΦ(⃗x, ⃗x ′)d⃗x ′
+=
+ik0
+(2π)2
+�
+R4
+�
+S(⃗x, ⃗ξ ′, z, ω)ei⃗ξ ′·⃗x ′d⃗ξ ′δ(⃗x ′)Φ(⃗x, ⃗x ′)d⃗x ′
+=
+ik0
+(2π)2 Φ(⃗x, 0)
+�
+R2
+�
+S(⃗x, ⃗ξ ′, z, ω)d⃗ξ ′ = 0.
+(25)
+In the last step we have leveraged the definition of Φ(⃗x, ⃗x ′, z) (17,18) which holds that when ⃗x′ = 0, the
+potential vanishes, that is, Φ(⃗x, 0) = 0. This statement is true whether we treat the dielectric constant
+fluctuations as a deterministic or a random process under the Markov approximation (18). Finally, to
+put (24) in the same form as the coherent Transport of Intensity Equation (TIE) [20], we require the
+definitions
+ρ(⃗x, z, ω) ≡
+�
+R2
+�
+S(⃗x, ⃗ξ, z, ω)d⃗ξ
+1
+k0
+�
+R2
+⃗ξ �
+S(⃗x, ⃗ξ, z, ω)d⃗ξ ≡ ρ(⃗x, z, ω) ◦ ⃗v(⃗x, z, ω)
+(26)
+where ◦ denotes the Hadamard product. Equation (26) therefore defines four generalized intensities
+ρν(⃗x, z, ω) ≡
+�
+R2 �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ, ν = 0, 1, 2, 3.
+In addition, (26) implicitly defines each element of
+⃗v(⃗x, z, ω) as
+⃗vν(⃗x, z, ω) ≡
+�
+R2 ⃗ξ �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ
+k0ρν(⃗x, z, ω)
+=
+�
+R2 ⃗ξ �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ
+k0
+�
+R2 �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ
+, ν = 0, 1, 2, 3
+(27)
+in which case (24) becomes
+∂zρ(⃗x, z, ω) + ∇X · [ρ(⃗x, z, ω) ◦ ⃗v(⃗x, z, ω)] = 0,
+(28)
+which is the vector TIE for partially coherent light. Note that whereas in our prior analysis [20], the above
+expression was a single scalar equation, in this case both ρ(⃗x, z, ω) and the product ρ(⃗x, z, ω) ◦⃗v(⃗x, z, ω)
+are column vectors each comprising four elements. Thus ρν and vν represent, respectively, the ν−th
+component of the Stokes 4-vector (ν = 0, 1, 2, 3) and, as we will show in section (III C), the rate of
+change with z of the transverse location of the ν−th component of the Stokes 4-vector. Hence, all eight
+components ρν and vν are real, the total beam intensity ρ0 must be non-negative, and the remaining
+seven components can be positive, negative, or zero. Just as for Stokes parameters for monochromatic
+light, the components of the Stokes 3-vector (ρ1, ρ2, ρ3) are proportional to intensity but are not limited
+to non-negative values.
+A.
+Mixture Model for Partially Coherent Stokes Parameters
+Equations (28) are transport equations governing each of the generalized Stokes parameters during
+propagation in a (possibly) inhomogeneous medium. By definition, the Stokes parameters were defined
+as an expectation over realizations of the X, Y components of the electric field Fourier amplitudes,
+EX(⃗x, z, ω), EY (⃗x, z, ω).
+Both the quantity being transported, ρ(⃗x, z, ω) and the “velocity” of that
+transport, ⃗v(⃗x, z, ω) carry simple physical interpretations when placed in the context of a particular
+model for the electric field as will be shown next.
+To aid in this interpretation, recall the electric field “transverse wave” model (3). Until now we have
+imposed no further model structure on the form of EX(⃗x, z, ω), EY (⃗x, z, ω). Leveraging prior work [20]
+for linearly-polarized, coherent beams, let the complex Fourier amplitudes take the form
+EX(⃗x, z, ω) = ρ1/2
+m (⃗x, z, ω)eiφm(⃗x,z,ω) cos(γm(⃗x, z, ω))
+EY (⃗x, z, ω) = ρ1/2
+m (⃗x, z, ω)eiφm(⃗x,z,ω) sin(γm(⃗x, z, ω))
+(29)
+7
+
+where the subscript m denotes “model”. The electric field is therefore described by a magnitude, phase,
+and spatially dependent linear polarization angle γm(⃗x, z, ω) which governs the ratio of Fourier amplitudes
+in the X and Y directions. It must be noted, however, that the model structure (29) presumes a polarized
+beam and should therefore be viewed as conditional on the beam being completely polarized. Such a
+model is clearly not appropriate for unpolarized light.
+While the first Stokes parameter �s0 is independent of the state of polarization, the same cannot be
+said of �sν, ν = 1 · · · 3. For partially polarized light it is therefore appropriate to use the mixture model
+[29]
+�
+�
+�
+�
+s′
+0(⃗x, ⃗x′, z, ω)
+s′
+1(⃗x, ⃗x′, z, ω)
+s′
+2(⃗x, ⃗x′, z, ω)
+s′
+3(⃗x, ⃗x′, z, ω)
+�
+�
+�
+� = [1 − P]
+�
+�
+�
+�
+s′
+0(⃗x, ⃗x′, z, ω)
+0
+0
+0
+�
+�
+�
+� + P
+�
+�
+�
+�
+s′
+0(⃗x, ⃗x′, z, ω)
+s′
+1(⃗x, ⃗x′, z, ω|P)
+s′
+2(⃗x, ⃗x′, z, ω|P)
+s′
+3(⃗x, ⃗x′, z, ω|P)
+�
+�
+�
+�
+(30)
+where the weighting is given by the Degree Of Polarization (DOP)
+P(⃗x, z, ω) =
+�
+(s1)2 + (s2)2 + (s3)2
+s0
+���
+⃗x1=⃗x2=⃗x =
+�
+1 − 4Det [W]
+Tr [W]2
+���
+⃗x1=⃗x2=⃗x.
+(31)
+Equation (31) thus defines a positive quantity 0 ≤ P(⃗x, z, ω) ≤ 1 which specifies the fraction of the beam
+intensity at location {⃗x, z} and within an infinitesimal range of optical frequencies {ω, ω + dω} that is
+fully polarized. The degree to which a beam is polarized is therefore directly related to coherence among
+the components of the optical field at a given transverse location. We note also that while P(⃗x, z, ω)
+varies spatially and with frequency, as defined it does not depend on the separation ⃗x ′ between two
+points on the beam face.
+Given the above definitions, we can better understand the quantities in the partially coherent TIE
+(28) as will be described next in sections (III B) and (III C). These definitions will also allow us to define
+a polarization gradient in the partially coherent case as will be shown in section (III D).
+B.
+Interpretation of the Partially Coherent Vector TIE
+To understand the meaning of ρ(⃗x, z, ω) we expand the first component of �
+S(⃗x, ⃗ξ, z, ω) which is just
+�s0 ′(⃗x, ⃗ξ, z, ω) and use the fact that the Wigner distributions can be expressed as Fourier transforms to
+obtain
+ρ0(⃗x, z, ω) =
+�
+R2 �s0
+′(⃗x, ⃗ξ, z, ω)d⃗ξ =
+�
+R4 (WXX(⃗x, ⃗x′, z, ω) + WY Y (⃗x, ⃗x′, z, ω)) e−i⃗ξ·⃗x′d⃗x ′d⃗ξ
+=
+�
+R2 (WXX(⃗x, ⃗x ′, z, ω) + WY Y (⃗x, ⃗x ′, z, ω)) δ(⃗x ′)d⃗x ′
+= SXX(⃗x, z, ω) + SY Y (⃗x, z, ω)
+(32)
+where we have invoked the definition of the delta function as the Fourier Transform of unity. Recall also
+that by definition, in transformed coordinates, at ⃗x ′ = 0,
+Wij(⃗x, 0, z, ω) = Sij(⃗x, z, ω), i, j = X, Y
+(33)
+so that the first component of ρ(⃗x, z, ω) is by definition the auto-spectral density of the field at transverse
+location ⃗x, z as demonstrated by (32). A similar analysis of the other components of ρ(⃗x, z, ω) shows
+that the full four-component Stokes vector is given by
+ρ(⃗x, z, ω) =
+�
+R2
+�
+�
+�
+�
+�
+�s′
+0
+P �s′
+1
+P �s′
+2
+P �s′
+3
+�
+�
+�
+�
+� d⃗ξ =
+�
+�
+�
+�
+SXX(⃗x, z, ω) + SY Y (⃗x, z, ω)
+P (SXX(⃗x, z, ω) − SY Y (⃗x, z, ω))
+P (SXY (⃗x, z, ω) + SY X(⃗x, z, ω))
+iP (SXY (⃗x, z, ω) − SY X(⃗x, z, ω))
+�
+�
+�
+� =
+�
+�
+�
+�
+s0
+Ps1
+Ps2
+Ps3
+�
+�
+�
+�
+⃗x1=⃗x2≡⃗x
+(34)
+Thus, the quantities being transported can be written simply as sums and differences of the transverse
+spectral densities that is, the generalized Stokes parameters for partially coherent light.
+8
+
+For example, substituting the model (29) for the first Stokes parameter
+s0 = ⟨EX(⃗x, z, ω)E∗
+X(⃗x, z, ω)⟩ + ⟨EY (⃗x, z, ω)E∗
+Y (⃗x, z, ω)⟩
+=
+�
+ρm(⃗x, z, ω) cos2(γm(⃗x, z, ω) + ρm(⃗x, z, ω) sin2(γm(⃗x, z, ω)
+�
+= ⟨ρm(⃗x, z, ω)⟩ ≡ ρ0(⃗x, z, ω).
+(35)
+Hence the first generalized Stokes parameter ρ0(⃗x, z, ω) is proportional to the expected beam spectral
+intensity at location {⃗x, z} in our model. Performing the same substitution for the remaining components
+in (34) we have
+ρ(⃗x, z, ω) =
+�
+�
+�
+�
+⟨ρm(⃗x, z, ω)⟩
+P(⃗x, z, ω) ⟨ρm(⃗x, z, ω) cos(2γm(⃗x, z, ω))⟩
+P(⃗x, z, ω) ⟨ρm(⃗x, z, ω) sin(2γm(⃗x, z, ω))⟩
+0
+�
+�
+�
+� .
+(36)
+The model structure (29) suggests a simple and well-known interpretation of the generalized Stokes
+parameters for linearly polarized light.
+However, for unpolarized light only the first component of
+ρ(⃗x, z, ω) is non-zero. The final intensity component is given by ρ3(⃗x, z, ω) = 0 due to the fact that we
+are considering only linear polarization.
+C.
+Interpretation of the Transport Velocity
+The components of ⃗v(⃗x, z, ω) in Eq. (28) are referred to as “velocities” as they were noted in the
+coherent case to govern the change in optical path in the transverse direction per unit change in the
+direction of propagation [11, 20]. We present a basic derivation supporting this interpretation in Appendix
+(A). This terminology is also in keeping with the wave mechanics interpretation of the transport equation,
+where the “transport” of intensity occurs in the transverse plane as the beam progresses in z. The first
+velocity term in this partially coherent case is obtained via (27),
+⃗v0(⃗x, z, ω) = 1
+k0
+�
+R2 ⃗ξ �s′
+0(⃗x, ⃗ξ, z, ω)d⃗ξ
+�
+R2 �s′
+0(⃗x, ⃗ξ, z, ω)d⃗ξ
+=
+�
+R2 ⃗ξ
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ
+k0ρ0(⃗x, z, ω)
+(37)
+The non-negativity of �s′
+0 allows us to interpret (37) as an average spatial frequency in the transverse
+plane (see e.g., Boashash [2]) normalized by k0. In fact, in [2] it was noted that the generalized definition
+of frequency is an average change in phase per unit change in the independent variable, which in the
+present context corresponds to the transverse distance ⃗x. Thus, (37) implicitly defines a phase gradient
+∇Xφ(⃗x, z, ω) ≡
+�
+R2 ⃗ξ �s′
+0(⃗x, ⃗ξ, z, ω)d⃗ξ
+�
+R2 �s′
+0(⃗x, ⃗ξ, z, ω)d⃗ξ
+(38)
+so that ⃗v0(⃗x, z, ω) = k−1
+0 ∇Xφ(⃗x, z, ω), which is precisely the definition for velocity found in the monochro-
+matic case [20]. This definition of generalized phase was also used in [34] for scalar, partially coherent
+electric fields. Here, we have shown this definition to be appropriate for vector electric fields as well. In-
+deed, we will show momentarily that these definitions are not only appropriate for transport of intensity,
+ρ0(⃗x, z, ω), but for transport of two other generalized Stokes parameters.
+Now consider again the mixture model (30). Substituting these definitions into (37) yields for the
+denominator SXX(⃗x, z, ω)+SY Y (⃗x, z, ω) = ρ0(⃗x, z, ω). For the numerator, considering first the polarized
+9
+
+portion, we can expand using Eqn. (19) to give
+�
+R2
+⃗ξ
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ
+=
+�
+R4
+⃗ξ
+��
+EX(⃗x − ⃗x′/2, z, ω)E∗
+X(⃗x + ⃗x′/2, z, ω)
+�
++
+�
+EY (⃗x − ⃗x′/2, z, ω)E∗
+Y (⃗x + ⃗x′/2, z, ω)
+��
+e−i⃗ξ·⃗x′d⃗ξd⃗x′
+=
+�
+R2
+��
+EX(⃗x − ⃗x′/2, z, ω)E∗
+X(⃗x + ⃗x′/2, z, ω)
+�
++
+�
+EY (⃗x − ⃗x′/2, z, ω)E∗
+Y (⃗x + ⃗x′/2, z, ω)
+�� �
+R2
+⃗ξe−i⃗ξ·⃗x′d⃗ξd⃗x′
+= i
+�
+R2
+��
+EX(⃗x − ⃗x′/2, z, ω)E∗
+X(⃗x + ⃗x′/2, z, ω)
+�
++
+�
+EY (⃗x − ⃗x′/2, z, ω)E∗
+Y (⃗x + ⃗x′/2, z, ω)
+��
+δ′(⃗x′)d⃗x′
+= i
+�
+R2 ∇X′
+��
+EX(⃗x − ⃗x′/2, z, ω)E∗
+X(⃗x + ⃗x′/2, z, ω)
+�
++
+�
+EY (⃗x − ⃗x′/2, z, ω)E∗
+Y (⃗x + ⃗x′/2, z, ω)
+��
+δ(⃗x′)d⃗x′
+= i∇X′
+��
+EX(⃗x − ⃗x′/2, z, ω)E∗
+X(⃗x + ⃗x′/2, z, ω)
+�
++
+�
+EY (⃗x − ⃗x′/2, z, ω)E∗
+Y (⃗x + ⃗x′/2, z, ω)
+�� �����
+⃗x′→0
+= ⟨ρm(⃗x, z)∇Xφm(⃗x, z)⟩ .
+(39)
+where the final line substitutes Eq. (29) for EX, EY and takes the required gradient ∇X′. In the second
+to last line we leveraged the identity in Eq. (7) of [3] concerning integrals over derivatives of the delta
+function. The end result is that under the mixture model, and assuming amplitude and transverse phase
+gradient are statistically independent,
+⟨∇Xφm(⃗x, z, ω)⟩ = ∇Xφ(⃗x, z, ω)
+1
+k0
+⟨∇Xφm(⃗x, z, ω)⟩ = ⃗v0(⃗x, z, ω)
+(40)
+The definition (38) therefore can be interpreted as the expected model phase and the corresponding
+“velocity” as the expected change in optical path in the transverse direction per unit change in the
+direction of propagation. Notably, this result is independent of the DOP. We may perform a similar
+analysis on the remaining components of ⃗v(⃗x, z, ω), finding that
+⃗v1(⃗x, z, ω) =
+�
+R2 ⃗ξ �s′
+1(⃗x, ⃗ξ, z, ω)d⃗ξ
+k0ρ1(⃗x, z, ω)
+= P(⃗x, z, ω) ⟨ρm∇Xφm(⃗x, z, ω) cos(2γm(⃗x, z, ω))⟩
+k0P(⃗x, z, ω) ⟨ρm cos(2γm(⃗x, z, ω))⟩
+= ⃗v0(⃗x, z, ω)
+⃗v2(⃗x, z, ω) =
+�
+R2 ⃗ξ �s′
+2(⃗x, ⃗ξ, z, ω)d⃗ξ
+k0ρ2(⃗x, z, ω)
+= P(⃗x, z, ω) ⟨ρm∇Xφm(⃗x, z, ω) sin(2γm(⃗x, z, ω))⟩
+k0P(⃗x, z, ω) ⟨ρm sin(2γm(⃗x, z, ω))⟩
+= ⃗v0(⃗x, z, ω). (41)
+Thus, in the case of linear polarization, all three non-zero generalized Stokes parameters are transported
+with the same transverse velocity.
+∂zρν(⃗x, z, ω) + ∇X · [ρν(⃗x, z, ω)v0(⃗x, z, ω)] = 0, ν = 0, 1, 2.
+(42)
+This is sensible as it suggests that, in expectation, both the beam intensity and properties related to the
+state of linear polarization are moving together in the transverse plane during propagation. Importantly,
+this also suggests that while the transport model requires an equation governing ⃗v0 (see section IV), it
+does not require separate equations for ⃗v1, ⃗v2.
+It it worth mentioning the relationship between the velocity vector and the well-known Poynting vector.
+Begin by noting that the numerator in (27) is of the form of the average Poynting vector for partially
+coherent light,
+P(⃗x, z) ≡ 1
+k0
+�
+R2
+⃗ξ �s′
+0(⃗x, ⃗ξ, z, ω)d⃗ξ,
+(43)
+so that ˜P(⃗x, z) ≡ ⃗v(⃗x, z, ω) is, in fact, the normalized Poynting vector as defined in [21] (Eq. 8 of the
+cited work). Also in [21] is was suggested that the normalized Poynting vector be decomposed via the
+Helmholtz decomposition theorem as
+⃗v(⃗x, z, ω) ≡ ⃗vS(⃗x, z, ω) + ⃗vV (⃗x, z, ω)
+= k−1
+0 ∇XφS(⃗x, z, ω) + k−1
+0 ∇X × ⃗φV (⃗x, z, ω)
+(44)
+10
+
+where S and V refer to scalar and vector contributions, respectively.
+If we let the scalar phase be
+φS(⃗x, z, ω) = φm(⃗x, z, ω) and define
+φV (⃗x, z, ω) =
+�dγ(⃗x, ω)
+dy
+z, −dγ(⃗x, ω)
+dx
+z, 0
+�
+(45)
+then we have that ⃗Ω(⃗x, z, ω) = [∇ × φV (⃗x, z, ω)]X. The partially coherent vector TIE would then be
+written
+∂zρ(⃗x, z, ω) + ∇X · [ρ(⃗x, z, ω)⃗vS(⃗x, z, ω)] + ∇X · [ρ(⃗x, z, ω)⃗vV (⃗x, z, ω)] = 0
+(46)
+which would clearly suggest that ∇X ·
+�
+ρ(⃗x, z, ω)⃗Ω(⃗x, z, ω)
+�
+= 0 given Eq. (28). In fact, we will show in
+the next section that this relationship indeed holds for linearly polarized light.
+D.
+Definition and Interpretation of the Polarization Angle Gradient
+Since we are considering a linearly polarized vector beam ρ3(⃗x, z, ω) = 0. However, although the first
+term in (24) is zero, the second term inside the transverse divergence operator is (following the same
+model interpretation and simplification as in (39))
+1
+k0
+�
+R2
+⃗ξ �s′
+3(⃗x, ⃗ξ, z, ω)d⃗ξ = −
+�
+ρm(⃗x, z, ω)P(⃗x, z, ω)
+k0
+∇Xγm(⃗x, z, ω)
+�
+= −ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)
+(47)
+so that by (24) we have the condition
+∇X ·
+�
+ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)
+�
+= 0
+(48)
+as we implied at the end of the previous section. This relationship was also found to hold in the coherent
+case [20] and will be leveraged later in the derivation. However, also note that in arriving at (48) we have
+produced a definition for the spatial gradient of the polarization angle that is appropriate for partially
+coherent light, given by
+⃗Ω(⃗x, z, ω) =
+−iP(⃗x, z, ω)
+�
+R2 ⃗ξ
+�
+�
+WXY (⃗x, ⃗ξ, z, ω) − �
+WY X(⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ
+k0ρ0(⃗x, z, ω)
+= P(⃗x, z, ω)
+k0
+⟨∇Xγm(⃗x, z, ω)⟩ .
+(49)
+The DOP is appropriately part of the definition (49) as one cannot define ⃗Ω for an unpolarized beam.
+Note also that in accordance with the works of Salem et al. [28] and Korotkova et al. [14] we allow that
+the DOP can vary in frequency ω, as well as the transverse location ⃗x, and the direction of propagation
+z.
+Just as with the normalized phase gradient, Eqn. (49) is expressed as a suitable average wavevector over
+the distribution �s′
+3. Similarly, it can be interpreted as a transverse, spatial gradient in the angle (phase)
+between the X and Y components. In the limit of a fully coherent beam we recover our deterministic
+definition. Importantly, as the beam de-polarizes we will see the terms involving ⃗Ω(⃗x, z, ω) vanish, a
+point we elaborate on in the next section.
+We note that a definition of polarization angle for partially coherent beams was proposed previously
+by Korotkova et al. [25], [13]
+θ(⃗x, z, ω) ≡ 1
+2 arctan
+�
+2Re {SXY (⃗x, z, ω)}
+SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)
+�
+.
+(50)
+This definition is somewhat consistent with our model structure (29) in the sense that substituting the
+polarized electric field (29) into (50) gives
+θ(⃗x, z, ω) ≡ 1
+2 arctan
+�s2(⃗x, z, ω)
+s1(⃗x, z, ω)
+�
+= 1
+2 arctan
+� ⟨sin(2γm(⃗x, z, ω))⟩
+⟨cos(2γm(⃗x, z, ω))⟩
+�
+.
+(51)
+However, this definition only yields our model polarization angle in the deterministic case as the ratio of
+expectations of the sin, cos terms does not equal the mean of the ratio in general. Moreover, the DOP
+11
+
+does not appear explicitly in the expression as it does in (49) and instead must be viewed as a part of
+the quantity SXY (⃗x, z, ω).
+More importantly, as we will see in section (IV), the quantity that governs the transverse movement
+of intensity is the polarization angle gradient as opposed to the angle itself. As we have just shown,
+the gradient is obtained via the first moment, Eq. (47). The resulting expression (49) accounts for the
+spatial frequencies ⃗ξ and yields directly the expected value of polarization angle gradient. On the other
+hand, differentiating (50) with respect to the transverse coordinates yields
+∇Xθ(⃗x, z, ω)
+= [SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)]∇XSXY (⃗x, z, ω) − SXY (⃗x, z, ω)∇X[SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)]
+4SXY (⃗x, z, ω)2 + [SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)]2
+.
+(52)
+As with Eq. (51), this definition of phase gradient results in products and sums of expectations of sin, cos
+functions so that the result equals ⟨∇Xγm(⃗x, z, ω)⟩ only in the deterministic case where no averaging
+operations are performed.
+Finally, we note that our definition (49) arose quite naturally as a consequence of 1) the paraxial
+transport equation for the generalized Stokes parameters, Eq. (24) and 2) the model Eq. (29) governing
+the expected amplitude, phase, and polarization angle of the electric field at location (⃗x, z) and frequency
+ω. The definition is also consistent with the definition of partially coherent phase used in the literature
+(Eqn. 38), and will therefore be used in what follows for the expected polarization angle gradient of our
+vector beam.
+E.
+Conservation of the generalized Stokes parameters
+To conclude this section, we point out an interesting result concerning the conservation of the gener-
+alized Stokes parameters during propagation through a (possibly) inhomogeneous medium. Since each
+of the parameters sν obey the same transport equation (28), the integrals of these quantities in the
+transverse plane are constant and are therefore conserved during propagation. To see that this is true,
+expand Eq. (28), recall the definition of the material derivative D(·)/Dz ≡ ∂z(·) + (⃗v · ∇X)(·), and
+then convert to Lagrangian coordinates whereby the spatial coordinates ⃗x are written as functions of z
+and the initial value, that is, ⃗x → ⃗xz(⃗x0) (which we will abbreviate as simply ⃗xz and the corresponding
+gradient ∇Xz). These steps can be written
+∂zρν(⃗x, z, ω) + ∇Xρν(⃗x, z, ω) · ⃗vν(⃗x, z, ω) + ρν(⃗x, z, ω) [∇X · ⃗vν(⃗x, z, ω)] = 0
+Dρν(⃗x, z, ω)
+Dz
++ ρν(⃗x, z, ω) [∇X · ⃗vν(⃗x, z, ω)] = 0
+dρν(⃗xz, ω)
+dz
++ ρν(⃗xz, ω) [∇Xz · ⃗vν(⃗xz, ω)] = 0,
+ν = 0, 1, 2, 3.
+(53)
+The resulting ordinary differential equation possesses the solution
+ρ(⃗xz, ω) = ρ(⃗x0, ω) exp
+�
+−
+� z
+s=0
+∇Xs · ⃗v(⃗xs, ω)d⃗xs
+�
+.
+(54)
+Therefore, the integral of the generalized Stokes parameters at any point along the propagation path is
+related to their initial values via the divergence of the velocity field. It can also be shown (see Appendix
+B, ref. [6]) that the expression (54) can be written alternatively as
+ρ(⃗xz, ω) = det |J⃗x0(⃗xz)|−1ρ(⃗x0, ω).
+(55)
+where the Jacobian J⃗x0(⃗xz) is the derivative of the coordinate functions ⃗xz(⃗x0, z) with respect to the
+fixed (initial) coordinates ⃗x0. If the Lagrangian coordinate mappings ⃗xz(⃗x0, z) are one-to-one, smooth
+functions of ⃗x0, this expression can be shown via the change of variables theorem to be equivalent to the
+integral formulation [1]
+�
+X
+ρ(⃗xz, ω)d⃗xz =
+�
+X
+ρ(⃗x0, ω)d⃗x0
+(56)
+12
+
+(see also Villani Chapter 11 [31]). Thus, it can be stated that the integral of each of the generalized
+Stokes parameters over the transverse plane is conserved on propagation. Interestingly, this conclusion
+was reached by an entirely different approach in [13]. However, in that work the claim was made with
+respect to propagation in free-space whereas here we see the integrals of the Stokes parameters are
+conserved even in an inhomogeneous medium.
+IV.
+MOMENTUM EQUATION FOR PARTIALLY-COHERENT, LINEARLY-POLARIZED
+LIGHT
+The above discussion defined both intensity and velocity in the general case of a propagating, linearly
+polarized, partially coherent optical field. Each of the generalized Stokes parameters were shown to be
+transported with a single velocity. In this section, we will derive the equation that governs this velocity.
+As we described in section (II), the process for obtaining the continuity equation can be viewed as
+taking the zeroth “moment” of the fundamental transport equation with respect to wavenumber, that is,
+integrating 23; (see again [18], [7], and [12]). Thus, to derive the partially coherent momentum equation
+we multiply (23) by ⃗ξ and integrate over wavenumber. The result is written
+∂z
+�
+R2
+⃗ξ �
+S(⃗x, ⃗ξ, z, ω)d⃗ξ + ∇X · 1
+k0
+�
+R2(⃗ξ ⊗ ⃗ξ) �
+S(⃗x, ⃗ξ, z, ω)d⃗ξ =
+�
+R2
+�
+R2 ik0⃗ξ �
+S(⃗x, ⃗ξ′, z, ω)Φ(⃗x, ⃗ξ − ⃗ξ′, z)d⃗ξ′d⃗ξ.
+(57)
+The first challenge associated with (57) is the term on the right-hand-side, integration of Wigner poten-
+tial. This term can be greatly simplified if we are willing to restrict the model to a weakly inhomogeneous
+medium where the transverse gradient in dielectric constant is small at a given transverse location ⃗x. In
+this case, we explore a series solution for the terms inside the potential function, for example,
+ϵ1(⃗x + ⃗x′
+2 , z) − ϵ1(⃗x − ⃗x′
+2 , z) ≈ ϵ1(⃗x, z) ± ∇Xϵ1(x, z) · ⃗x′ + · · ·
+(58)
+so that the potential functions (17,18) become
+Φ(⃗x, ⃗x′, z) ≈ ⃗x′ · ∇X
+�1
+2ϵ1(⃗x, z)
+�
++ O(⃗x′3)
+Φ(⃗x, ⃗x′, z) ≈ ⃗x′ · ∇X
+�
+−ik0
+4 A(0, z)
+�
++ O(⃗x′2)
+(59)
+Using these expressions, and representing the terms inside the transverse gradient generically as g(⃗x)
+(to accommodate a deterministic or stochastic medium), we we see that the inner double integral on the
+right hand side of Eq. (57) can be expanded using the Fourier representation of the potential as
+� 1
+2π
+�2 �
+R2
+�
+R2 ik0 �
+S(⃗x, ⃗ξ′, z, ω)
+�
+ei(⃗ξ−⃗ξ′)·⃗x′⃗x′ · ∇Xg(⃗x, z)d⃗x′
+�
+d⃗ξ′
+= −
+� 1
+2π
+�2 �
+R2
+�
+R2 k0 �
+S(⃗x, ⃗ξ′, z, ω)
+�
+∇ξ′ei(⃗ξ−⃗ξ′)·⃗x′ · ∇Xg(⃗x, z)d⃗x′
+�
+d⃗ξ′
+= −
+� 1
+2π
+�2 �
+R2 k0 �
+S(⃗x, ⃗ξ′, z, ω)∇ξ′
+��
+R2 ei(⃗ξ−⃗ξ′)·⃗x′d⃗x′
+�
+· ∇Xg(⃗x, z)d⃗ξ′
+= −
+�
+R2 k0∇⃗ξ′ δ(⃗ξ − ⃗ξ′) �
+S(⃗x, ⃗ξ′, z, ω) · ∇Xg(⃗x, z)d⃗ξ′
+= −
+�
+R2 k0δ(⃗ξ − ⃗ξ′)∇⃗ξ′ �
+S(⃗x, ⃗ξ′, z, ω) · ∇Xg(⃗x, z)d⃗ξ′
+= −k0∇⃗ξ �
+S(⃗x, ⃗ξ, z, ω) · ∇Xg(⃗x, z)
+(60)
+where we have used the trick found in [19]( Eq.
+5.29), which is to note that ∇ξ′
+�
+ei(⃗ξ−⃗ξ′)·⃗x′�
+=
+−i⃗x′ei(⃗ξ−⃗ξ′)·⃗x′. The expression (60) is the “linearized” Wigner potential (see, for example, [33]) and
+transforms Eqn. (57) into
+∂z
+�
+R2
+⃗ξ �
+S(⃗x, ⃗ξ, z, ω)d⃗ξ + ∇X · 1
+k0
+�
+R2(⃗ξ ⊗ ⃗ξ) �
+S(⃗x, ⃗ξ, z, ω)d⃗ξ = −k0∇Xg(⃗x, z)
+�
+R2
+�
+⃗ξ · ∇⃗ξ �
+S(⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ.
+(61)
+13
+
+We have already shown via Eq. (26) that the first term in (61) can be written
+∂z
+�
+R2
+⃗ξ �s′ν(⃗x, ⃗ξ, z, ω)d⃗ξ = k0∂z [ρν(⃗x, z, ω)⃗v0(⃗x, z, ω)] , ν = 0, · · · , 2
+(62)
+while for the third Stokes parameter we showed previously in Eqn. (47) that
+∂z
+�
+R2
+⃗ξ �s′
+3(⃗x, ⃗ξ, z, ω)d⃗ξ = −k0∂z
+�
+ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)
+�
+.
+(63)
+The integral in the last term can also be simplified via integration by parts. Presuming that the Stokes
+parameters vanish at the boundaries of integration (as |⃗ξ| → ∞), the result is simply
+�
+R2
+�
+⃗ξ · ∇⃗ξ �
+S(⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ = −ρ(⃗x, z, ω)
+(64)
+for the first three Stokes parameters (ν = 0, 1, 2), while the integral disappears for the third since, in the
+absence of ellipticity, ρ3(⃗x, z, ω) = 0. With these simplifications, Eqn. (61) becomes
+∂z [ρν(⃗x, z, ω)⃗v0(⃗x, z, ω)] + ∇X · 1
+k2
+0
+�
+R2(⃗ξ ⊗ ⃗ξ) �s′ν(⃗x, ⃗ξ, z, ω)d⃗ξ = ρν(⃗x, z, ω)∇Xg(⃗x, z), ν = 0, 1, 2
+−∂z
+�
+ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)
+�
++ ∇X · 1
+k2
+0
+�
+R2(⃗ξ ⊗ ⃗ξ) �s′
+3(⃗x, ⃗ξ, z, ω)d⃗ξ = 0
+(65)
+Equation (65) is a vector equation describing the evolution of the “momentum” ρ ◦ ⃗v for each of the
+generalized Stokes parameters and it allows us to develop expressions for the unknowns ⃗v0 and ⃗Ω. As
+we noted in section (III C), because there is only a single velocity associated with the first 3 equations,
+we need only focus on the expression (65) for ν = 0 and the last expression where ν = 3. We also note
+that the right hand side of the second expression is zero for the linearized potential since ρ3(⃗x, z, ω) = 0.
+As we will show, this leads to the conclusion that the polarization angle gradient will not change on
+propagation. Depolarization during propagation would therefore be a consequence of higher-order terms
+in the expansion of the Wigner potential or retention of the electric field divergence in the starting wave
+equation (see e.g., [5]).
+Recall the continuity equation (28) was obtained as the zeroth moment of the wave-vector with respect
+to the Stokes parameters (Eqn. 23) and contained the “velocity”, which is the first moment of the wave-
+vector with respect to the Stokes parameters. In turn, (65) contains the second moment of the wave-vector
+with respect to the Stokes parameters. As might be expected, if we were to construct the expression for
+the second moment it would contain the integral of third order wave-vector terms and so on. This is the
+so-called “closure problem” (see [12] Eqs. 3.3-3.5 and surrounding discussion of [17]).
+Obtaining closure requires an expression for this second moment (the term involving the outer product)
+in terms of already defined quantities. In our context, this means writing the integral involving the outer
+product in terms of ρ and ⃗v. This can be accomplished by again leveraging our electric field model (29).
+Note that using this particular field model as an expression to fix the closure problem was employed in
+both [17, 18]. To see how this can be done, expand the second term in (65) as was done in [18]. Doing
+so requires us to expand the Wigner-averaged outer product into its four constituent terms which are of
+the two basic forms.
+Using the first of equations (65) as an example, the outer product is seen to produce the following four
+integrals,
+�
+R2 ξ2
+ii
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ,
+i ∈ X, Y
+�
+R2 ξiξj
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ,
+i, j ∈ X, Y.
+(66)
+Both the unpolarized and polarized portions of the mixture model can be simplified in the same manner
+as was done for the velocity terms, Eq.
+(39).
+Substituting in the definition of either the unpolar-
+ized/polarized Wigner Transforms �
+WXX(⃗x, ⃗ξ, z, ω) and �
+WY Y (⃗x, ⃗ξ, z, ω), the first of the needed moments
+14
+
+in (66) can be simplified as
+�
+R2 ξ2
+XX
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ
+= − ∂2
+∂x′2
+�
+EX
+�
+⃗x − ⃗x′
+2 , z, ω
+�
+E∗
+X
+�
+⃗x + ⃗x′
+2 , z, ω
+�
++ EY
+�
+⃗x − ⃗x′
+2 , z, ω
+�
+E∗
+Y
+�
+⃗x + ⃗x′
+2 , z, ω
+�� ����� x′ → 0
+y′ → 0
+.
+(67)
+Using our model for the electric field, Eq. (29), substituting into the above and taking the derivative
+and subsequent limits (and noting the procedure in the “y” direction is the same) yields
+�
+R2 ξ2
+XX
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ = k2
+0ρ0(v2
+x + Ω2
+x) + (∂xρ0)2
+4ρ0
+− ∂2
+xρ0
+4
+�
+R2 ξ2
+Y Y
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ = k2
+0ρ0(v2
+y + Ω2
+y) + (∂yρ0)2
+4ρ0
+− ∂2
+yρ0
+4
+(68)
+where vx,y and Ωx,y are, respectively, the “x” and “y” components of those vectors. In this and several
+subsequent expressions we omit the arguments (⃗x, z, ω) from the model quanitities for brevity.
+The
+“mixed” averages follow a similar procedure and are found to be
+�
+R2ξXξY
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ = k2
+0ρ0(vxvy + ΩxΩy) + ∂xρ0∂yρ0
+4ρ0
+− ∂xyρ0
+4
+(69)
+so that we may finally write
+1
+k2
+0
+�
+R2
+�
+⃗ξ ⊗ ⃗ξ
+� �
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ =
+ρ0⃗v ⊗ ⃗v + ρ0⃗Ω ⊗ ⃗Ω +
+1
+4k2
+0
+�
+�
+(∂xρ0)2
+ρ2
+0
+− ∂2
+xρ0
+ρ0
+∂xρ0∂yρ0
+ρ2
+0
+− ∂xyρ0
+ρ0
+∂xρ0∂yρ0
+ρ2
+0
+− ∂xyρ0
+ρ0
+(∂yρ0)2
+ρ2
+0
+−
+∂2
+yρ0
+ρ0
+�
+� ρ0
+(70)
+Repeating this same process for the third Stokes parameter (second equation in (65)) yields the equations
+∂z
+� ρ0⃗v
+−ρ0⃗Ω
+�
++
+�
+� ∇X ·
+�
+ρ0⃗v ⊗ ⃗v + ρ0R + ρ0⃗Ω ⊗ ⃗Ω
+�
+−∇X ·
+�
+ρ0⃗v ⊗ ⃗Ω + ρ0⃗Ω ⊗ ⃗v
+�
+�
+� =
+� ρ0∇Xg(⃗x, z)
+0
+�
+(71)
+where we have defined the matrix
+R =
+1
+4k2
+0
+�
+�
+(∂xρ0)2
+ρ2
+0
+− ∂2
+xρ0
+ρ0
+∂xρ0∂yρ0
+ρ2
+0
+− ∂xyρ0
+ρ0
+∂xρ0∂yρ0
+ρ2
+0
+− ∂xyρ0
+ρ0
+(∂yρ0)2
+ρ2
+0
+−
+∂2
+yρ0
+ρ0
+�
+�
+(72)
+A convenient simplification occurs upon expansion of the outer product via ∇·( ⃗B ⊗ ⃗A) = ⃗A(∇· ⃗B)+( ⃗B ·
+∇) ⃗A. This identity, along with Eqs. (42) and (48), specifically ∂zρ0+∇X ·(ρ0⃗v) = 0 and ∇X ·(ρ0P⃗Ω) = 0,
+transforms (71) into
+� ρ0D⃗v/Dz
+ρ0D⃗Ω/Dz
+�
++
+�
+� ∇X ·
+�
+ρ0R + ρ0⃗Ω ⊗ ⃗Ω
+�
+�
+ρ0⃗Ω · ∇X
+�
+⃗v
+�
+� =
+� ρ0∇Xg(⃗x, z)
+0
+�
+.
+(73)
+where we have again leveraged the material derivative defined earlier in section (III E). The only com-
+ponents of (73) that are influenced by the DOP are those involving the polarization angle gradient. To
+simplify further, note that the divergence of ρ0(⃗x, z, ω)R(⃗x, z, ω) is
+∇X · [ρ0(⃗x, z, ω)R(⃗x, z, ω)] = − 1
+2k2
+0
+ρ0(⃗x, z, ω)∇X
+�
+∇2
+Xρ1/2
+0
+(⃗x, z, ω)
+ρ1/2
+0
+(⃗x, z, ω)
+�
+.
+(74)
+15
+
+Expanding the outer product and invoking (48), the first of Eqns (73) can be written
+D⃗v0(⃗x, z, ω)
+Dz
+= −
+�
+⃗Ω(⃗x, z, ω) · ∇X
+�
+⃗Ω(⃗x, z, ω) + ∇Xg(⃗x, z) +
+1
+2k2
+0
+∇X
+�
+∇2
+Xρ1/2
+0
+(⃗x, z, ω)
+ρ1/2
+0
+(⃗x, z, ω)
+�
+.
+(75)
+which is of precisely the same form as the coherent momentum equation [20]. In the case of a deterministic
+medium, the gradient potential forcing term is ∇Xϵ1/2. In [20] the commonly used medium assumptions
+were used whereby, ⟨ϵ⟩ = 1 and ϵ1 = n2(⃗x, z)−1. In this case Eq. (75) matches exactly that found in [20]
+for P = 1 (fully polarized light). For a stochastic, homogeneous medium where g(⃗x, z) = −ik0A(⃗x, z)/4
+we have
+∇Xg(⃗x, z) = −ik0
+4 ∇XA(0, z)
+= −ik0
+4
+�
+R2 i⃗ξSNN(⃗ξ, 0)ei⃗ξ·⃗0d⃗ξ
+= k0
+4
+�
+R2
+⃗ξSNN(⃗ξ, 0)d⃗ξ
+(76)
+Taking as a simple model for the refractive index fluctuations as Eqn.
+(38) of [5], we have that
+∇Xg(⃗x, z) = 0, i.e., the gradient of the refractive index covariance is zero. Thus, the time-averaged
+transverse location is unchanged due to the turbulence. Note, the model (75) can also be used to derive
+known results for “beam wander” in the coherent case following the approach in [11]. In that case,
+an expression for the transverse coordinate variance is developed and is seen to be proportional to the
+variance of the refractive index gradient (as opposed to the gradient of the variance) and one recovers
+the established result (see [11] for details).
+As in our prior work, it is advantageous to switch to Lagrangian coordinates ⃗x → ⃗xz(⃗x0) which are
+functions of propagation distance z and initial value ⃗x0. With this transformation, and noting that the
+“velocities” are the coordinate derivatives with respect to z, (75) becomes
+d⃗v0(⃗xz, ω)
+dz
+= d2⃗xz(ω)
+dz2
+= −
+�
+⃗Ω(⃗xz, ω) · ∇Xz
+�
+⃗Ω(⃗xz, ω) + ∇Xzg(⃗xz) +
+1
+2k2
+0
+∇Xz
+�
+∇2
+Xzρ1/2
+0
+(⃗xz, ω)
+ρ1/2
+0
+(⃗xz, ω)
+�
+.
+(77)
+This is a sensible model as it suggests that a fully depolarized beam will follow a path that is influenced
+only by diffraction (last term in 77) and refractive index fluctuations. However, if the beam is polarized
+and if the polarization angle distribution possesses non-zero first and second spatial derivatives, there
+is an additional effect that must be considered, given by the first term on the right hand side of (75).
+In the case of diffraction only, the model (77) recovers the known result for the beam path in both the
+Gaussian beam [11]) and Airy beam [20]) situations.
+The second equation in (73) governs the evolution of the polarization angle. Following the derivation
+in Appendix (C), this expression reduces to
+D⃗Ω
+Dz +
+�
+⃗Ω · ∇X
+�
+⃗v = 0
+(78)
+or in Lagrangian coordinates
+d⃗Ω(⃗xz, ω)
+dz
+= 0.
+(79)
+Thus, for partially coherent light, the time-average polarization gradient as defined in (49) will remain
+unchanged during propagation when observed in Lagrangian coordinates. This result is consistent with
+our prior work which held that in the fully coherent case, the polarization angle remained unchanged on
+propagation [20]. By definition, depolarization will decrease ⃗Ω, ultimately resulting in ⃗Ω = 0 for P = 0.
+However, no mechanism exists in (79) to cause this to occur. While it is known that the atmosphere can
+serve as a depolarizing element, under the MA model such an effect can only be observed by retaining
+higher-order spatial variations in the refractive index (stemming ultimately from retaining the electric
+field divergence term in the wave equation) [5, 11]. In both of the cited works retention of such terms
+has almost no influence on beam propagation but for very long distances.
+This distance scales as
+O(k2
+0ℓ3
+0/σ2
+ϵ ) ∼ 1010m, even when considering strong turbulence (refractive index variance σ2
+η ∼ 10−10),
+16
+
+small inhomogeneities (length scale ℓ0 ∼ 10−3m), and 1.55µm wavelength light [5]. Other works have
+predicted depolarization will arise due to asymmetries in the initial spatial correlations across the beam
+face [25]. We have not included this effect in our model given that lasers do not naturally possess such
+asymmetry and creating such a beam would be challenging.
+In summary, we conclude that whether we are considering coherent, monochromatic, or partially
+coherent, polychromatic wave propagation, the basic governing equations are the same.
+ρ0(⃗xz, ω) = | det J⃗x0(⃗xz)|−1ρ0(⃗x0, ω).
+d2⃗xz
+dz2 = −
+�
+⃗Ω(⃗xz, ω) · ∇Xz
+�
+⃗Ω(⃗xz, ω) + ∇Xzg(⃗xz) +
+1
+2k2
+0
+∇Xz
+�
+∇2
+Xzρ1/2
+0
+(⃗xz, ω)
+ρ1/2
+0
+(⃗xz, ω)
+�
+d⃗Ω(⃗xz, ω)
+dz
+= 0
+(80)
+It is the definition of quantities in the expression that is different in the two cases. These can be
+summarized as follows:
+Coherent, monochromatic
+ρ(⃗x, z) = EX(⃗x, z)E∗
+X(⃗x, z) + EY (⃗x, z)E∗
+Y (⃗x, z)
+⃗v(⃗x, z) = 1
+k0
+∇Xφ(⃗x, z)
+⃗Ω(⃗x, z) = 1
+k0
+∇Xγ(⃗x, z)
+(81)
+Partially coherent, polychromatic
+ρ0(⃗x, z, ω) =
+�
+R2
+�
+�
+WXX(⃗x, ξ, z, ω) + �
+WY Y (⃗x, ξ, z, ω)
+�
+d⃗ξ = SXX(⃗x, z, ω) + SY Y (⃗x, z, ω)
+⃗v0(⃗x, z, ω) = 1
+k0
+⟨∇Xφ(⃗x, z, ω)⟩ =
+�
+R2 ⃗ξ
+�
+�
+WXX(⃗x, ⃗ξ, z, ω) + �
+WY Y (⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ
+k0ρ0(⃗x, z)
+⃗Ω(⃗x, z, ω) = P(⃗x, z, ω)
+k0
+⟨∇Xγ(⃗x, z, ω)⟩ =
+−iP
+�
+R2 ⃗ξ
+�
+�
+WXY (⃗x, ⃗ξ, z, ω) − �
+WY X(⃗x, ⃗ξ, z, ω)
+�
+d⃗ξ
+k0ρ0(⃗x, z)
+(82)
+where we can see that the latter reduce to the former in the coherent limit.
+V.
+EXAMPLES
+In our previous work we showed that for a particular functional form of polarization angle gradient
+the beam would “accelerate” in the transverse direction following a curved path and we have shown in
+(80) that this is also true in the partially coherent case. Using a new definition of polarization angle
+gradient (49), we see the the beam will follow the same coherent path as in [20] for P = 1 but will follow
+a different path as the DOP decreases P < 1.
+Consider first the simple case of a beam with Gaussian intensity distribution of equal amplitude in
+the x and y components. As in [14] (see Eqn. 3.6 in the cited work) we take P(⃗x, z) = P as a constant
+representing the correlation coefficient between the x and y components of the electric field and obeying
+0 ≤ P ≤ 1. In our prior work we set P = 1 and used the polarization profile
+γ(y0) = π
+2
+(y0 − a)2
+a2
++ π
+8
+(83)
+which varies in y0 only and not in x0. Thus, the polarization gradient terms becomes simply
+⃗Ω(y0) =
+�
+0, P π(y0 − a)
+a2
+�
+(84)
+17
+
+and the second of Eqs. (80) becomes the ordinary differential equations
+d2xz
+dz2 = x0
+d2yz
+dz2 = −P2
+k2
+0
+dγ(y0)
+dy0
+d2γ(y0)
+dy2
+0
+(85)
+subject to the initial velocity dx0/dz = dy0/dz = 0 and initial displacements x0 and y0. This expression
+can be solved to give the the partially coherent beam path
+xz = x0
+yz = P2π2z2
+2k2
+0a4 (a − y0) + y0.
+(86)
+The prediction is that for fully polarized light, P = 1, the beam will follow the same path obtained in
+[20] while, for unpolarized light, P = 0, and the beam will simply follow a straight path. Figure (1) shows
+a family of such curves associated with the beam center, y0 = 0, for different degrees of polarization.
+Also shown is the polarization angle profile across the beam face, plotted as a function of the transverse
+coordinates ⃗x0. The result is the expected behavior. As the DOP is reduced, so too is the degree to which
+FIG. 1: (a) Transverse displacement of the beam center as the DOP is reduced from unity to 0 in
+increments of 0.2. The model parameters were the same as those used in our prior work [20] and (b)
+The associated polarization angle profile, Eq. (83)
+the path of the beam center is altered. We also note that in this example the effects of diffraction could
+have been included just as they were in our coherent model [20]. The result is the standard Gaussian
+beam spreading about the path predicted in Eqn. (86).
+As a second example, we consider the case where both bending and diffraction are retained in the model.
+Specifically, we tailor the polarization profile so that the polarization-induced bending acts in opposition
+to the diffractive effect, thereby mitigating the degree to which the beam spreads on propagation. Using
+the polarization profile
+γ(y0) = κπ
+2
+y2
+0
+a2 − π
+8
+(87)
+we solve the coupled system of Equations (80) using an initial Gaussian intensity profile
+ρ0(⃗x0, ω) = I(ω)e−(x2
+0+y2
+0)/(2σ2)
+(88)
+with width σ. The solution is obtained analytically in this case using the approach outlined in [20] which
+18
+
+(a)
+X10-3
+5
+Transverse displacement y(z) meters
+= 0.8
+3
+2
+P = 0.6
+P = 0.4
+P = 0.2
+0
+P = 0.0
+.1
+0
+5
+10
+15
+20
+25
+30
+Propagation distance z meters(b)
+2
+(mm)
+0
+1
+-2
+-2
+-1
+0
+1
+2
+(uw) °xleads to the transverse path
+xz = x0
+�
+z2 + σ4k2
+0
+σ2k0
+yz = −(Pκ)2π2z2
+2k2
+0a4
+y0 + y0
+�
+z2 + σ4k2
+0
+σ2k0
+=
+��
+1 +
+z2
+k2
+0σ4
+�1/2
+− π2
+2
+(Pκ)2
+k2
+0a4 z2
+�
+y0.
+(89)
+The corresponding determinant of the Jacobian is therefore
+det J⃗x0(⃗xz) = 1 +
+z2
+σ4k2
+0
+− (Pκ)2π2z2�
+z2 + σ4k2
+0
+2a4σ2k3
+0
+.
+(90)
+and the desired intensity profile on propagation is given by the first of Eqs. (80). The polarization profile
+(87) has the effect of bending the outer edges of the beam, y0 = ±a/2, toward the center thus mitigating
+the outward spreading caused by the diffractive term. These results can be seen visually in Fig. (2) for
+a fully polarized beam of wavelength 1.55 µm, width σ = 1.2 mm and extending over transverse spatial
+dimension a = 4.17 mm. In this fully polarized case we set κ = 1.
+FIG. 2: (a) Normal beam diffraction profile over the range z = [0, 250] m, (b) diffraction profile
+associated with the polarization angle distribution given by Eq. (87) and for P = 1. Here the spreading
+due to diffraction is compensated by the bending due to the polarization angle gradient. These effects
+cancel each other out at range zc ≈ (2a4k0)/ (Pπσ)2 (obtained by setting yz = 0). Beyond this range
+the beam super diffracts and the spreading is proportional to (z − zc)2. Plots (c) and (d) use the same
+polarization profile but with P = 0.75 and P = 0.5 respectively.
+The normalized beam intensity is plotted for z = [0, 250] m in order to clearly show the effects of the
+polarization gradient. By altering the polarization of the light across the beam face we can control, to
+some extent, the degree to which the beam diffracts. Of course, in this example the diffractive effect
+causes transverse spreading that is linear in z while the bending we show here is quadratic. Thus, at a
+19
+
+(a)
+50
+4
+P=0
+6'0
+ Transverse Displacement (mm)
+0.8
+¥2
+0.7
+0
+0.6
+0
+0.5
+-10
+0.4
+0.3
+-30
+0.2
+-40
+0.1
+-50
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+240
+Propagation Distance (m)(q)
+50
+P=1
+40
+0.9
+Transverse Displacement (mm)
+0.8
+2
+0.7
+0
+0.6
+0
+0.5
+-10
+0.4
+-20 
+0.3
+-30
+0.2
+-40
+0.1
+-50
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+240
+Propagation Distance (m)(c)
+50
+40
+P=0.75
+0.9
+ Transverse Displacement (mm)
+0.8
+2
+0.7
+0
+0.6
+0
+0.5
+-10
+0.4
+-20 
+0.3
+30
+0.2
+-40
+0.1
+-50
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+240
+Propagation Distance (m)(d)
+50
+40
+P=0.5
+0.9
+ Transverse Displacement (mm)
+0.8
+2
+0.7
+0
+0.6
+0
+0.5
+-10
+0.4
+-20 
+0.3
+30
+0.2
+-40
+0.1
+-50
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+220
+240
+Propagation Distance (m)distance zc, obtained by setting yz = 0 in Eqn. (89), the rays are predicted to come to a focus. Then
+zc =
+√
+2
+π2
+k0
+(Pκ)2
+� a4
+σ2
+� �
+1 +
+�
+1 + π4(Pκ)4
+�σ
+a
+�8
+�1/2
+(91)
+and in the limit zc ≫ k0σ2 we find the particularly simple form
+zc = 2a4k0
+(Pπσ)2 .
+(92)
+(Note also this limit is also obtained when π4(Pκ)4(σ/a)8 << 1, a condition that will always be met
+when the beam width 2σ is less than the output aperture dimension 2a). Beyond z > zc, however,
+the beam diverges strongly and the intensity rapidly diminishes. Thus, using this polarization profile
+diffraction can be mitigated, but only over a prescribed range. As the beam depolarizes the degree
+to which diffraction is mitigated will change, however, by increasing the constant κ in the polarization
+gradient such that Pκ = 1, we expect to see the same results as in Fig. (2b). In practice, of course,
+there will be a limit to how large a polarization gradient can be produced. As the DOP decreases (with
+κ = 1) we see the expected results in Figs. (2c-d), namely that the beam gradually moves toward the
+standard diffraction profile associated with an incoherent beam, Fig. (2a).
+Fig. (3) shows a plot of zc from (91) for fixed polarization gradient (κ = 1) as a function of degree
+of polarization P in the range 0.01 ≤ P ≤ 1. The zc ∝ P−1 behavior continues indefinitely as P → 0,
+that is, regardless of the degree of polarization, there exists a z−value for which the effects of diffraction
+could be completely compensated. Of course in practice, there is a limit to κ as one can not generate
+a beam with an infinitely sharp polarization gradient. The practical limit will for such a beam will be
+determined by the resolution of the device used in its creation, e.g., the spatial light modulators used in
+reference [20].
+The results in Fig. 3 are consistent with the following interpretation of the mixture model in (30). The
+term ”mixture” cannot be taken literally to mean that the beam comprises a collection of photons in
+which a fraction P are fully polarized and fraction (1 − P) are completely unpolarized. If that were true
+then the polarization gradient could only compensate diffraction for the polarized photons and the beam
+spread at z = zc would necessarily increase as P decreased. Instead, we found that the beam width at
+the focus, in the y− direction for this example, is independent of P. Hence, we interpret (30) to mean
+that every photon in the beam is described by Stokes vector
+�
+S =
+�
+�
+�
+�
+s′
+0(⃗x, ⃗x′, z, ω)
+Ps′
+1(⃗x, ⃗x′, z, ω|P)
+Ps′
+2(⃗x, ⃗x′, z, ω|P)
+Ps′
+3(⃗x, ⃗x′, z, ω|P)
+�
+�
+�
+� .
+(93)
+Every photon thus exhibits properties of partial polarization with degree of polarization P.
+VI.
+SUMMARY
+We have generalized our prior, coherent vector beam model to the partially coherent case by introducing
+a mixture model for the generalized Stokes vector and by subsequently choosing appropriate definitions
+of amplitude, phase, and polarization angle gradient in describing propagation of the generalized Stokes
+parameters. The resulting “transport” model predicts the time-averaged optical path of a vector beam
+propagating through an inhomogeneous medium (e.g., an atmosphere). Importantly, the model reduces
+to that of our previous work on vector beams in the coherent limit [20] and matches the predictions
+found in [30] and [5] in the absence of a polarization gradient. Consequently, it predicts the same optical
+path as in our coherent model in the fully polarized case, while reducing to the standard “unaltered”
+path in the unpolarized situation.
+Notably, the model requires a new definition of polarization angle gradient that is consistent with
+more recent definitions of phase for partially coherent beams. These definitions become important in the
+governing equations as the movement of intensity during propagation is governed by phase and angle
+gradients as opposed to the phase and angle variables. As a by-product of the model development we
+were also able to demonstrate conservation of the generalized Stokes parameters on propagation, even in
+20
+
+FIG. 3: zc as a function of degree of polarization P for λ = 1.55 µm, a = 4.17 mm, and σ = 1.15 mm.
+For these parameters, zc ∝ P−1 and the trend continues indefinitely as P → 0.
+the case where the intervening medium possesses refractive index fluctuations. This is a generalization
+of the result established in [13].
+An obvious prediction of the model Eqn. (79) is that the state of linear polarization (as captured
+by ⃗Ω) remains unchanged on propagation from a Lagrangian viewpoint. Because the definition of ⃗Ω
+includes the DOP, this is also tantamount to the statement that the DOP is unchanged on propagation.
+The result is consistent with the result of Charnotskii [5] which found that no significant depolarization
+occurs due to the presence of a turbulent atmosphere. Had we included the higher order refractive index
+variations (by retaining the electric field divergence in the wave equation) and/or retained the full Wigner
+potential in forming Eqn. (61) (as opposed to using the linearized potential) this effect may have been
+observable in our model as well.
+We considered two such examples, one in which the curved path taken by the beam studied in [20] was
+slowly altered by changing the degree of polarization. In the coherent case, the path taken is the same as
+in the cited reference. As the degree of polarization is reduced so too is degree of beam bending, reducing
+to propagation in a straight line in the fully incoherent case. In the second example, we consider both
+the bending and diffracting beam. For that example we chose an initial polarization distribution that
+would mitigate the effects of diffraction over a predictable distance. The result suggests new class of
+“non-diffracting” beam. The effects of depolarization can be countered in this case by simply increasing
+the strength of the polarization gradient applied across the transverse dimension of the beam.
+In short, the model we have developed can be used to predict vector beam trajectories in a variety of
+propagation scenarios of interest, including through free-space and a potentially turbulent atmosphere.
+As such it provides a powerful tool in forecasting the utility of the newly discovered vector beam bending
+effect in application. We note that the model rests on the same two assumptions as were used in the
+coherent model development [20], namely, the slowly varying envelope approximation and a weakly
+inhomogeneous medium.
+VII.
+ACKNOWLEDGMENTS
+The authors would like to thank the Office of Naval Research Code 33, for support under
+PE#0601153N, award #N0001422WX01660
+Appendix A: Interpretation of Transport Velocity
+In this section we present a more traditional optics interpretation of the transverse velocity ⃗v =
+k−1
+0 ∇Xφ for coherent light. In particular, we show how this quantity can be related to the familiar
+concept of “optical path length” (OPL). This allows us to properly view the transverse phase gradient
+21
+
+107
+106
+105
+(w)
+104
+103
+102
+10-2
+10-1
+100
+DOP (P)of our electric field as the rate at which intensity is moved in the transverse plane. The interpretation
+will remain valid so long as the paraxial assumption holds.
+To begin, we note that the OPL in the context of a propagating wave is defined (see, for example,
+Saleh [27]) as
+Φ(⃗x, z) = z + φ(⃗x, z)/(2k0)
+(A1)
+that is, as the phase argument of the wave normalized by the wavenumber. The quantity φ/(2k0) can
+be viewed as a perturbation to the optical path resulting from refractive index fluctuations, diffraction,
+and, in this context, from polarization angle gradients as well.
+Consider now the expression for total path length L(⃗xz) in Lagrangian coordinates. For a differential
+length element parameterized by propagation distance, dℓ(z)
+L(⃗xz) =
+� z
+0
+dℓ(ζ)
+=
+� z
+0
+(dζ2 + dx2
+ζ + dy2
+ζ)1/2 =
+� z
+0
+�
+1 +
+�dxζ
+dζ
+�2
++
+�dyζ
+dζ
+�2�1/2
+dζ
+≈
+� z
+0
+dζ + 1
+2
+� z
+0
+�dxζ
+dζ
+�2
+dζ + 1
+2
+� z
+0
+�dyζ
+dζ
+�2
+dζ
+(A2)
+where in the final step we have made use of the fact that for ϵ1, ϵ2 ≪ 1, (1 + ϵ1 + ϵ2)1/2 ≈ 1 + 1
+2ϵ1 + 1
+2ϵ2
+which is equivalent to the paraxial assumption. Equating this expression to the OPL given in Eqn. (A1)
+gives
+L(⃗xz) = z + φ(⃗xz)
+2k0
+=
+� z
+0
+dζ + 1
+2
+� z
+0
+�dxζ
+dζ
+�2
+dζ + 1
+2
+� z
+0
+�dyζ
+dζ
+�2
+dζ
+(A3)
+so that
+φ(⃗xz)
+2k0
+= 1
+2
+� z
+0
+�dxζ
+dζ
+�2
+dζ + 1
+2
+� z
+0
+�dyζ
+dζ
+�2
+dζ
+(A4)
+Differentiating with respect to z yields
+1
+k0
+�∂φ(⃗xz)
+∂xz
+∂xz
+∂z + ∂φ(⃗xz)
+∂yz
+∂yz
+∂z
+�
+=
+�dxz
+dz
+�2
++
+�dyz
+dz
+�2
+.
+(A5)
+which can be written
+1
+k0
+∇Xzφ(⃗xz) · ⃗v(⃗xz) = ⃗v(⃗xz) · ⃗v(⃗xz)
+(A6)
+thus
+⃗v(⃗xz) = 1
+k0
+∇Xzφ(⃗xz).
+(A7)
+This demonstrates that the change in optical path length in the transverse plane per unit change in ˆz
+(i.e. the transverse “velocity”) is equal to the transverse phase gradient normalized by the wavenumber.
+Eq. (A7) was presented in [23] (Eq. 11 of that work) and is referred to appropriately as the “flow vector
+of spectral density”. The connection between ⃗v(⃗xz) and transverse coordinate changes was also leveraged
+in [11] although not explicitly derived as we have here. The key assumption used to arrive at this result
+is that changes in the transverse direction are much smaller than changes in the direction of propagation
+and is tantamount to the same paraxial assumption we have used throughout our model development.
+While the above analysis is performed for the coherent case, we have demonstrated that one can re-
+define the model quantities (amplitude, phase, and polarization angle) as appropriate averages in the
+partially coherent case. Thus, the above interpretation still holds and the transverse phase gradient can
+be interpreted as the average change in optical path in the transverse direction per unit change in the
+direction of propagation.
+22
+
+Appendix B: Derivation of Equation (55)
+The identity required of Eqn.
+(55) requires relating the determinant of the Jacobian of the La-
+grangian coordinate mappings xz(x0, y0), yz(x0, y0) to the divergence of the associated velocity field
+uz(xz, yz), vz(xz, yz). Begin with the determinant
+det(J⃗x0(⃗xz)) = dxz
+dx0
+dyz
+dy0
+− dyz
+dx0
+dxz
+dy0
+,
+(B1)
+take the derivative with respect to z and recognize that uz ≡ dxz/dz, vz ≡ dyz/dz. Then
+d
+dz det(J⃗x0(⃗xz)) = duz
+dx0
+dyz
+dy0
++ dvz
+dy0
+dxz
+dx0
+− dvz
+dx0
+dxz
+dy0
+− duz
+dy0
+dyz
+dx0
+(B2)
+Both uz(xz, yz), vz(xz, yz) are functions of the Lagrangian coordinates, hence we can apply the chain
+rule to obtain
+duz
+dx0
+= ∂uz
+∂xz
+∂xz
+∂x0
++ ∂uz
+∂yz
+∂yz
+dx0
+duz
+dy0
+= ∂uz
+∂xz
+∂xz
+dy0
++ duz
+∂yz
+∂yz
+∂y0
+dvz
+dx0
+= ∂vz
+∂xz
+∂xz
+∂x0
++ ∂vz
+∂yz
+∂yz
+∂x0
+dvz
+dy0
+= ∂vz
+dxz
+∂xz
+∂y0
++ ∂vz
+∂yz
+∂yz
+∂y0
+.
+(B3)
+Substituting (B3) into (B2) and simplifying yields the ordinary differential equation
+d det(J⃗x0(⃗xz))
+dz
+= (∇Xs · ⃗v(⃗xs) det(J⃗x0(⃗xz)),
+(B4)
+with solution
+det (J⃗x0(⃗xz)) = exp
+�� z
+s=0
+∇Xs · ⃗v(⃗xs)ds
+�
+(B5)
+This relationship allows (54) to be written as (55) thus completing the derivation.
+Appendix C: Derivation of Equation (79)
+To derive Equation (79) we begin with Eqn. (71)
+∂z
+�
+ρ0⃗Ω
+�
++ ∇X ·
+�
+ρ0⃗v ⊗ ⃗Ω + ρ0⃗Ω ⊗ ⃗v
+�
+= 0
+(C1)
+Using two applications of the vector identity ∇X ·
+�
+⃗B ⊗ ⃗A
+�
+= ⃗A
+�
+∇X · ⃗B
+�
++
+�
+⃗B · ∇X
+�
+⃗A and grouping
+terms we have
+ρ0[∂z⃗Ω + (⃗v · ∇X)⃗Ω] + ⃗Ω
+�
+((((((((
+∂z(ρ0) + ∇X · ρ0⃗v
+�
++ ⃗v
+�
+∇X · [ρo⃗Ω]
+�
++
+�
+ρ0⃗Ω · ∇X
+�
+⃗v = 0
+(C2)
+where Eqn. (48) has allowed us to cancel the second to last term while continuity of the first Stokes
+parameter (i.e., the intensity) cancels the third term.
+Noting the first term in brackets is the total
+derivative of ⃗Ω, allows us to write Eqn. (C2) in Eulerian coordinates as
+D⃗Ω
+Dz +
+�
+⃗Ω · ∇X
+�
+⃗v = 0
+(C3)
+However, further simplification is possible by re-writing (C2) as
+∂z⃗Ω +
+�
+(⃗v · ∇X) Ω +
+�
+⃗Ω · ∇X
+�
+⃗v
+�
+= 0
+∂z⃗Ω + ∇X
+�
+⃗Ω · ⃗v
+�
+= 0.
+(C4)
+23
+
+where we have leveraged the identity ∇X
+�
+⃗A · ⃗B
+�
+= ( ⃗A·∇X) ⃗B+( ⃗B·∇X) ⃗A+ ⃗A×(∇X × ⃗B)+ ⃗B×(∇X × ⃗A)
+and note that both ⃗Ω and ⃗v are expressible as gradients of scalars, hence the cross-product terms vanish.
+Recalling that ⃗Ω ≡ k−1
+0 ∇Xγ we can write
+k−1
+0
+[∂z(∇Xγ) + ∇X(∇Xγ · ⃗v)] = 0
+= k−1
+0 ∇X [∂zγ + ∇Xγ · ⃗v] = 0
+= k−1
+0 ∇X
+Dγ
+Dz = 0.
+(C5)
+If we now make the switch to Lagrangian coordinates, the total derivatives become ordinary derivatives
+d⃗Ω(⃗xz, ω)
+dz
+= 0.
+(C6)
+which is Eqn. (79).
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+[29] B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher. Measuring the Stokes polarization parameters.
+American Journal of Physics, 75(2):163–168, 2007.
+[30] V. I. Tatarskii. Light propagation in a medium with random refractive index inhomogeneities in the Markov
+random process approximation. Sov. Phys. JETP, 29:1133–1138, 1969.
+[31] C. Villani. Optimal Transport: Old and New. Springer-Verlag, Berlin, 2008.
+[32] E. P. Wigner. On the quantum correction for thermodynamic equilibrium. Physical Review, 40:749–759,
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+[33] D. Yin, M. Tang, and S. Jin.
+The Gaussian beam method for the Wigner equation with discontinuous
+potentials. Inverse Problems and Imaging, 7(3):1051–1074, 2013.
+[34] J. Zhou, Y. Liu, Y. Ke, H. Luo, and S. Wen. Generation of Airy vortex and Airy vector beams based on the
+modulation of dynamic and geometric phases. Optics Letters, 40(13):3193–3196, 2015.
+[35] C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi. Transport of intensity phase retrieval and computational
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+25
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf,len=1243
+page_content='Transport Model for the Propagation of Partially-Coherent,  Polarization-Gradient Vector Beams  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Nichols and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Nickel  Naval Research Laboratory  4555 Overlook Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  SW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Washington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 20375  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Bucholtz  Jacobs Technology, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  2551 Dulles View Drive  Herndon, VA 20171  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Rohde  University of Virginia  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  of Electrical and Computer Engineering  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  of Biomedical Engineering  415 Lane Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Charlottesville, VA  In a recent work [20], we predicted and experimentally validated a new physical mechanism for  altering the propagation path of a monochromatic beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Specifically, we showed that by properly  tailoring the spatial distribution of the linear state of polarization transverse to the direction of  propagation, the beam followed a curved trajectory in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Here we extend the model to the  partially coherent, polychromatic case by redefining the beam amplitude, phase, and polarization  angle as appropriate statistical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In particular, we propose an entirely new definition  of linear polarization gradient as an average over the third generalized Stokes parameter in the  spatial frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the new model, the beam curvature matches that of our previous work  in the fully coherent case, but is predicted to gradually vanish as the beam loses coherence and  becomes depolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The model also clearly predicts, however, that there does not exist a natural  mechanism in free space or due to atmospheric turbulence that will cause significant depolarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Simulated beam trajectories are shown for varying levels of initial partial coherence and for different  polarization profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' A new class of non-diffracting beams is also suggested by way of example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Lastly, as a byproduct of the derivation we generalize a previously known result concerning the  free-space propagation of generalized Stokes parameters [13] by showing the result also holds for  propagation in inhomogeneous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  INTRODUCTION  Recently we demonstrated that through proper choice of spatial distribution of linear polarization  angle, a coherent, monochromatic beam follows a curved trajectory in free space [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The effect is similar  in magnitude to that of Airy beam bending (both scale as k−2  0  where k0 is the free-space wavenumber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  However, in our approach it is the centroid of the beam that experiences a transverse acceleration  as opposed to the Airy beam case where the main intensity lobe transversely accelerates, but where  the centroid of the beam does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  The physics of the vector beam bending were predicted via a  transport model, whereby the parameters defining the optical field (intensity and phase) were shown to  be governed by the transport of intensity equation (TIE) and an eikonal equation, expressing conservation  of transverse linear momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The model was re-formulated in Lagrangian coordinates in order to solve  for the transverse beam location which, under appropriate choice of polarization gradient, was shown to  bend in the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' These predictions were subsequently verified in experiment [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In this work we extend the model to the more general case of partially coherent light, showing that with  appropriate definitions of optical phase, amplitude, and polarization gradient the model still predicts a  bending effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' For perfectly coherent light, the governing equations reduce to the previously derived  coherent case, while for incoherent light, the effect disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The resulting model also predicts that the  state of linear polarization (including the degree of polarization) will remain unchanged on propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Collectively, these model predictions represent the main contribution of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  A second, but related contribution is to provide a new definition of polarization angle gradient that  is appropriate for modeling partially coherent vector beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Our definition is based on the normalized  integral of the third generalized Stokes parameter and incorporates the degree of polarization directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  While this definition falls naturally from the vector transport equations, it is also consistent with recent  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='03994v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='optics]  10 Jan 2023  work on scalar, partially coherent beams where a similar definition of generalized phase was proposed  [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In what follows, we show that this scalar definition is also easily extended to the vector case and is  also a natural consequence of the transport model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Finally, in the process of deriving this result, we generalize a previously known result from Korotkova  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In that work, the author showed that the transverse integral of the generalized Stokes parameters  were conserved under free-space propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We confirm this result using an entirely different approach  and extend it to the case of propagation in an inhomogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Our approach is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We begin with the wave equation in the general case of a weakly-inhomogenous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  We  make the paraxial assumption via the slowly-varying envelope approximation to obtain Helmholtz  equations for each field component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the instance where the medium warrants a probabilistic  description, we additionally invoke the Markov Approximation [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' To accommodate the probabilistic nature of the partially coherent problem, the fields are  represented in the coherency matrix formalism where each matrix element is an expectation of  a product of finite-time Fourier transforms [24] of field components at two different transverse  spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Importantly, we retain the dependence on spatially-separated points throughout  the calculation in the form of Wigner Transforms with respect to the transverse spatial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  This information is vital in our model as the state of polarization is not uniform in the transverse  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We obtain a vector transport equation for the generalized Stokes parameters which is the partially  coherent, vector counterpart to the transport of intensity equations obtained in our earlier paper  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' It is entirely reasonable to expect that, in the case of partial polarization, individual com-  ponents of the Stokes 4-vector must obey continuity equations while, in the fully-polarized case,  continuity of the intensity is sufficient to characterize beam propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We extend the method of moments developed for the scalar problem [7, 18] to the vector case,  leading to the coupled transport equations describing the evolution of expected amplitude, phase,  and polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Importantly, the model reduces to our monochromatic, coherent model in  those respective limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  The resulting model extends our prior coherent model variables (phase, polarization angle, and intensity)  to the partially coherent case by taking appropriate averages over the Wigner spatial frequency and with  these new definitions, predict the expected beam path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  TRANSPORT MODELING OF PARTIALLY COHERENT, POLARIZED LIGHT  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Spectral Coherency Representation  We start with the vector wave equation for electric-field vector E(x, t)  ∇2E(x, t) − ϵ(x)  c2  0  ∂2E(x, t)  ∂t2  = 0  (1)  which presumes an inhomogeneous, isotropic medium characterized by dimensionless dielectric constant  ϵ(x) > 1, where x denotes position in three-dimensional Cartesian space, where c0 = 1/√ϵ0µ0 is the  speed of light in vacuum, and where ϵ0 and µ0 are the permittivity and permeability, respectively, of free  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We assume the inhomogeneity is weak and define  ϵ1(x) = ϵ(x) − ⟨ϵ⟩  ⟨ϵ⟩  << 1  (2)  where ⟨ϵ⟩ is the average value of ϵ(x) taken over the entire region of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  We assume the wave propagates in the +z-direction and we model the electric field vector as a sta-  tionary random process, existing for times −T ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Since (1) is linear in E(x, t), we may consider a  superposition of monochromatic, transverse-wave solutions  E(x, t) = 1  2π  � ∞  −∞  {EX(x, ω)T , EY (x, ω)T , 0} e−ik0zeiωtdω,  −T ≤ t ≤ T  (3)  2  represented here by the Fourier integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Phase accumulation in the direction of propagation is captured  by the e−ik0z term with wavenumber k0 = (ω/c0) ⟨ϵ⟩1/2 = (2π/λ) ⟨ϵ⟩1/2 at free-space wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The  quantities {EX(x, ω)T , EY (x, ω)T , 0}e−ik0z are therefore the associated Fourier amplitudes at frequency  ω and possess units of electric field per Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Given the above construction, these are given by  e−ik0zEX,Y (x, ω)T =  � T  −T  EX,Y (x, t)e−iωtdt  (4)  as discussed in [24] (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The explicit dependence of the Fourier amplitudes on T will be retained  for the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The electric field and its Fourier amplitudes are distinguished by their arguments (t  and ω resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Substitute (3) into (1) and make the slowly-varying envelope approximation in which it is assumed  that changes in the amplitude of the field over z−distances of a wavelength are small compared to the  amplitude itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Then |∂zz(·)| ≪ |k0∂z(·)| and the net result is that spatial gradients in x become  gradients only in the transverse plane leading to the parabolic equations  �  −i2k0∂z + ∇2  X + k2  0ϵ1(⃗x, z)  �  {EX(⃗x, z, ω)T , EY (⃗x, z, ω)T } = 0  �  i2k0∂z + ∇2  X + k2  0ϵ1(⃗x, z)  �  {E∗  X(⃗x, z, ω)T , E∗  Y (⃗x, z, ω)T } = 0  (5)  which must be satisfied by each vector component of Fourier amplitudes and their complex conjugates  (denoted with ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Here we have denoted the position in the transverse plane as the vector ⃗x ≡ {x, y} to  distinguish it from the propagation direction z (x ≡ {⃗x, z}), and have denoted the transverse gradient  operator ∇X(·) ≡ {∂x(·), ∂y(·)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Note that (5) comprises four separate equations in, respectively, EX, EY , E∗  X, E∗  Y which we will call (a-  d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Now perform the following operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Multiply (a) and (b) evaluated at ⃗x1 by E∗  X(⃗x2, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Multiply  (a) and (b) evaluated at ⃗x1 by E∗  Y (⃗x2, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Multiply (c) and (d) evaluated at ⃗x2 by EX(⃗x1, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Multiply (c) and (d) evaluated at ⃗x2 by EY (⃗x1, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Note that in each of the above operations all four  multiplications are performed from the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The results, in order, are eight equations which we denote  (i)-(viii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  If we then subtract (v) from (i), (vii) from (ii), (vi) from (iii), and (viii) from (iv) we find (taking into  account the chain rule for the derivatives in z):  �  i2k0∂z +  �  ∇2  X2 − ∇2  X1  �  + k2  0 (ϵ1(⃗x2, z) − ϵ1(⃗x1, z))  �  Ei(⃗x1, z, ω)T E∗  j (⃗x2, z, ω)T =0, i, j ∈ {X, Y }  (6)  where the notation ∇2  Xl indicates that the Laplacian operation is to be performed as a function of the  transverse variable designated ⃗xl for l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Taking the expected value of (6) over realizations of the  electric fields and performing the limiting operation  �  Ei(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  j (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  ≡ lim  T →∞  �  E  �Ei(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)T E∗  j (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)T  2T  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' j ∈ {X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Y }  (7)  then yields  i2k∂zWij(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) +  �  ∇2  X2 − ∇2  X1  �  Wij(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  + k2  0 [ϵ1(⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) − ϵ1(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)] Wij(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' j ∈ {X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Y }  (8)  where each Wij(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) is an element of the spectral density matrix  Wij ∈ W(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) =  � ⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ ⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩  ⟨EY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ ⟨EY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩  �  (9)  and possesses units of electric field squared per Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Note that multiplying (9) by ϵ0/2 gives W(⃗x1, ⃗x2, z, ω)  in units of power spectral density in Watts/Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Equation (8) treats the medium properties as deterministic, such as propagation through a medium  of known refractive index profile, in which case it is common to set ⟨ϵ⟩ = 1 and ϵ1(⃗x, z) = n2(⃗x, z) − 1  where n(⃗x, z) is the refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Alternatively, we can assume the medium is described by its  statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Treating ϵ1(⃗x, z) as a zero-mean, Gaussian random variable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', as in a tur-  bulent atmosphere) requires averaging over both the field amplitudes and the medium properties in  taking the expectation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Define these properties via the covariance ⟨ϵ1(⃗x2, z)ϵ1(⃗x1, z′)⟩ =  �  R3 SNN(⃗ξ, κ)ei⃗ξ·(⃗x2−⃗x1)+iκ(z−z′)d⃗ξdκ , that is, as the inverse Fourier Transform of the three-dimensional  3  spectral density associated with fluctuations in the dielectric constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  We then invoke the well-  established Markov Approximation (MA) of Tatarskii [30], [9] (see extension to the vector electric field  case in [5]) which assumes the medium is “delta” correlated in the direction of propagation so that  ⟨ϵ1(⃗x2, z)ϵ1(⃗x1, z′)⟩ ≈ δ(z − z′)A(⃗x2 − ⃗x1, z′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Under the MA, Tatarskii [30] showed that the required  average in (6) can be written  �  [ϵ1(⃗x2, z) − ϵ1(⃗x1, z)] Ei(⃗x1, z, ω)T E∗  j (⃗x2, z, ω)T  �  ≈ ik0  2 [A(0, z) − A(⃗x2 − ⃗x1, z)] Wij(⃗x1, ⃗x2, z, ω)  (10)  so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (8) becomes  i2k∂zWij(⃗x1, ⃗x2, z, ω) +  �  ∇2  X2 − ∇2  X1  �  Wij(⃗x1, ⃗x2, z, ω)  + ik3  0  2 [A(0, z) − A(⃗x2 − ⃗x1, z)] Wij(⃗x1, ⃗x2, z, ω) = 0  (11)  As we will show, the choice of either a deterministic (8) or a stochastic (11) medium can be easily  accommodated in the transport model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Also note that the last term in both (8) and (11) possess units  of [m] so that kA(·) is dimensionless, as is ϵ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Note that in [5], the approximation ϵ1 ≈ 2η is invoked  which would have led to the pre-factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (10) being written as 2ik0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Equation (8) or (11) therefore contain four separate equations, each governing a different element of  the spectral density matrix (see also Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Note that at the spatial location ⃗x1 = ⃗x2 ≡ ⃗x  the expressions ⟨Ei(⃗x1, z, ω)E∗  j (⃗x2, z, ω)⟩ reduce to the more familiar power spectral density functions  Sij(⃗x, z, ω) =  �  Ei(⃗x, z, ω)E∗  j (⃗x, z, ω)  �  , i, j ∈ {X, Y }  (12)  that is, Sij(⃗x, z, ω) = Wij(⃗x, ⃗x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Generalize Stokes Parameter Representation  It will be convenient to describe propagation in terms of generalized Stokes parameters [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Define  s0(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = ⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ + ⟨EY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩  = WXX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + WY Y (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  s1(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = ⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ − ⟨EY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩  = WXX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) − WY Y (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  s2(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = 2Re{⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩} = ⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ + ⟨EY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩  = WXY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + WY X(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  s3(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = −2Im{⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩} = i (⟨EX(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ − ⟨EY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩)  = i (WXY (⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) − WY X(⃗x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (13)  Unlike conventional Stokes parameters which are evaluated at a single point in space, these generalized  parameters depend on the field relationships at two points ⃗x1 and ⃗x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As ⃗x2 → ⃗x1 ≡ ⃗x we recover the  standard definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' If the above holds, we can also write  W(⃗x1, ⃗x2, z, ω) = 1  2  � s0(⃗x1, ⃗x2, z, ω) + s1(⃗x1, ⃗x2, z, ω) s2(⃗x1, ⃗x2, z, ω) − is3(⃗x1, ⃗x2, z, ω)  s2(⃗x1, ⃗x2, z, ω) + is3(⃗x1, ⃗x2, z, ω) s0(⃗x1, ⃗x2, z, ω) − s1(⃗x1, ⃗x2, z, ω)  �  (14)  which matches the result of [22] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='54) and [26] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5), the only difference being the sign  convention chosen for exp (iωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Wigner Representation  Because (8,11) are linear in Wij(⃗x1, ⃗x2, z, ω) we can consider still other forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' For example, (8) is also  seen to govern the generalized Stokes parameters,  �  i2k0∂z +  �  ∇2  X2 − ∇2  X1  �  + k2  0 (ϵ1(⃗x2) − ϵ1(⃗x1))  �  sν = 0,  ν = 0, · · · , 3  (15)  4  where sν = sν (⃗x1, ⃗x2, z, ω) and where we have omitted the arguments for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Applying the coordi-  nate transformation ⃗x1,2 → ⃗x ∓ ⃗x ′/2 and noting that, in this new system the operator (∇2  X2 − ∇2  X1) →  2∇X · ∇X′ [23], we have  �  i2k0∂z + 2(∇X · ∇X′) + 2k2  0Φ(⃗x, ⃗x ′, z)  �  s′  ν = 0,  ν = 0, · · · , 3  (16)  where we again use the shorthand notation s′  ν = s′  ν(⃗x, ⃗x ′, z, ω) and where  Φ(⃗x, ⃗x ′, z) = 1  2  �  ϵ1(⃗x + ⃗x ′  2 , z) − ϵ1(⃗x − ⃗x ′  2 , z)  �  (17)  will be referred to as the Wigner potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In free space Φ(⃗x, ⃗x′, z) = 0 while in the case of a  stochastic medium, the Wigner potential of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (16) becomes  Φ(⃗x, ⃗x′, z) = ik0  4 [A(0, z) − A(⃗x′, z)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (18)  In these differential coordinates the spectral coherency matrix then becomes  Wij(⃗x, ⃗x ′, z, ω) ≡  �  Ei  �  ⃗x − ⃗x ′  2 , z, ω  �  E∗  j  �  (⃗x + ⃗x ′  2 , z, ω  ��  , i, j = X, Y  (19)  Equation (16) matches that of Charnotskii [5] (Equation 22 in the cited work) and states that the  generalized Stokes parameters obey a parabolic equation of the same basic structure as the complex  electric field amplitude in the deterministic, coherent case (see also, [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Now consider the spatial Fourier Transform of (19) with respect to the variable ⃗x ′ and the resulting  Fourier Transform pair  �  Wij(⃗x, ⃗ξ, z, ω) =  �  R2 Wij(⃗x, ⃗x ′, z, ω)e−i⃗ξ·⃗x′d⃗x ′  Wij(⃗x, ⃗x ′, z, ω) =  � 1  2π  �2 �  R2  �  Wij(⃗x, ⃗ξ, z, ω)ei⃗ξ·⃗x ′d⃗ξ  (20)  where the hat �  W denotes the spatial Fourier transform, the operator  �  Rn denotes the n−dimensional  integral over the space of real numbers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=',  �  R2 ≡  � ∞  −∞  � ∞  −∞), and where the differential element in  transverse real space and reciprocal 2-space are denoted by, respectively, d⃗x ′ ≡ dx′dy′ and d⃗ξ ≡ dξxdξy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Note that, unlike in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (3), we do not need to restrict the limits of integration in defining the Fourier  Transform as the beam spatial correlations are assumed to be naturally limited in transverse extent, that  is |Wij(⃗x, ±∞, z, ω)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The quantity �  Wij(⃗x, ⃗ξ, z, ω) is thus the Wigner transform [32] associated with  complex arguments Ei(⃗x, z, ω), Ej(⃗x, z, ω), i, j ∈ X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In the expressions (20),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Wij(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) is the spatial covariance between the two field components i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' j  as a function of separation ⃗x ′ in the transverse plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' at transverse position ⃗x and downrange position  z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' and within a small optical spectral range {ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω + dω},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' while �  Wij(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) is the covariance between  Fourier amplitudes of electric field components i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' j associated with transverse spatial frequency ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' at  downrange position z and within a small optical spectral range {ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω + dω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Importantly, this latter  quantity is proportional to the average phase difference between field components i, j with transverse  separation ⃗x′ and corresponding spatial frequency ⃗ξ (see Priestly section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='1 [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This point will be  re-visited in section (III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  We replace the spectral density matrix in coordinates ⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x ′ with its Wigner representation to obtain  i2k0  � 1  2π  �2  ∂z  �  R2 �sν  ′ei⃗ξ·⃗x′d⃗ξ + 2(∇X · ∇X′)  � 1  2π  �2 �  R2 �sν  ′ei⃗ξ·⃗x′d⃗ξ + 2k2  0Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)sν  ′ = 0  � 1  2π  �2 �  R2  �  ∂z�sν  ′ + 1  k0  (⃗ξ · ∇X)�sν  ′ −  �  R2 ik0Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)sν  ′e−i⃗ξ·⃗x′d⃗x′  �  ei⃗ξ·⃗x′d⃗ξ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ν = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 3  (21)  where now the generalized Stokes parameters sν ′ have been replaced by their respective spatial Fourier  transforms �sν ′ with respect to the differential coordinate ⃗x ′ and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' are written in terms of the  5  Wigner distribution functions  �s0  ′(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = �  WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + �  WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �s1  ′(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = �  WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) − �  WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �s2  ′(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = �  WXY (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + �  WY X(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �s3  ′(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = i  �  �  WXY (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) − �  WY X(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  (22)  which comprise the same result as Luis’ [16] Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Recognizing that the term in square brackets  must be zero in order to satisfy (21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' we obtain the vector transport equations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  ∂z �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + 1  k0  (⃗ξ · ∇X) �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) =  �  R2 ik0Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)e−i⃗ξ·⃗x′d⃗x′  = − ik0  (2π)2  �  R4 Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)ei(⃗ξ′−⃗ξ)·⃗x′d⃗ξ′d⃗x′  = − ik0  (2π)2  �  R2 Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ − ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)d⃗ξ′  (23)  where the 4-vector �  S ≡ {�s0 ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' �s1 ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' �s2 ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' �s3 ′}T and S is the associated inverse FT with respect to wavevector  ⃗ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Equation (23) thus denotes four separate transport equations, each governing one component �s ′  ν, ν =  0 · · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The term involving the vector potential can be written in several useful forms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' the second line  and matches the results of [33] (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='4) and [12] (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='1), albeit with a different convention for the  Fourier Transform, while the final convolution form is found in [4] (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Expression (23) is the fundamental vector transport equation that governs the evolution of the gener-  alized Stokes parameters in the direction of propagation, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the scalar case (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', analysis of the 1D  Schr¨odinger equation [33] or scalar optical wave propagation [10], [23]), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (23) is often taken as the  starting point in predicting the evolution of �  WXX(⃗x, ⃗ξ, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Here we have extended these prior works  to the full vector electric field in order to account for spatially dependent polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Related works on  the full vector transport equations for the Wigner distribution include [26] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='6), and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' However  we believe this work to be the first extension of such analyses to partially polarized optical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  PARTIALLY-COHERENT, VECTOR TRANSPORT OF INTENSITY  In this section we extend the transport-of-intensity equations (TIEs) from the partially coherent scalar  case, as described in [23, 35], to the partially-coherent vector case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' To arrive at such a model, we follow  an approach used in quantum mechanics for obtaining the so-called “hydrodynamic” model, so named  for the resemblance of the model to the continuity and momentum equations commonly found in fluid  mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Specifically, we take “moments” of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (23) with respect to the wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The first two  such moments are obtained by multiplying (23) by unity and ⃗ξ respectively, and then integrating over the  wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The resulting expressions govern the conservation of mass (intensity), momentum (phase),  and energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This approach has been used in analysis of the 1-D Schr¨odinger equation (see  [4, 7, 12]) and has only very recently been extended to the 2-D Pauli equation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' While the dominant  application area has been quantum mechanics, Marcuvitz [18] has suggested the approach as a general  technique for studying wave propagation in the scalar case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In this work, we extend this “method of moments” to the vector transport equation (23) for partially  coherent propagation problems in optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The first such conservation equation is obtained by directly  integrating (23) (that is, taking the “zeroth” moment) which yields  ∂z  �  R2  �  S(⃗x, ⃗ξ, z, ω)d⃗ξ + ∇X · 1  k0  �  R2  ⃗ξ �  S(⃗x, ⃗ξ, z, ω)d⃗ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (24)  To see that the Wigner potential term integrates to zero,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' replace the generalized Stokes parameters by  their inverse spatial Fourier transforms (in frequency variable ⃗ξ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' in which case integrating the right  6  hand side of (23) over wavevector ⃗ξ can be carried out as  =  �  R6  ik0  (2π)2 �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)ei⃗ξ ′·⃗x′Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)e−i⃗ξ·⃗x ′d⃗x ′d⃗ξ ′d⃗ξ  =  ik0  (2π)2  �  R4  �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)ei⃗ξ ′·⃗x ′d⃗ξ ′  �  R2 e−i⃗ξ·⃗x ′d⃗ξΦ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x ′)d⃗x ′  =  ik0  (2π)2  �  R4  �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)ei⃗ξ ′·⃗x ′d⃗ξ ′δ(⃗x ′)Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x ′)d⃗x ′  =  ik0  (2π)2 Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 0)  �  R2  �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)d⃗ξ ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (25)  In the last step we have leveraged the definition of Φ(⃗x, ⃗x ′, z) (17,18) which holds that when ⃗x′ = 0, the  potential vanishes, that is, Φ(⃗x, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This statement is true whether we treat the dielectric constant  fluctuations as a deterministic or a random process under the Markov approximation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Finally, to  put (24) in the same form as the coherent Transport of Intensity Equation (TIE) [20], we require the  definitions  ρ(⃗x, z, ω) ≡  �  R2  �  S(⃗x, ⃗ξ, z, ω)d⃗ξ  1  k0  �  R2  ⃗ξ �  S(⃗x, ⃗ξ, z, ω)d⃗ξ ≡ ρ(⃗x, z, ω) ◦ ⃗v(⃗x, z, ω)  (26)  where ◦ denotes the Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Equation (26) therefore defines four generalized intensities  ρν(⃗x, z, ω) ≡  �  R2 �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ, ν = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In addition, (26) implicitly defines each element of  ⃗v(⃗x, z, ω) as  ⃗vν(⃗x, z, ω) ≡  �  R2 ⃗ξ �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ  k0ρν(⃗x, z, ω)  =  �  R2 ⃗ξ �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ  k0  �  R2 �sν ′(⃗x, ⃗ξ, z, ω)d⃗ξ  , ν = 0, 1, 2, 3  (27)  in which case (24) becomes  ∂zρ(⃗x, z, ω) + ∇X · [ρ(⃗x, z, ω) ◦ ⃗v(⃗x, z, ω)] = 0,  (28)  which is the vector TIE for partially coherent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Note that whereas in our prior analysis [20], the above  expression was a single scalar equation, in this case both ρ(⃗x, z, ω) and the product ρ(⃗x, z, ω) ◦⃗v(⃗x, z, ω)  are column vectors each comprising four elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus ρν and vν represent, respectively, the ν−th  component of the Stokes 4-vector (ν = 0, 1, 2, 3) and, as we will show in section (III C), the rate of  change with z of the transverse location of the ν−th component of the Stokes 4-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Hence, all eight  components ρν and vν are real, the total beam intensity ρ0 must be non-negative, and the remaining  seven components can be positive, negative, or zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Just as for Stokes parameters for monochromatic  light, the components of the Stokes 3-vector (ρ1, ρ2, ρ3) are proportional to intensity but are not limited  to non-negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Mixture Model for Partially Coherent Stokes Parameters  Equations (28) are transport equations governing each of the generalized Stokes parameters during  propagation in a (possibly) inhomogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' By definition, the Stokes parameters were defined  as an expectation over realizations of the X, Y components of the electric field Fourier amplitudes,  EX(⃗x, z, ω), EY (⃗x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Both the quantity being transported, ρ(⃗x, z, ω) and the “velocity” of that  transport, ⃗v(⃗x, z, ω) carry simple physical interpretations when placed in the context of a particular  model for the electric field as will be shown next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  To aid in this interpretation, recall the electric field “transverse wave” model (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Until now we have  imposed no further model structure on the form of EX(⃗x, z, ω), EY (⃗x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Leveraging prior work [20]  for linearly-polarized, coherent beams, let the complex Fourier amplitudes take the form  EX(⃗x, z, ω) = ρ1/2  m (⃗x, z, ω)eiφm(⃗x,z,ω) cos(γm(⃗x, z, ω))  EY (⃗x, z, ω) = ρ1/2  m (⃗x, z, ω)eiφm(⃗x,z,ω) sin(γm(⃗x, z, ω))  (29)  7  where the subscript m denotes “model”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The electric field is therefore described by a magnitude, phase,  and spatially dependent linear polarization angle γm(⃗x, z, ω) which governs the ratio of Fourier amplitudes  in the X and Y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' It must be noted, however, that the model structure (29) presumes a polarized  beam and should therefore be viewed as conditional on the beam being completely polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Such a  model is clearly not appropriate for unpolarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  While the first Stokes parameter �s0 is independent of the state of polarization, the same cannot be  said of �sν, ν = 1 · · · 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' For partially polarized light it is therefore appropriate to use the mixture model  [29]  �  �  �  �  s′  0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  s′  1(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  s′  2(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  s′  3(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  �  �  � = [1 − P]  �  �  �  �  s′  0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  0  0  0  �  �  �  � + P  �  �  �  �  s′  0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  s′  1(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω|P)  s′  2(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω|P)  s′  3(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω|P)  �  �  �  �  (30)  where the weighting is given by the Degree Of Polarization (DOP)  P(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) =  �  (s1)2 + (s2)2 + (s3)2  s0  ���  ⃗x1=⃗x2=⃗x =  �  1 − 4Det [W]  Tr [W]2  ���  ⃗x1=⃗x2=⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (31)  Equation (31) thus defines a positive quantity 0 ≤ P(⃗x, z, ω) ≤ 1 which specifies the fraction of the beam  intensity at location {⃗x, z} and within an infinitesimal range of optical frequencies {ω, ω + dω} that is  fully polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The degree to which a beam is polarized is therefore directly related to coherence among  the components of the optical field at a given transverse location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We note also that while P(⃗x, z, ω)  varies spatially and with frequency, as defined it does not depend on the separation ⃗x ′ between two  points on the beam face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Given the above definitions, we can better understand the quantities in the partially coherent TIE  (28) as will be described next in sections (III B) and (III C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' These definitions will also allow us to define  a polarization gradient in the partially coherent case as will be shown in section (III D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Interpretation of the Partially Coherent Vector TIE  To understand the meaning of ρ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) we expand the first component of �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) which is just  �s0 ′(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) and use the fact that the Wigner distributions can be expressed as Fourier transforms to  obtain  ρ0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) =  �  R2 �s0  ′(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)d⃗ξ =  �  R4 (WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)) e−i⃗ξ·⃗x′d⃗x ′d⃗ξ  =  �  R2 (WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)) δ(⃗x ′)d⃗x ′  = SXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + SY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  (32)  where we have invoked the definition of the delta function as the Fourier Transform of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Recall also  that by definition, in transformed coordinates, at ⃗x ′ = 0,  Wij(⃗x, 0, z, ω) = Sij(⃗x, z, ω), i, j = X, Y  (33)  so that the first component of ρ(⃗x, z, ω) is by definition the auto-spectral density of the field at transverse  location ⃗x, z as demonstrated by (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' A similar analysis of the other components of ρ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) shows  that the full four-component Stokes vector is given by  ρ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) =  �  R2  �  �  �  �  �  �s′  0  P �s′  1  P �s′  2  P �s′  3  �  �  �  �  � d⃗ξ =  �  �  �  �  SXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + SY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  P (SXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) − SY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω))  P (SXY (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + SY X(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω))  iP (SXY (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) − SY X(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω))  �  �  �  � =  �  �  �  �  s0  Ps1  Ps2  Ps3  �  �  �  �  ⃗x1=⃗x2≡⃗x  (34)  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' the quantities being transported can be written simply as sums and differences of the transverse  spectral densities that is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' the generalized Stokes parameters for partially coherent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  8  For example, substituting the model (29) for the first Stokes parameter  s0 = ⟨EX(⃗x, z, ω)E∗  X(⃗x, z, ω)⟩ + ⟨EY (⃗x, z, ω)E∗  Y (⃗x, z, ω)⟩  =  �  ρm(⃗x, z, ω) cos2(γm(⃗x, z, ω) + ρm(⃗x, z, ω) sin2(γm(⃗x, z, ω)  �  = ⟨ρm(⃗x, z, ω)⟩ ≡ ρ0(⃗x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (35)  Hence the first generalized Stokes parameter ρ0(⃗x, z, ω) is proportional to the expected beam spectral  intensity at location {⃗x, z} in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Performing the same substitution for the remaining components  in (34) we have  ρ(⃗x, z, ω) =  �  �  �  �  ⟨ρm(⃗x, z, ω)⟩  P(⃗x, z, ω) ⟨ρm(⃗x, z, ω) cos(2γm(⃗x, z, ω))⟩  P(⃗x, z, ω) ⟨ρm(⃗x, z, ω) sin(2γm(⃗x, z, ω))⟩  0  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (36)  The model structure (29) suggests a simple and well-known interpretation of the generalized Stokes  parameters for linearly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  However, for unpolarized light only the first component of  ρ(⃗x, z, ω) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The final intensity component is given by ρ3(⃗x, z, ω) = 0 due to the fact that we  are considering only linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Interpretation of the Transport Velocity  The components of ⃗v(⃗x, z, ω) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (28) are referred to as “velocities” as they were noted in the  coherent case to govern the change in optical path in the transverse direction per unit change in the  direction of propagation [11, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We present a basic derivation supporting this interpretation in Appendix  (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This terminology is also in keeping with the wave mechanics interpretation of the transport equation,  where the “transport” of intensity occurs in the transverse plane as the beam progresses in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The first  velocity term in this partially coherent case is obtained via (27),  ⃗v0(⃗x, z, ω) = 1  k0  �  R2 ⃗ξ �s′  0(⃗x, ⃗ξ, z, ω)d⃗ξ  �  R2 �s′  0(⃗x, ⃗ξ, z, ω)d⃗ξ  =  �  R2 ⃗ξ  �  �  WXX(⃗x, ⃗ξ, z, ω) + �  WY Y (⃗x, ⃗ξ, z, ω)  �  d⃗ξ  k0ρ0(⃗x, z, ω)  (37)  The non-negativity of �s′  0 allows us to interpret (37) as an average spatial frequency in the transverse  plane (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', Boashash [2]) normalized by k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In fact, in [2] it was noted that the generalized definition  of frequency is an average change in phase per unit change in the independent variable, which in the  present context corresponds to the transverse distance ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus, (37) implicitly defines a phase gradient  ∇Xφ(⃗x, z, ω) ≡  �  R2 ⃗ξ �s′  0(⃗x, ⃗ξ, z, ω)d⃗ξ  �  R2 �s′  0(⃗x, ⃗ξ, z, ω)d⃗ξ  (38)  so that ⃗v0(⃗x, z, ω) = k−1  0 ∇Xφ(⃗x, z, ω), which is precisely the definition for velocity found in the monochro-  matic case [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This definition of generalized phase was also used in [34] for scalar, partially coherent  electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Here, we have shown this definition to be appropriate for vector electric fields as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In-  deed, we will show momentarily that these definitions are not only appropriate for transport of intensity,  ρ0(⃗x, z, ω), but for transport of two other generalized Stokes parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Now consider again the mixture model (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Substituting these definitions into (37) yields for the  denominator SXX(⃗x, z, ω)+SY Y (⃗x, z, ω) = ρ0(⃗x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' For the numerator, considering first the polarized  9  portion, we can expand using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (19) to give  �  R2  ⃗ξ  �  �  WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + �  WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  d⃗ξ  =  �  R4  ⃗ξ  ��  EX(⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  +  �  EY (⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  ��  e−i⃗ξ·⃗x′d⃗ξd⃗x′  =  �  R2  ��  EX(⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  +  �  EY (⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �� �  R2  ⃗ξe−i⃗ξ·⃗x′d⃗ξd⃗x′  = i  �  R2  ��  EX(⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  +  �  EY (⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  ��  δ′(⃗x′)d⃗x′  = i  �  R2 ∇X′  ��  EX(⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  +  �  EY (⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  ��  δ(⃗x′)d⃗x′  = i∇X′  ��  EX(⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  X(⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  +  �  EY (⃗x − ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)E∗  Y (⃗x + ⃗x′/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �� �����  ⃗x′→0  = ⟨ρm(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)∇Xφm(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (39)  where the final line substitutes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (29) for EX, EY and takes the required gradient ∇X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the second  to last line we leveraged the identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (7) of [3] concerning integrals over derivatives of the delta  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The end result is that under the mixture model, and assuming amplitude and transverse phase  gradient are statistically independent,  ⟨∇Xφm(⃗x, z, ω)⟩ = ∇Xφ(⃗x, z, ω)  1  k0  ⟨∇Xφm(⃗x, z, ω)⟩ = ⃗v0(⃗x, z, ω)  (40)  The definition (38) therefore can be interpreted as the expected model phase and the corresponding  “velocity” as the expected change in optical path in the transverse direction per unit change in the  direction of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Notably, this result is independent of the DOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We may perform a similar  analysis on the remaining components of ⃗v(⃗x, z, ω), finding that  ⃗v1(⃗x, z, ω) =  �  R2 ⃗ξ �s′  1(⃗x, ⃗ξ, z, ω)d⃗ξ  k0ρ1(⃗x, z, ω)  = P(⃗x, z, ω) ⟨ρm∇Xφm(⃗x, z, ω) cos(2γm(⃗x, z, ω))⟩  k0P(⃗x, z, ω) ⟨ρm cos(2γm(⃗x, z, ω))⟩  = ⃗v0(⃗x, z, ω)  ⃗v2(⃗x, z, ω) =  �  R2 ⃗ξ �s′  2(⃗x, ⃗ξ, z, ω)d⃗ξ  k0ρ2(⃗x, z, ω)  = P(⃗x, z, ω) ⟨ρm∇Xφm(⃗x, z, ω) sin(2γm(⃗x, z, ω))⟩  k0P(⃗x, z, ω) ⟨ρm sin(2γm(⃗x, z, ω))⟩  = ⃗v0(⃗x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (41)  Thus, in the case of linear polarization, all three non-zero generalized Stokes parameters are transported  with the same transverse velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  ∂zρν(⃗x, z, ω) + ∇X · [ρν(⃗x, z, ω)v0(⃗x, z, ω)] = 0, ν = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (42)  This is sensible as it suggests that, in expectation, both the beam intensity and properties related to the  state of linear polarization are moving together in the transverse plane during propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Importantly,  this also suggests that while the transport model requires an equation governing ⃗v0 (see section IV), it  does not require separate equations for ⃗v1, ⃗v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  It it worth mentioning the relationship between the velocity vector and the well-known Poynting vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Begin by noting that the numerator in (27) is of the form of the average Poynting vector for partially  coherent light,  P(⃗x, z) ≡ 1  k0  �  R2  ⃗ξ �s′  0(⃗x, ⃗ξ, z, ω)d⃗ξ,  (43)  so that ˜P(⃗x, z) ≡ ⃗v(⃗x, z, ω) is, in fact, the normalized Poynting vector as defined in [21] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 8 of the  cited work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Also in [21] is was suggested that the normalized Poynting vector be decomposed via the  Helmholtz decomposition theorem as  ⃗v(⃗x, z, ω) ≡ ⃗vS(⃗x, z, ω) + ⃗vV (⃗x, z, ω)  = k−1  0 ∇XφS(⃗x, z, ω) + k−1  0 ∇X × ⃗φV (⃗x, z, ω)  (44)  10  where S and V refer to scalar and vector contributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  If we let the scalar phase be  φS(⃗x, z, ω) = φm(⃗x, z, ω) and define  φV (⃗x, z, ω) =  �dγ(⃗x, ω)  dy  z, −dγ(⃗x, ω)  dx  z, 0  �  (45)  then we have that ⃗Ω(⃗x, z, ω) = [∇ × φV (⃗x, z, ω)]X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The partially coherent vector TIE would then be  written  ∂zρ(⃗x, z, ω) + ∇X · [ρ(⃗x, z, ω)⃗vS(⃗x, z, ω)] + ∇X · [ρ(⃗x, z, ω)⃗vV (⃗x, z, ω)] = 0  (46)  which would clearly suggest that ∇X ·  �  ρ(⃗x, z, ω)⃗Ω(⃗x, z, ω)  �  = 0 given Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In fact, we will show in  the next section that this relationship indeed holds for linearly polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Definition and Interpretation of the Polarization Angle Gradient  Since we are considering a linearly polarized vector beam ρ3(⃗x, z, ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' However, although the first  term in (24) is zero, the second term inside the transverse divergence operator is (following the same  model interpretation and simplification as in (39))  1  k0  �  R2  ⃗ξ �s′  3(⃗x, ⃗ξ, z, ω)d⃗ξ = −  �  ρm(⃗x, z, ω)P(⃗x, z, ω)  k0  ∇Xγm(⃗x, z, ω)  �  = −ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)  (47)  so that by (24) we have the condition  ∇X ·  �  ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)  �  = 0  (48)  as we implied at the end of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This relationship was also found to hold in the coherent  case [20] and will be leveraged later in the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' However, also note that in arriving at (48) we have  produced a definition for the spatial gradient of the polarization angle that is appropriate for partially  coherent light, given by  ⃗Ω(⃗x, z, ω) =  −iP(⃗x, z, ω)  �  R2 ⃗ξ  �  �  WXY (⃗x, ⃗ξ, z, ω) − �  WY X(⃗x, ⃗ξ, z, ω)  �  d⃗ξ  k0ρ0(⃗x, z, ω)  = P(⃗x, z, ω)  k0  ⟨∇Xγm(⃗x, z, ω)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (49)  The DOP is appropriately part of the definition (49) as one cannot define ⃗Ω for an unpolarized beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Note also that in accordance with the works of Salem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' [28] and Korotkova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' [14] we allow that  the DOP can vary in frequency ω, as well as the transverse location ⃗x, and the direction of propagation  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Just as with the normalized phase gradient, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (49) is expressed as a suitable average wavevector over  the distribution �s′  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Similarly, it can be interpreted as a transverse, spatial gradient in the angle (phase)  between the X and Y components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the limit of a fully coherent beam we recover our deterministic  definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Importantly, as the beam de-polarizes we will see the terms involving ⃗Ω(⃗x, z, ω) vanish, a  point we elaborate on in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  We note that a definition of polarization angle for partially coherent beams was proposed previously  by Korotkova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' [25], [13]  θ(⃗x, z, ω) ≡ 1  2 arctan  �  2Re {SXY (⃗x, z, ω)}  SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (50)  This definition is somewhat consistent with our model structure (29) in the sense that substituting the  polarized electric field (29) into (50) gives  θ(⃗x, z, ω) ≡ 1  2 arctan  �s2(⃗x, z, ω)  s1(⃗x, z, ω)  �  = 1  2 arctan  � ⟨sin(2γm(⃗x, z, ω))⟩  ⟨cos(2γm(⃗x, z, ω))⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (51)  However, this definition only yields our model polarization angle in the deterministic case as the ratio of  expectations of the sin, cos terms does not equal the mean of the ratio in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Moreover, the DOP  11  does not appear explicitly in the expression as it does in (49) and instead must be viewed as a part of  the quantity SXY (⃗x, z, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  More importantly, as we will see in section (IV), the quantity that governs the transverse movement  of intensity is the polarization angle gradient as opposed to the angle itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As we have just shown,  the gradient is obtained via the first moment, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The resulting expression (49) accounts for the  spatial frequencies ⃗ξ and yields directly the expected value of polarization angle gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' On the other  hand, differentiating (50) with respect to the transverse coordinates yields  ∇Xθ(⃗x, z, ω)  = [SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)]∇XSXY (⃗x, z, ω) − SXY (⃗x, z, ω)∇X[SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)]  4SXY (⃗x, z, ω)2 + [SXX(⃗x, z, ω) − SY Y (⃗x, z, ω)]2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (52)  As with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (51), this definition of phase gradient results in products and sums of expectations of sin, cos  functions so that the result equals ⟨∇Xγm(⃗x, z, ω)⟩ only in the deterministic case where no averaging  operations are performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Finally, we note that our definition (49) arose quite naturally as a consequence of 1) the paraxial  transport equation for the generalized Stokes parameters, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (24) and 2) the model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (29) governing  the expected amplitude, phase, and polarization angle of the electric field at location (⃗x, z) and frequency  ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The definition is also consistent with the definition of partially coherent phase used in the literature  (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 38), and will therefore be used in what follows for the expected polarization angle gradient of our  vector beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Conservation of the generalized Stokes parameters  To conclude this section, we point out an interesting result concerning the conservation of the gener-  alized Stokes parameters during propagation through a (possibly) inhomogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Since each  of the parameters sν obey the same transport equation (28), the integrals of these quantities in the  transverse plane are constant and are therefore conserved during propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' To see that this is true,  expand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (28), recall the definition of the material derivative D(·)/Dz ≡ ∂z(·) + (⃗v · ∇X)(·), and  then convert to Lagrangian coordinates whereby the spatial coordinates ⃗x are written as functions of z  and the initial value, that is, ⃗x → ⃗xz(⃗x0) (which we will abbreviate as simply ⃗xz and the corresponding  gradient ∇Xz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' These steps can be written  ∂zρν(⃗x, z, ω) + ∇Xρν(⃗x, z, ω) · ⃗vν(⃗x, z, ω) + ρν(⃗x, z, ω) [∇X · ⃗vν(⃗x, z, ω)] = 0  Dρν(⃗x, z, ω)  Dz  + ρν(⃗x, z, ω) [∇X · ⃗vν(⃗x, z, ω)] = 0  dρν(⃗xz, ω)  dz  + ρν(⃗xz, ω) [∇Xz · ⃗vν(⃗xz, ω)] = 0,  ν = 0, 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (53)  The resulting ordinary differential equation possesses the solution  ρ(⃗xz, ω) = ρ(⃗x0, ω) exp  �  −  � z  s=0  ∇Xs · ⃗v(⃗xs, ω)d⃗xs  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (54)  Therefore, the integral of the generalized Stokes parameters at any point along the propagation path is  related to their initial values via the divergence of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' It can also be shown (see Appendix  B, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' [6]) that the expression (54) can be written alternatively as  ρ(⃗xz, ω) = det |J⃗x0(⃗xz)|−1ρ(⃗x0, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (55)  where the Jacobian J⃗x0(⃗xz) is the derivative of the coordinate functions ⃗xz(⃗x0, z) with respect to the  fixed (initial) coordinates ⃗x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' If the Lagrangian coordinate mappings ⃗xz(⃗x0, z) are one-to-one, smooth  functions of ⃗x0, this expression can be shown via the change of variables theorem to be equivalent to the  integral formulation [1]  �  X  ρ(⃗xz, ω)d⃗xz =  �  X  ρ(⃗x0, ω)d⃗x0  (56)  12  (see also Villani Chapter 11 [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus, it can be stated that the integral of each of the generalized  Stokes parameters over the transverse plane is conserved on propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Interestingly, this conclusion  was reached by an entirely different approach in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' However, in that work the claim was made with  respect to propagation in free-space whereas here we see the integrals of the Stokes parameters are  conserved even in an inhomogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  MOMENTUM EQUATION FOR PARTIALLY-COHERENT, LINEARLY-POLARIZED  LIGHT  The above discussion defined both intensity and velocity in the general case of a propagating, linearly  polarized, partially coherent optical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Each of the generalized Stokes parameters were shown to be  transported with a single velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In this section, we will derive the equation that governs this velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  As we described in section (II), the process for obtaining the continuity equation can be viewed as  taking the zeroth “moment” of the fundamental transport equation with respect to wavenumber, that is,  integrating 23;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (see again [18], [7], and [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus, to derive the partially coherent momentum equation  we multiply (23) by ⃗ξ and integrate over wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The result is written  ∂z  �  R2  ⃗ξ �  S(⃗x, ⃗ξ, z, ω)d⃗ξ + ∇X · 1  k0  �  R2(⃗ξ ⊗ ⃗ξ) �  S(⃗x, ⃗ξ, z, ω)d⃗ξ =  �  R2  �  R2 ik0⃗ξ �  S(⃗x, ⃗ξ′, z, ω)Φ(⃗x, ⃗ξ − ⃗ξ′, z)d⃗ξ′d⃗ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (57)  The first challenge associated with (57) is the term on the right-hand-side, integration of Wigner poten-  tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This term can be greatly simplified if we are willing to restrict the model to a weakly inhomogeneous  medium where the transverse gradient in dielectric constant is small at a given transverse location ⃗x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In  this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' we explore a series solution for the terms inside the potential function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  ϵ1(⃗x + ⃗x′  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) − ϵ1(⃗x − ⃗x′  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) ≈ ϵ1(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) ± ∇Xϵ1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) · ⃗x′ + · · ·  (58)  so that the potential functions (17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='18) become  Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) ≈ ⃗x′ · ∇X  �1  2ϵ1(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  �  + O(⃗x′3)  Φ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗x′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) ≈ ⃗x′ · ∇X  �  −ik0  4 A(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  �  + O(⃗x′2)  (59)  Using these expressions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' and representing the terms inside the transverse gradient generically as g(⃗x)  (to accommodate a deterministic or stochastic medium),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' we we see that the inner double integral on the  right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (57) can be expanded using the Fourier representation of the potential as  � 1  2π  �2 �  R2  �  R2 ik0 �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  ei(⃗ξ−⃗ξ′)·⃗x′⃗x′ · ∇Xg(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)d⃗x′  �  d⃗ξ′  = −  � 1  2π  �2 �  R2  �  R2 k0 �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  ∇ξ′ei(⃗ξ−⃗ξ′)·⃗x′ · ∇Xg(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)d⃗x′  �  d⃗ξ′  = −  � 1  2π  �2 �  R2 k0 �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)∇ξ′  ��  R2 ei(⃗ξ−⃗ξ′)·⃗x′d⃗x′  �  ∇Xg(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)d⃗ξ′  = −  �  R2 k0∇⃗ξ′ δ(⃗ξ − ⃗ξ′) �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) · ∇Xg(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)d⃗ξ′  = −  �  R2 k0δ(⃗ξ − ⃗ξ′)∇⃗ξ′ �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) · ∇Xg(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)d⃗ξ′  = −k0∇⃗ξ �  S(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) · ∇Xg(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  (60)  where we have used the trick found in [19]( Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='29), which is to note that ∇ξ′  �  ei(⃗ξ−⃗ξ′)·⃗x′�  =  −i⃗x′ei(⃗ξ−⃗ξ′)·⃗x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The expression (60) is the “linearized” Wigner potential (see, for example, [33]) and  transforms Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (57) into  ∂z  �  R2  ⃗ξ �  S(⃗x, ⃗ξ, z, ω)d⃗ξ + ∇X · 1  k0  �  R2(⃗ξ ⊗ ⃗ξ) �  S(⃗x, ⃗ξ, z, ω)d⃗ξ = −k0∇Xg(⃗x, z)  �  R2  �  ⃗ξ · ∇⃗ξ �  S(⃗x, ⃗ξ, z, ω)  �  d⃗ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (61)  13  We have already shown via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (26) that the first term in (61) can be written  ∂z  �  R2  ⃗ξ �s′ν(⃗x, ⃗ξ, z, ω)d⃗ξ = k0∂z [ρν(⃗x, z, ω)⃗v0(⃗x, z, ω)] , ν = 0, · · · , 2  (62)  while for the third Stokes parameter we showed previously in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (47) that  ∂z  �  R2  ⃗ξ �s′  3(⃗x, ⃗ξ, z, ω)d⃗ξ = −k0∂z  �  ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (63)  The integral in the last term can also be simplified via integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Presuming that the Stokes  parameters vanish at the boundaries of integration (as |⃗ξ| → ∞), the result is simply  �  R2  �  ⃗ξ · ∇⃗ξ �  S(⃗x, ⃗ξ, z, ω)  �  d⃗ξ = −ρ(⃗x, z, ω)  (64)  for the first three Stokes parameters (ν = 0, 1, 2), while the integral disappears for the third since, in the  absence of ellipticity, ρ3(⃗x, z, ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' With these simplifications, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (61) becomes  ∂z [ρν(⃗x, z, ω)⃗v0(⃗x, z, ω)] + ∇X · 1  k2  0  �  R2(⃗ξ ⊗ ⃗ξ) �s′ν(⃗x, ⃗ξ, z, ω)d⃗ξ = ρν(⃗x, z, ω)∇Xg(⃗x, z), ν = 0, 1, 2  −∂z  �  ρ0(⃗x, z, ω)⃗Ω(⃗x, z, ω)  �  + ∇X · 1  k2  0  �  R2(⃗ξ ⊗ ⃗ξ) �s′  3(⃗x, ⃗ξ, z, ω)d⃗ξ = 0  (65)  Equation (65) is a vector equation describing the evolution of the “momentum” ρ ◦ ⃗v for each of the  generalized Stokes parameters and it allows us to develop expressions for the unknowns ⃗v0 and ⃗Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As  we noted in section (III C), because there is only a single velocity associated with the first 3 equations,  we need only focus on the expression (65) for ν = 0 and the last expression where ν = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We also note  that the right hand side of the second expression is zero for the linearized potential since ρ3(⃗x, z, ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  As we will show, this leads to the conclusion that the polarization angle gradient will not change on  propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Depolarization during propagation would therefore be a consequence of higher-order terms  in the expansion of the Wigner potential or retention of the electric field divergence in the starting wave  equation (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Recall the continuity equation (28) was obtained as the zeroth moment of the wave-vector with respect  to the Stokes parameters (Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 23) and contained the “velocity”, which is the first moment of the wave-  vector with respect to the Stokes parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In turn, (65) contains the second moment of the wave-vector  with respect to the Stokes parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As might be expected, if we were to construct the expression for  the second moment it would contain the integral of third order wave-vector terms and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This is the  so-called “closure problem” (see [12] Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='3-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5 and surrounding discussion of [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Obtaining closure requires an expression for this second moment (the term involving the outer product)  in terms of already defined quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In our context, this means writing the integral involving the outer  product in terms of ρ and ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This can be accomplished by again leveraging our electric field model (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Note that using this particular field model as an expression to fix the closure problem was employed in  both [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' To see how this can be done, expand the second term in (65) as was done in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Doing  so requires us to expand the Wigner-averaged outer product into its four constituent terms which are of  the two basic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Using the first of equations (65) as an example, the outer product is seen to produce the following four  integrals,  �  R2 ξ2  ii  �  �  WXX(⃗x, ⃗ξ, z, ω) + �  WY Y (⃗x, ⃗ξ, z, ω)  �  d⃗ξ,  i ∈ X, Y  �  R2 ξiξj  �  �  WXX(⃗x, ⃗ξ, z, ω) + �  WY Y (⃗x, ⃗ξ, z, ω)  �  d⃗ξ,  i, j ∈ X, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (66)  Both the unpolarized and polarized portions of the mixture model can be simplified in the same manner  as was done for the velocity terms, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Substituting in the definition of either the unpolar-  ized/polarized Wigner Transforms �  WXX(⃗x, ⃗ξ, z, ω) and �  WY Y (⃗x, ⃗ξ, z, ω), the first of the needed moments  14  in (66) can be simplified as  �  R2 ξ2  XX  �  �  WXX(⃗x, ⃗ξ, z, ω) + �  WY Y (⃗x, ⃗ξ, z, ω)  �  d⃗ξ  = − ∂2  ∂x′2  �  EX  �  ⃗x − ⃗x′  2 , z, ω  �  E∗  X  �  ⃗x + ⃗x′  2 , z, ω  �  + EY  �  ⃗x − ⃗x′  2 , z, ω  �  E∗  Y  �  ⃗x + ⃗x′  2 , z, ω  �� ����� x′ → 0  y′ → 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (67)  Using our model for the electric field, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (29), substituting into the above and taking the derivative  and subsequent limits (and noting the procedure in the “y” direction is the same) yields  �  R2 ξ2  XX  �  �  WXX(⃗x, ⃗ξ, z, ω) + �  WY Y (⃗x, ⃗ξ, z, ω)  �  d⃗ξ = k2  0ρ0(v2  x + Ω2  x) + (∂xρ0)2  4ρ0  − ∂2  xρ0  4  �  R2 ξ2  Y Y  �  �  WXX(⃗x, ⃗ξ, z, ω) + �  WY Y (⃗x, ⃗ξ, z, ω)  �  d⃗ξ = k2  0ρ0(v2  y + Ω2  y) + (∂yρ0)2  4ρ0  − ∂2  yρ0  4  (68)  where vx,y and Ωx,y are, respectively, the “x” and “y” components of those vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In this and several  subsequent expressions we omit the arguments (⃗x, z, ω) from the model quanitities for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  The  “mixed” averages follow a similar procedure and are found to be  �  R2ξXξY  �  �  WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + �  WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  d⃗ξ = k2  0ρ0(vxvy + ΩxΩy) + ∂xρ0∂yρ0  4ρ0  − ∂xyρ0  4  (69)  so that we may finally write  1  k2  0  �  R2  �  ⃗ξ ⊗ ⃗ξ  � �  �  WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + �  WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  d⃗ξ =  ρ0⃗v ⊗ ⃗v + ρ0⃗Ω ⊗ ⃗Ω +  1  4k2  0  �  �  (∂xρ0)2  ρ2  0  − ∂2  xρ0  ρ0  ∂xρ0∂yρ0  ρ2  0  − ∂xyρ0  ρ0  ∂xρ0∂yρ0  ρ2  0  − ∂xyρ0  ρ0  (∂yρ0)2  ρ2  0  −  ∂2  yρ0  ρ0  �  � ρ0  (70)  Repeating this same process for the third Stokes parameter (second equation in (65)) yields the equations  ∂z  � ρ0⃗v  −ρ0⃗Ω  �  +  �  � ∇X ·  �  ρ0⃗v ⊗ ⃗v + ρ0R + ρ0⃗Ω ⊗ ⃗Ω  �  −∇X ·  �  ρ0⃗v ⊗ ⃗Ω + ρ0⃗Ω ⊗ ⃗v  �  �  � =  � ρ0∇Xg(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  0  �  (71)  where we have defined the matrix  R =  1  4k2  0  �  �  (∂xρ0)2  ρ2  0  − ∂2  xρ0  ρ0  ∂xρ0∂yρ0  ρ2  0  − ∂xyρ0  ρ0  ∂xρ0∂yρ0  ρ2  0  − ∂xyρ0  ρ0  (∂yρ0)2  ρ2  0  −  ∂2  yρ0  ρ0  �  �  (72)  A convenient simplification occurs upon expansion of the outer product via ∇·( ⃗B ⊗ ⃗A) = ⃗A(∇· ⃗B)+( ⃗B ·  ∇) ⃗A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This identity, along with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (42) and (48), specifically ∂zρ0+∇X ·(ρ0⃗v) = 0 and ∇X ·(ρ0P⃗Ω) = 0,  transforms (71) into  � ρ0D⃗v/Dz  ρ0D⃗Ω/Dz  �  +  �  � ∇X ·  �  ρ0R + ρ0⃗Ω ⊗ ⃗Ω  �  �  ρ0⃗Ω · ∇X  �  ⃗v  �  � =  � ρ0∇Xg(⃗x, z)  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (73)  where we have again leveraged the material derivative defined earlier in section (III E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The only com-  ponents of (73) that are influenced by the DOP are those involving the polarization angle gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' To  simplify further, note that the divergence of ρ0(⃗x, z, ω)R(⃗x, z, ω) is  ∇X · [ρ0(⃗x, z, ω)R(⃗x, z, ω)] = − 1  2k2  0  ρ0(⃗x, z, ω)∇X  �  ∇2  Xρ1/2  0  (⃗x, z, ω)  ρ1/2  0  (⃗x, z, ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (74)  15  Expanding the outer product and invoking (48), the first of Eqns (73) can be written  D⃗v0(⃗x, z, ω)  Dz  = −  �  ⃗Ω(⃗x, z, ω) · ∇X  �  ⃗Ω(⃗x, z, ω) + ∇Xg(⃗x, z) +  1  2k2  0  ∇X  �  ∇2  Xρ1/2  0  (⃗x, z, ω)  ρ1/2  0  (⃗x, z, ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (75)  which is of precisely the same form as the coherent momentum equation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the case of a deterministic  medium, the gradient potential forcing term is ∇Xϵ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In [20] the commonly used medium assumptions  were used whereby, ⟨ϵ⟩ = 1 and ϵ1 = n2(⃗x, z)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In this case Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (75) matches exactly that found in [20]  for P = 1 (fully polarized light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' For a stochastic, homogeneous medium where g(⃗x, z) = −ik0A(⃗x, z)/4  we have  ∇Xg(⃗x, z) = −ik0  4 ∇XA(0, z)  = −ik0  4  �  R2 i⃗ξSNN(⃗ξ, 0)ei⃗ξ·⃗0d⃗ξ  = k0  4  �  R2  ⃗ξSNN(⃗ξ, 0)d⃗ξ  (76)  Taking as a simple model for the refractive index fluctuations as Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (38) of [5], we have that  ∇Xg(⃗x, z) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', the gradient of the refractive index covariance is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus, the time-averaged  transverse location is unchanged due to the turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Note, the model (75) can also be used to derive  known results for “beam wander” in the coherent case following the approach in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In that case,  an expression for the transverse coordinate variance is developed and is seen to be proportional to the  variance of the refractive index gradient (as opposed to the gradient of the variance) and one recovers  the established result (see [11] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  As in our prior work, it is advantageous to switch to Lagrangian coordinates ⃗x → ⃗xz(⃗x0) which are  functions of propagation distance z and initial value ⃗x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' With this transformation, and noting that the  “velocities” are the coordinate derivatives with respect to z, (75) becomes  d⃗v0(⃗xz, ω)  dz  = d2⃗xz(ω)  dz2  = −  �  ⃗Ω(⃗xz, ω) · ∇Xz  �  ⃗Ω(⃗xz, ω) + ∇Xzg(⃗xz) +  1  2k2  0  ∇Xz  �  ∇2  Xzρ1/2  0  (⃗xz, ω)  ρ1/2  0  (⃗xz, ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (77)  This is a sensible model as it suggests that a fully depolarized beam will follow a path that is influenced  only by diffraction (last term in 77) and refractive index fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' However, if the beam is polarized  and if the polarization angle distribution possesses non-zero first and second spatial derivatives, there  is an additional effect that must be considered, given by the first term on the right hand side of (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In the case of diffraction only, the model (77) recovers the known result for the beam path in both the  Gaussian beam [11]) and Airy beam [20]) situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  The second equation in (73) governs the evolution of the polarization angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Following the derivation  in Appendix (C), this expression reduces to  D⃗Ω  Dz +  �  ⃗Ω · ∇X  �  ⃗v = 0  (78)  or in Lagrangian coordinates  d⃗Ω(⃗xz, ω)  dz  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (79)  Thus, for partially coherent light, the time-average polarization gradient as defined in (49) will remain  unchanged during propagation when observed in Lagrangian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This result is consistent with  our prior work which held that in the fully coherent case, the polarization angle remained unchanged on  propagation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' By definition, depolarization will decrease ⃗Ω, ultimately resulting in ⃗Ω = 0 for P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  However, no mechanism exists in (79) to cause this to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' While it is known that the atmosphere can  serve as a depolarizing element, under the MA model such an effect can only be observed by retaining  higher-order spatial variations in the refractive index (stemming ultimately from retaining the electric  field divergence term in the wave equation) [5, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In both of the cited works retention of such terms  has almost no influence on beam propagation but for very long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  This distance scales as  O(k2  0ℓ3  0/σ2  ϵ ) ∼ 1010m, even when considering strong turbulence (refractive index variance σ2  η ∼ 10−10),  16  small inhomogeneities (length scale ℓ0 ∼ 10−3m), and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='55µm wavelength light [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Other works have  predicted depolarization will arise due to asymmetries in the initial spatial correlations across the beam  face [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We have not included this effect in our model given that lasers do not naturally possess such  asymmetry and creating such a beam would be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In summary, we conclude that whether we are considering coherent, monochromatic, or partially  coherent, polychromatic wave propagation, the basic governing equations are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  ρ0(⃗xz, ω) = | det J⃗x0(⃗xz)|−1ρ0(⃗x0, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  d2⃗xz  dz2 = −  �  ⃗Ω(⃗xz, ω) · ∇Xz  �  ⃗Ω(⃗xz, ω) + ∇Xzg(⃗xz) +  1  2k2  0  ∇Xz  �  ∇2  Xzρ1/2  0  (⃗xz, ω)  ρ1/2  0  (⃗xz, ω)  �  d⃗Ω(⃗xz, ω)  dz  = 0  (80)  It is the definition of quantities in the expression that is different in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' These can be  summarized as follows:  Coherent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' monochromatic  ρ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) = EX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)E∗  X(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) + EY (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)E∗  Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  ⃗v(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) = 1  k0  ∇Xφ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  ⃗Ω(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z) = 1  k0  ∇Xγ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  (81)  Partially coherent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' polychromatic  ρ0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) =  �  R2  �  �  WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + �  WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  d⃗ξ = SXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + SY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  ⃗v0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = 1  k0  ⟨∇Xφ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ =  �  R2 ⃗ξ  �  �  WXX(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) + �  WY Y (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  d⃗ξ  k0ρ0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  ⃗Ω(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) = P(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  k0  ⟨∇Xγ(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)⟩ =  −iP  �  R2 ⃗ξ  �  �  WXY (⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω) − �  WY X(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ⃗ξ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' ω)  �  d⃗ξ  k0ρ0(⃗x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' z)  (82)  where we can see that the latter reduce to the former in the coherent limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  EXAMPLES  In our previous work we showed that for a particular functional form of polarization angle gradient  the beam would “accelerate” in the transverse direction following a curved path and we have shown in  (80) that this is also true in the partially coherent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Using a new definition of polarization angle  gradient (49), we see the the beam will follow the same coherent path as in [20] for P = 1 but will follow  a different path as the DOP decreases P < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Consider first the simple case of a beam with Gaussian intensity distribution of equal amplitude in  the x and y components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As in [14] (see Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='6 in the cited work) we take P(⃗x, z) = P as a constant  representing the correlation coefficient between the x and y components of the electric field and obeying  0 ≤ P ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In our prior work we set P = 1 and used the polarization profile  γ(y0) = π  2  (y0 − a)2  a2  + π  8  (83)  which varies in y0 only and not in x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus, the polarization gradient terms becomes simply  ⃗Ω(y0) =  �  0, P π(y0 − a)  a2  �  (84)  17  and the second of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (80) becomes the ordinary differential equations  d2xz  dz2 = x0  d2yz  dz2 = −P2  k2  0  dγ(y0)  dy0  d2γ(y0)  dy2  0  (85)  subject to the initial velocity dx0/dz = dy0/dz = 0 and initial displacements x0 and y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This expression  can be solved to give the the partially coherent beam path  xz = x0  yz = P2π2z2  2k2  0a4 (a − y0) + y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (86)  The prediction is that for fully polarized light, P = 1, the beam will follow the same path obtained in  [20] while, for unpolarized light, P = 0, and the beam will simply follow a straight path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Figure (1) shows  a family of such curves associated with the beam center, y0 = 0, for different degrees of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Also shown is the polarization angle profile across the beam face, plotted as a function of the transverse  coordinates ⃗x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The result is the expected behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As the DOP is reduced, so too is the degree to which  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 1: (a) Transverse displacement of the beam center as the DOP is reduced from unity to 0 in  increments of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The model parameters were the same as those used in our prior work [20] and (b)  The associated polarization angle profile, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (83)  the path of the beam center is altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We also note that in this example the effects of diffraction could  have been included just as they were in our coherent model [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The result is the standard Gaussian  beam spreading about the path predicted in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (86).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  As a second example, we consider the case where both bending and diffraction are retained in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Specifically, we tailor the polarization profile so that the polarization-induced bending acts in opposition  to the diffractive effect, thereby mitigating the degree to which the beam spreads on propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Using  the polarization profile  γ(y0) = κπ  2  y2  0  a2 − π  8  (87)  we solve the coupled system of Equations (80) using an initial Gaussian intensity profile  ρ0(⃗x0, ω) = I(ω)e−(x2  0+y2  0)/(2σ2)  (88)  with width σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The solution is obtained analytically in this case using the approach outlined in [20] which  18  (a)  X10-3  5  Transverse displacement y(z) meters  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='8  3  2  P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='6  P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='4  P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='2  0  P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='1  0  5  10  15  20  25  30  Propagation distance z meters(b)  2  (mm)  0  1  2  2  1  0  1  2  (uw) °xleads to the transverse path  xz = x0  �  z2 + σ4k2  0  σ2k0  yz = −(Pκ)2π2z2  2k2  0a4  y0 + y0  �  z2 + σ4k2  0  σ2k0  =  ��  1 +  z2  k2  0σ4  �1/2  − π2  2  (Pκ)2  k2  0a4 z2  �  y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (89)  The corresponding determinant of the Jacobian is therefore  det J⃗x0(⃗xz) = 1 +  z2  σ4k2  0  − (Pκ)2π2z2�  z2 + σ4k2  0  2a4σ2k3  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (90)  and the desired intensity profile on propagation is given by the first of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The polarization profile  (87) has the effect of bending the outer edges of the beam, y0 = ±a/2, toward the center thus mitigating  the outward spreading caused by the diffractive term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' These results can be seen visually in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (2) for  a fully polarized beam of wavelength 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='55 µm, width σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='2 mm and extending over transverse spatial  dimension a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='17 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In this fully polarized case we set κ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 2: (a) Normal beam diffraction profile over the range z = [0, 250] m, (b) diffraction profile  associated with the polarization angle distribution given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (87) and for P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Here the spreading  due to diffraction is compensated by the bending due to the polarization angle gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' These effects  cancel each other out at range zc ≈ (2a4k0)/ (Pπσ)2 (obtained by setting yz = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Beyond this range  the beam super diffracts and the spreading is proportional to (z − zc)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Plots (c) and (d) use the same  polarization profile but with P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='75 and P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  The normalized beam intensity is plotted for z = [0, 250] m in order to clearly show the effects of the  polarization gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' By altering the polarization of the light across the beam face we can control, to  some extent, the degree to which the beam diffracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Of course, in this example the diffractive effect  causes transverse spreading that is linear in z while the bending we show here is quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=" Thus, at a  19  (a)  50  4  P=0  6'0  Transverse Displacement (mm)  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='8  ¥2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='3  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='2  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='1  50  0  0  20  40  60  80  100  120  140  160  180  200  220  240  Propagation Distance (m)(q)  50  P=1  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='9  Transverse Displacement (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='4  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='3  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='2  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='1  50  0  0  20  40  60  80  100  120  140  160  180  200  220  240  Propagation Distance (m)(c)  50  40  P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='9  Transverse Displacement (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='4  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='3  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='2  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='1  50  0  0  20  40  60  80  100  120  140  160  180  200  220  240  Propagation Distance (m)(d)  50  40  P=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='9  Transverse Displacement (mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='4  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='3  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='2  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='1  50  0  0  20  40  60  80  100  120  140  160  180  200  220  240  Propagation Distance (m)distance zc, obtained by setting yz = 0 in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (89), the rays are predicted to come to a focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Then  zc =  √  2  π2  k0  (Pκ)2  � a4  σ2  � �  1 +  �  1 + π4(Pκ)4  �σ  a  �8  �1/2  (91)  and in the limit zc ≫ k0σ2 we find the particularly simple form  zc = 2a4k0  (Pπσ)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (92)  (Note also this limit is also obtained when π4(Pκ)4(σ/a)8 << 1, a condition that will always be met  when the beam width 2σ is less than the output aperture dimension 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Beyond z > zc, however,  the beam diverges strongly and the intensity rapidly diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus, using this polarization profile  diffraction can be mitigated, but only over a prescribed range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As the beam depolarizes the degree  to which diffraction is mitigated will change, however, by increasing the constant κ in the polarization  gradient such that Pκ = 1, we expect to see the same results as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In practice, of course,  there will be a limit to how large a polarization gradient can be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As the DOP decreases (with  κ = 1) we see the expected results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (2c-d), namely that the beam gradually moves toward the  standard diffraction profile associated with an incoherent beam, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (3) shows a plot of zc from (91) for fixed polarization gradient (κ = 1) as a function of degree  of polarization P in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='01 ≤ P ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The zc ∝ P−1 behavior continues indefinitely as P → 0,  that is, regardless of the degree of polarization, there exists a z−value for which the effects of diffraction  could be completely compensated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Of course in practice, there is a limit to κ as one can not generate  a beam with an infinitely sharp polarization gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The practical limit will for such a beam will be  determined by the resolution of the device used in its creation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', the spatial light modulators used in  reference [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 3 are consistent with the following interpretation of the mixture model in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The  term ”mixture” cannot be taken literally to mean that the beam comprises a collection of photons in  which a fraction P are fully polarized and fraction (1 − P) are completely unpolarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' If that were true  then the polarization gradient could only compensate diffraction for the polarized photons and the beam  spread at z = zc would necessarily increase as P decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Instead, we found that the beam width at  the focus, in the y− direction for this example, is independent of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Hence, we interpret (30) to mean  that every photon in the beam is described by Stokes vector  �  S =  �  �  �  �  s′  0(⃗x, ⃗x′, z, ω)  Ps′  1(⃗x, ⃗x′, z, ω|P)  Ps′  2(⃗x, ⃗x′, z, ω|P)  Ps′  3(⃗x, ⃗x′, z, ω|P)  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (93)  Every photon thus exhibits properties of partial polarization with degree of polarization P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  SUMMARY  We have generalized our prior, coherent vector beam model to the partially coherent case by introducing  a mixture model for the generalized Stokes vector and by subsequently choosing appropriate definitions  of amplitude, phase, and polarization angle gradient in describing propagation of the generalized Stokes  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The resulting “transport” model predicts the time-averaged optical path of a vector beam  propagating through an inhomogeneous medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', an atmosphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Importantly, the model reduces  to that of our previous work on vector beams in the coherent limit [20] and matches the predictions  found in [30] and [5] in the absence of a polarization gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Consequently, it predicts the same optical  path as in our coherent model in the fully polarized case, while reducing to the standard “unaltered”  path in the unpolarized situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Notably, the model requires a new definition of polarization angle gradient that is consistent with  more recent definitions of phase for partially coherent beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' These definitions become important in the  governing equations as the movement of intensity during propagation is governed by phase and angle  gradients as opposed to the phase and angle variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As a by-product of the model development we  were also able to demonstrate conservation of the generalized Stokes parameters on propagation, even in  20  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 3: zc as a function of degree of polarization P for λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='55 µm, a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='17 mm, and σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='15 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  For these parameters, zc ∝ P−1 and the trend continues indefinitely as P → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  the case where the intervening medium possesses refractive index fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This is a generalization  of the result established in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  An obvious prediction of the model Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (79) is that the state of linear polarization (as captured  by ⃗Ω) remains unchanged on propagation from a Lagrangian viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Because the definition of ⃗Ω  includes the DOP, this is also tantamount to the statement that the DOP is unchanged on propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  The result is consistent with the result of Charnotskii [5] which found that no significant depolarization  occurs due to the presence of a turbulent atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Had we included the higher order refractive index  variations (by retaining the electric field divergence in the wave equation) and/or retained the full Wigner  potential in forming Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (61) (as opposed to using the linearized potential) this effect may have been  observable in our model as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  We considered two such examples, one in which the curved path taken by the beam studied in [20] was  slowly altered by changing the degree of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the coherent case, the path taken is the same as  in the cited reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' As the degree of polarization is reduced so too is degree of beam bending, reducing  to propagation in a straight line in the fully incoherent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In the second example, we consider both  the bending and diffracting beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' For that example we chose an initial polarization distribution that  would mitigate the effects of diffraction over a predictable distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The result suggests new class of  “non-diffracting” beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The effects of depolarization can be countered in this case by simply increasing  the strength of the polarization gradient applied across the transverse dimension of the beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  In short, the model we have developed can be used to predict vector beam trajectories in a variety of  propagation scenarios of interest, including through free-space and a potentially turbulent atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  As such it provides a powerful tool in forecasting the utility of the newly discovered vector beam bending  effect in application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' We note that the model rests on the same two assumptions as were used in the  coherent model development [20], namely, the slowly varying envelope approximation and a weakly  inhomogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors would like to thank the Office of Naval Research Code 33, for support under  PE#0601153N, award #N0001422WX01660  Appendix A: Interpretation of Transport Velocity  In this section we present a more traditional optics interpretation of the transverse velocity ⃗v =  k−1  0 ∇Xφ for coherent light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' In particular, we show how this quantity can be related to the familiar  concept of “optical path length” (OPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' This allows us to properly view the transverse phase gradient  21  107  106  105  (w)  104  103  102  10-2  10-1  100  DOP (P)of our electric field as the rate at which intensity is moved in the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The interpretation  will remain valid so long as the paraxial assumption holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  To begin, we note that the OPL in the context of a propagating wave is defined (see, for example,  Saleh [27]) as  Φ(⃗x, z) = z + φ(⃗x, z)/(2k0)  (A1)  that is, as the phase argument of the wave normalized by the wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The quantity φ/(2k0) can  be viewed as a perturbation to the optical path resulting from refractive index fluctuations, diffraction,  and, in this context, from polarization angle gradients as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Consider now the expression for total path length L(⃗xz) in Lagrangian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' For a differential  length element parameterized by propagation distance, dℓ(z)  L(⃗xz) =  � z  0  dℓ(ζ)  =  � z  0  (dζ2 + dx2  ζ + dy2  ζ)1/2 =  � z  0  �  1 +  �dxζ  dζ  �2  +  �dyζ  dζ  �2�1/2  dζ  ≈  � z  0  dζ + 1  2  � z  0  �dxζ  dζ  �2  dζ + 1  2  � z  0  �dyζ  dζ  �2  dζ  (A2)  where in the final step we have made use of the fact that for ϵ1, ϵ2 ≪ 1, (1 + ϵ1 + ϵ2)1/2 ≈ 1 + 1  2ϵ1 + 1  2ϵ2  which is equivalent to the paraxial assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Equating this expression to the OPL given in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (A1)  gives  L(⃗xz) = z + φ(⃗xz)  2k0  =  � z  0  dζ + 1  2  � z  0  �dxζ  dζ  �2  dζ + 1  2  � z  0  �dyζ  dζ  �2  dζ  (A3)  so that  φ(⃗xz)  2k0  = 1  2  � z  0  �dxζ  dζ  �2  dζ + 1  2  � z  0  �dyζ  dζ  �2  dζ  (A4)  Differentiating with respect to z yields  1  k0  �∂φ(⃗xz)  ∂xz  ∂xz  ∂z + ∂φ(⃗xz)  ∂yz  ∂yz  ∂z  �  =  �dxz  dz  �2  +  �dyz  dz  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (A5)  which can be written  1  k0  ∇Xzφ(⃗xz) · ⃗v(⃗xz) = ⃗v(⃗xz) · ⃗v(⃗xz)  (A6)  thus  ⃗v(⃗xz) = 1  k0  ∇Xzφ(⃗xz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (A7)  This demonstrates that the change in optical path length in the transverse plane per unit change in ˆz  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' the transverse “velocity”) is equal to the transverse phase gradient normalized by the wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (A7) was presented in [23] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' 11 of that work) and is referred to appropriately as the “flow vector  of spectral density”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The connection between ⃗v(⃗xz) and transverse coordinate changes was also leveraged  in [11] although not explicitly derived as we have here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' The key assumption used to arrive at this result  is that changes in the transverse direction are much smaller than changes in the direction of propagation  and is tantamount to the same paraxial assumption we have used throughout our model development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  While the above analysis is performed for the coherent case, we have demonstrated that one can re-  define the model quantities (amplitude, phase, and polarization angle) as appropriate averages in the  partially coherent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Thus, the above interpretation still holds and the transverse phase gradient can  be interpreted as the average change in optical path in the transverse direction per unit change in the  direction of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  22  Appendix B: Derivation of Equation (55)  The identity required of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (55) requires relating the determinant of the Jacobian of the La-  grangian coordinate mappings xz(x0, y0), yz(x0, y0) to the divergence of the associated velocity field  uz(xz, yz), vz(xz, yz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Begin with the determinant  det(J⃗x0(⃗xz)) = dxz  dx0  dyz  dy0  − dyz  dx0  dxz  dy0  ,  (B1)  take the derivative with respect to z and recognize that uz ≡ dxz/dz, vz ≡ dyz/dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Then  d  dz det(J⃗x0(⃗xz)) = duz  dx0  dyz  dy0  + dvz  dy0  dxz  dx0  − dvz  dx0  dxz  dy0  − duz  dy0  dyz  dx0  (B2)  Both uz(xz, yz), vz(xz, yz) are functions of the Lagrangian coordinates, hence we can apply the chain  rule to obtain  duz  dx0  = ∂uz  ∂xz  ∂xz  ∂x0  + ∂uz  ∂yz  ∂yz  dx0  duz  dy0  = ∂uz  ∂xz  ∂xz  dy0  + duz  ∂yz  ∂yz  ∂y0  dvz  dx0  = ∂vz  ∂xz  ∂xz  ∂x0  + ∂vz  ∂yz  ∂yz  ∂x0  dvz  dy0  = ∂vz  dxz  ∂xz  ∂y0  + ∂vz  ∂yz  ∂yz  ∂y0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (B3)  Substituting (B3) into (B2) and simplifying yields the ordinary differential equation  d det(J⃗x0(⃗xz))  dz  = (∇Xs · ⃗v(⃗xs) det(J⃗x0(⃗xz)),  (B4)  with solution  det (J⃗x0(⃗xz)) = exp  �� z  s=0  ∇Xs · ⃗v(⃗xs)ds  �  (B5)  This relationship allows (54) to be written as (55) thus completing the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Appendix C: Derivation of Equation (79)  To derive Equation (79) we begin with Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (71)  ∂z  �  ρ0⃗Ω  �  + ∇X ·  �  ρ0⃗v ⊗ ⃗Ω + ρ0⃗Ω ⊗ ⃗v  �  = 0  (C1)  Using two applications of the vector identity ∇X ·  �  ⃗B ⊗ ⃗A  �  = ⃗A  �  ∇X · ⃗B  �  +  �  ⃗B · ∇X  �  ⃗A and grouping  terms we have  ρ0[∂z⃗Ω + (⃗v · ∇X)⃗Ω] + ⃗Ω  �  ((((((((  ∂z(ρ0) + ∇X · ρ0⃗v  �  + ⃗v\x18\x18\x18\x18\x18\x18  �  ∇X · [ρo⃗Ω]  �  +  �  ρ0⃗Ω · ∇X  �  ⃗v = 0  (C2)  where Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (48) has allowed us to cancel the second to last term while continuity of the first Stokes  parameter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=', the intensity) cancels the third term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Noting the first term in brackets is the total  derivative of ⃗Ω, allows us to write Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (C2) in Eulerian coordinates as  D⃗Ω  Dz +  �  ⃗Ω · ∇X  �  ⃗v = 0  (C3)  However, further simplification is possible by re-writing (C2) as  ∂z⃗Ω +  �  (⃗v · ∇X) Ω +  �  ⃗Ω · ∇X  �  ⃗v  �  = 0  ∂z⃗Ω + ∇X  �  ⃗Ω · ⃗v  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (C4)  23  where we have leveraged the identity ∇X  �  ⃗A · ⃗B  �  = ( ⃗A·∇X) ⃗B+( ⃗B·∇X) ⃗A+ ⃗A×(∇X × ⃗B)+ ⃗B×(∇X × ⃗A)  and note that both ⃗Ω and ⃗v are expressible as gradients of scalars, hence the cross-product terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  Recalling that ⃗Ω ≡ k−1  0 ∇Xγ we can write  k−1  0  [∂z(∇Xγ) + ∇X(∇Xγ · ⃗v)] = 0  = k−1  0 ∇X [∂zγ + ∇Xγ · ⃗v] = 0  = k−1  0 ∇X  Dγ  Dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (C5)  If we now make the switch to Lagrangian coordinates, the total derivatives become ordinary derivatives  d⃗Ω(⃗xz, ω)  dz  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  (C6)  which is Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
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+page_content=' Inverse Problems and Imaging, 7(3):1051–1074, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
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+page_content=' Ke, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Luo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Wen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Generation of Airy vortex and Airy vector beams based on the  modulation of dynamic and geometric phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Optics Letters, 40(13):3193–3196, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
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+page_content=' Tian, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content=' Waller, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
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+page_content=' Optics and Lasers in Engineering, 71:20–  32, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
+page_content='  25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E2T4oBgHgl3EQfmAd3/content/2301.03994v1.pdf'}
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+Cold plasma waves in the chiral Maxwell-Carroll-Field-Jackiw electrodynamics
+Filipe S. Ribeiro
+a,∗ Pedro D. S. Silva
+a,† and Manoel M. Ferreira Jr.
+b‡
+aPrograma de P´os-gradua¸c˜ao em F´ısica, Universidade Federal do Maranh˜ao,
+Campus Universit´ario do Bacanga, S˜ao Lu´ıs (MA), 65080-805, Brazil and
+bDepartamento de F´ısica, Universidade Federal do Maranh˜ao,
+Campus Universit´ario do Bacanga, S˜ao Lu´ıs (MA), 65080-805, Brazil
+In this work, we study the propagation and absorption of plasma waves in the chiral Maxwell-
+Carroll-Field-Jackiw electrodynamics (MCJF). The Maxwell equations are rewritten for a cold,
+uniform, and collisionless fluid plasma model, allowing us to determine the new refractive indices and
+propagating modes. The case of transversal propagation is examined considering a purely timelike
+CFJ background that plays the role of the magnetic conductivity chiral parameter. We find four
+distinct refractive indices associated with RCP and LCP waves. For each index, the propagation
+and absorption zones are illustrated for some specific parameter values. The optical behavior is
+investigated by means of the rotatory power (RP) and dichroism coefficient. The existence of a
+negative refraction zone enhances the rotatory power. It is also observed RP sign reversal, a feature
+of rotating plasmas.
+PACS numbers: 11.30.Cp, 41.20.Jb, 41.90.+e, 42.25.Lc
+I.
+INTRODUCTION
+The study of electromagnetic (EM) waves propagation
+[1, 2] in cold magnetized plasma is based on magneto-
+ionic theory [3–8], developed by E. Appleton [9] and D.
+Hartree [10] between 1929 and 1932 to describe the radio
+waves propagation in the ionosphere, in the context of the
+usual electrodynamics [11]. EM waves in plasmas have
+been studied in other scenarios recently, as in logarithmic
+nonlinear electrodynamics [12].
+The chiral magnetic effect (CME) is the macroscopic
+generation of an electric current in the presence of a mag-
+netic field, stemming from an asymmetry between the
+number density of left- and right-handed chiral fermions
+[13–17]. It has been extensively investigated in several
+distinct contexts, such as quark-gluon plasmas [18–20],
+cosmology [21], neutron stars [22, 23], and electroweak
+interactions [24].
+The CME plays a very relevant role
+in Weyl semimetals, where it is usually connected to
+the chiral anomaly associated with Weyl nodal points
+[25], the absence of the Weyl nodes [26], anisotropic ef-
+fects stemming from tilted Weyl cones [27], the CME
+and anomalous transport in Weyl semimetals [28], quan-
+tum oscillations arising from the CME [29], computa-
+tion of the electromagnetic fields produced by an elec-
+tric charge near a topological Weyl semimetal with two
+Weyl nodes [30], renormalization evaluations for Weyl
+semimetals and Dirac materials [31], and solutions of ax-
+ion electrodynamics [32].
+The CME current can be classically described by the
+axion Lagrangian [32–38],
+L = −1
+4F µνFµν + θ(E · B),
+(1)
+∗ filipe.ribeiro@discente.ufma.br, filipe99ribeiro@hotmail.com
+† pedro.dss@discente.ufma.br, pdiego.10@hotmail.com
+‡ manojr.ufma@gmail.com, manoel.messias@ufma.br
+where θ is the axion field. In this context, the Maxwell
+equations are
+∇ · E = ρ − ∇θ · B,
+(2)
+∇ × B − ∂tE = j + (∂tθ)B + ∇θ × E,
+(3)
+where the terms involving θ derivatives find association
+with condensed matter effects [38]. Indeed, ∇θ · B repre-
+sents an anomalous charge density, while ∇θ×B appears
+in the anomalous Hall effect, and (∂tθ)B plays the role
+of the chiral magnetic current. In the case the axion field
+does not depend on the space coordinates, ∇θ = 0, the
+Maxwell equations (2) and (3) read
+∇ · E = ρ,
+∇ × B − ∂tE = j + (∂tθ)B,
+(4)
+where (∂tθ)B, the chiral magnetic current, may also
+be addressed as a term of Maxwell-Carroll-Field-Jackiw
+(MCFJ) theory. A classical electrodynamics scenario en-
+dowed with a chiral magnetic current has been investi-
+gated considering symmetric and antisymmetric conduc-
+tivity [39]. The latter case has also been addressed in
+Ref. [40].
+The MCFJ model [41] is the CPT-odd part of the U (1)
+gauge sector of the Standard Model Extension (SME)
+[42]. It is described by the Lagrangian density
+L = −1
+4F µνFµν − 1
+4ϵµναβ (kAF )µ AνFαβ − AµJµ,
+(5)
+with (kAF )µ being the 4-vector background which con-
+trols the Lorentz violation. This theory has been investi-
+gated in multiple respects [43], encompassing radiative
+evaluations [44, 45], topological defects solutions [46],
+supersymmetric generalizations [47], classical solutions,
+quantum aspects and unitarity analysis [48]. It may also
+be connected with the CME in the sense that it provides
+a modified Amp`ere’s law,
+∇ × B − ∂E
+∂t = J + k0
+AF B + kAF × E,
+(6)
+arXiv:2301.02183v1  [physics.plasm-ph]  5 Jan 2023
+
+ID2
+containing the magnetic current, JB = k0
+AF B, with the
+component k0
+AF playing the role of the magnetic conduc-
+tivity.
+The SME photon sector is also composed of a CPT-
+even term constituted of a rank 4 Lorentz-violating tensor
+[49], whose components may be properly parametrized in
+terms of dimensionless 3 × 3 matrices, κDE, κDB, κHE,
+and κHB, which allow to write generalized constitutive
+relations between the fields (D, E) and (H, B),
+�
+D
+H
+�
+=
+�
+�
+ϵ1 + κDE
+κDB
+κHE
+µ−11 + κHB
+�
+�
+�
+E
+B
+�
+,
+(7)
+similar to the ones that hold in continuous medium elec-
+trodynamics, see Eqs. (8a) and (8b). Here, D is the elec-
+tric displacement, while H is the magnetic field. This
+CPT-even electrodynamics was investigated in several
+contexts, involving consistency aspects [50], finite tem-
+perature and boundary effects [51].
+Lorentz-violating
+electrodynamics in continuous matter [52, 53] has been
+a topic of interest in the latest years due to its potential
+to describe interesting effects of the phenomenology of
+new materials, such as Weyl semimetals [54]. A classical
+field theory description of wave propagation, refractive
+indices, and optical effects in a continuous medium de-
+scribed by the MCFJ electrodynamics (with usual consti-
+tutive relations), including its Lorentz-violating higher-
+order derivative version [55], was discussed in Ref. [56].
+Chiral media are endowed with parity violation [57–
+60], being described by parity-odd models, as bi-isotropic
+[61] and bi-anisotropic electrodynamics [62–67], whose
+constitutive relations read
+D = ˆϵ E + ˆα B,
+(8a)
+H = ˆβ E + ˆζ B,
+(8b)
+and ˆϵ = [ϵij], ˆα = [αij], ˆβ = [βij], and ˆζ = [ζij] represent,
+in principle, 3 × 3 complex matrices.
+The bi-isotropic
+relations involve the diagonal isotropic tensors, ϵij = ϵδij,
+αij = αδij, βij = βδij.
+In chiral scenarios, LCP and
+RCP waves travel at distinct phase velocities, implying
+birefringence and optical rotation [68]. This phenomenon
+stems from the natural optical activity of the medium
+or can be induced by the action of external fields (e.
+g., Faraday effect [69–71]), and it is measured in terms
+of the rotation angle per unit length or rotatory power
+(RP) [72]. Magneto-optical effects are used to investigate
+features of new materials, such as topological insulators
+[73–79] and graphene compounds [80].
+The RP is a probe to examine the optical behavior of
+several distinct systems, for instance, crystals [81, 82],
+organic compounds [57, 83], graphene phenomena at ter-
+ahertz band [84], and gas of fast-spinning molecules [85].
+The optical rotation may depend on the frequency (RP
+dispersion) and undergo reversion (anomalous RP dis-
+persion) [86–88]. It also finds interesting applications in
+chiral metamaterials [89–91], chiral semimetals [92, 93],
+in the determination of the rotation direction of pulsars
+[94], and in rotating plasmas, which constitutes a sce-
+nario where RP sign reversal also takes place [95]. Re-
+cently, RP reversal was also reported in a bi-isotropic
+dielectric in the presence of chiral magnetic current [96].
+Furthermore, in the presence of absorption, dichroism
+is another useful tool for the optical characterization of
+matter. It occurs when LCP and RCP light waves are
+absorbed by the medium at different degrees. It has been
+used to distinguish between Dirac and Weyl semimetals
+[97], perform enantiomeric discrimination [98, 99], and
+for developing graphene-based devices at terahertz fre-
+quencies [100].
+Another feature of chiral systems is the possible oc-
+currence of negative refraction and negative refractive
+index, which was first proposed by Veselago in 1968 [101]
+and experimentally observed in 2000 [102, 103]. Later,
+other experiments confirmed the negative refraction by
+using Snell’s law [104, 105].
+This unusual property
+was achieved in constructed metamaterials with both
+negative electric permittivity and magnetic permeability
+[106, 107]. The negative refractive index also appears in
+quark-gluon plasmas [108, 109], magnetoelectric materi-
+als [110], metasurfaces [111], chiral bi-anisotropic meta-
+materials [112, 113], and new materials, such as Dirac
+semimetals [114, 115].
+In chiral plasmas described by
+generalized bi-isotropic constitutive relations [116, 117],
+the negative refractive index can occur within some fre-
+quency band and is not necessarily associated with si-
+multaneously negative electric permittivity and negative
+magnetic permeability, being attributed to the chirality
+parameter introduced in the constitutive relations,
+Di = εijEj + iξcBi,
+Hi = µ−1Bi + iξcEi,
+(9)
+where εij, µ, and ξc are the plasma electric permit-
+tivity tensor, the magnetic permeability, and the con-
+stant chirality parameter. Plasmas metamaterials have
+been investigated as new media endowed with interesting
+properties, such as negative refraction and nonlinearities
+[118, 119].
+In this work, we are interested in examining the
+wave propagation in a magnetized cold plasma ruled
+by the MCFJ model, a chiral route distinct from the
+bi-isotropic/anisotropic electrodynamics of the relations
+(9). We carry out our analysis considering the timelike
+Lorentz-violating background component, which plays
+the role of chiral magnetic conductivity. The refractive
+indices are evaluated and optical effects, such as bire-
+fringence and dichroism, are examined, which could be
+useful to trace analogies with other material properties.
+We also find that the chiral conductivity yields negative
+refraction in specific frequency bands, enhancing the ro-
+tatory power and dichroism signals.
+This paper is outlined as follows. In Sec. II, we briefly
+review some aspects of the MCFJ model. In Sec. III, the
+main properties of propagation in usual cold magnetized
+plasmas are presented. The dispersion relations and re-
+fractive indices for cold plasmas in chiral electrodynamics
+
+3
+are addressed in Sec. IV. The optical effects are examined
+in Sec. V. Finally, we summarize our results in Sec. VI.
+II.
+BASICS ON MCFJ ELECTRODYNAMICS
+The Carroll-Field-Jackiw model was proposed as a
+gauge invariant CPT-odd electrodynamics constrained
+by birefringence data of distant galaxies [41]. It was later
+incorporated as the CPT-odd sector of the SME [42], and
+it has been investigated in several respects [43, 44]. In
+matter, it is described by the following Lagrangian den-
+sity1 [56]:
+L = −1
+4GµνFµν − 1
+4ϵµναβ (kAF )µ AνFαβ − AµJµ, (10)
+yielding the MCFJ equation of motion,
+∂ρGρκ + ϵβκµν (kAF )β Fµν = Jκ .
+(11)
+Here, (kAF )µ =
+�
+k0
+AF , kAF
+�
+is a constant 4-vector back-
+ground responsible for the Lorentz violation, and
+Fµν = ∂µAν − ∂νAµ,
+Gµν = 1
+2χµναβFαβ,
+(12)
+are the usual U(1) vacuum and continuous matter field
+strength, respectively.
+The 4-rank tensor, χµναβ, de-
+scribes the medium constitutive tensor [120], whose com-
+ponents provide the electric and magnetic responses of
+the medium. Indeed, the electric permittivity and mag-
+netic permeability tensor components are written as
+ϵij ≡ χ0ij0 and µ−1
+lk
+≡
+1
+4ϵijlχijmnϵmnk, respectively.
+For isotropic polarization and magnetization, it holds
+ϵij = ϵδij and µ−1
+ij = µδij, providing the usual isotropic
+constitutive relations,
+D = ϵE,
+H = µB.
+(13)
+A straightforward calculation from Eq. (11) yields
+∇ · D = J0 − kAF · B,
+(14)
+∇ × H − ∂D
+∂t = J + k0
+AF B + kAF × E,
+(15)
+where Gi0 = Di and Gij = −ϵijkHk. The homogeneous
+Maxwell equations are given by
+∇ · B = 0,
+∇ × E + ∂B
+∂t = 0.
+(16)
+By using a plane-wave ansatz for the electromagnetic
+fields, the MCFJ equations (14)-(16) read:
+1 We use natural units h = c = 1 and the Minkowski metric sig-
+nature gµν = diag (1, −1, −1, −1).
+ik · D + kAF · B = J0,
+(17a)
+ik × H + iωD − k0
+AF B − kAF × E = J,
+(17b)
+k · B = 0,
+k × E − ωB = 0,
+(17c)
+where k is the wave vector and ω is the (angular) wave
+frequency.
+In the presence of anisotropy, the permittivity and per-
+meability are represented by rank 2 tensors, εij and µij,
+which may also depend on the frequency (for a dispersive
+medium). For an anisotropic medium, the constitutive
+relations (13) are replaced by [1, 2],
+Di = εij(ω)Ej,
+Bi = µij(ω)Hj.
+(18)
+For non-magnetic media with isotropic magnetic perme-
+ability, it holds µij(ω) = µ0, where µ0 is the vacuum
+permeability. Considering the constitutive relations (18),
+the modified Amp`ere-Maxwell’s law, Eq.(17b), and Fara-
+day’s law, Eq.(17c), in the absence of sources, we obtain
+a modified wave equation for the electric field,
+ki �
+kjEj�
+− k2Ei = −ω2µ0¯εij (ω) Ej,
+(19)
+where we define the extended permittivity tensor,
+¯εij(ω) = εij(ω) − ik0
+AF
+ω2 ϵikjkk − iϵikj
+kk
+AF
+ω .
+(20)
+Using the definition to the refractive index, n = k/ω, the
+modified wave equation becomes
+MijEj = 0,
+(21)
+with Mij given by
+Mij = n2δij −ninj − εij
+ε0
+− i
+ω
+�
+V0ϵikjnk + ϵikjV k�
+, (22)
+in which ε0 is the vacuum electric permittivity, and
+V0 = k0
+AF /ε0,
+V k = kk
+AF /ε0.
+(23)
+appear as the components of a redefined background,
+V µ =
+�
+V0, V i�
+.
+The nontrivial solutions for the elec-
+tric field require a vanishing determinant of the matrix
+Mij, det Mij = 0, which provides the dispersion relations
+that describe the wave propagation in the medium.
+In this work, we will study plasma waves propagation
+for a chiral (parity-odd) medium, which means restrain-
+ing our investigation to the case of a purely timelike
+Lorentz-violating background vector, (kAF )µ =
+�
+k0
+AF , 0
+�
+.
+The latter also plays the role of chiral magnetic conduc-
+tivity. In this scenario, the wave equation (21) becomes
+�
+n2δij − ninj − εij
+ε0
+− iV0
+ω ϵikjnk
+�
+Ej = 0.
+(24)
+
+4
+III.
+THE USUAL MAGNETIZED COLD
+PLASMA
+In this work we will adopt the fluid theory approach
+in the cold plasma limit [3–7]:
+∂n
+∂t + ∇ · (nu) = 0,
+(25)
+∂u
+∂t + u · ∇u = q
+m (E + u × B0) ,
+(26)
+where n is the electron number density, u is the electron
+fluid velocity field, q and m are the electron charge and
+mass, respectively, and B0 is the equilibrium magnetic
+field. For simplicity, the ions are supposed to be infinitely
+massive, which is appropriate for high-frequency waves.
+Furthermore, thermal and collisional effects are also dis-
+regarded. The linearized version of the magnetized cold
+plasmas [7] consider fluctuations around average quanti-
+ties, n0 and B0, which are constant in time and space.
+Thus, the plasma quantities read
+n = n0 + δn,
+(27a)
+u = δu,
+(27b)
+E = δE
+(27c)
+B = B0 + δB,
+(27d)
+with δn, δu, δE and δB being first order plane wave
+magnitude perturbations. Following the usual procedure
+[3–6], assuming B0 = B0ˆz, we write the corresponding
+dielectric tensor,
+εij(ω) = ε0
+�
+�
+S
+−iD 0
+iD
+S
+0
+0
+0
+P
+�
+� ,
+(28)
+where
+S = 1−
+ω2
+p
+(ω2 − ω2c), D =
+ωcω2
+p
+ω (ω2 − ω2c), P = 1− ω2
+p
+ω2 , (29)
+and
+ωp = n0q2
+mϵ0
+,
+ωc = |q|B0
+m
+,
+(30)
+are the plasma and cyclotron frequencies, respectively.
+From the Maxwell theory, two distinct refractive in-
+dices are obtained,
+n± =
+�
+1 −
+ω2p
+ω (ω ± ωc),
+(31)
+which provide right-handed circularly polarized (RCP)
+and left-handed circularly polarized (LCP) modes,
+ELCP =
+i
+√
+2
+�
+1
+i
+�
+,
+ERCP =
+i
+√
+2
+�
+1
+−i
+�
+,
+(32)
+for the propagating modes associated to n±, respectively.
+This is the standard result of wave propagation in the
+usual magnetized cold plasma. We recall that a cutoff
+happens whenever the refractive index, n, goes to zero.
+On the other hand, a resonance occurs if n tends to infin-
+ity. From the indices (31), we obtain the following cutoff
+frequencies:
+ω± = 1
+2
+��
+ω2c + 4ω2p ∓ ωc
+�
+,
+(33)
+where ω± is related to n±, respectively.
+A very usual effect in magnetized plasmas is the circu-
+lar birefringence2, which causes the rotation of the plane
+of polarization of a linearly polarized wave that propa-
+gates within the medium.
+Thus the linearly polarized
+wave emerges from the medium with an electric field
+whose polarization is rotated relative to its initial lin-
+ear configuration. Such a phenomenon can be properly
+explained by decomposing the initial wave into two cir-
+cularly polarized waves (RCP and LCP) that travel with
+different phase velocities. In this case, the rotation an-
+gle of the electric field can be expressed as the difference
+between the refractive indices associated with the RCP
+and LCP waves [68, 72]:
+θ = πL
+λ0
+(Re [nRCP ] − Re [nLCP ]) ,
+(34)
+where λ0 is the vacuum wavelength of the incident wave.
+The rotation power δ = θ/L (phase difference per unit
+length), is given as
+δ = −ω
+2 (Re [nLCP ] − Re [nRCP ]) .
+(35)
+In a cold magnetized plasma, the refractive indices (31)
+provide the following rotatory power:
+δ = −ω
+2 Re
+�
+�
+�
+1 −
+ω2p
+ω (ω + ωc) −
+�
+1 −
+ω2p
+ω (ω − ωc)
+�
+� .
+(36)
+The behavior of the RP (36) in terms of the frequency
+ω is depicted in Fig. 1. One notices that there is a di-
+vergence at ωc, being positive for ω < ωc and negative
+for ω > ωc. It tends to zero at the high-frequency limit
+ω >> (ωp, ωc), where it decays as
+δ ≈ −ω2
+pωc
+2ω2 .
+(37)
+Associated with the imaginary part of the refractive in-
+dex, one can also examine dichroism, an optical effect
+2 In plasmas, the birefringence is usually a consequence of the Fara-
+day effect, occurring due to the presence of the external field
+B0, which generates distinct phase velocities for the propagating
+modes [70].
+
+5
+that occurs when circularly polarized waves are absorbed
+by the medium at different degrees [56, 58, 59]. Thus
+dichroism coefficient refers to the difference in absorp-
+tion of LCP and RCP waves, being given by:
+δd = −ω
+2 (Im[nLCP ] − Im[nRCP ]) .
+(38)
+which, for the refractive indices (31), implies
+δd = −ω
+2 Im
+�
+�
+�
+1 −
+ω2p
+ω (ω + ωc) −
+�
+1 −
+ω2p
+ω (ω − ωc)
+�
+� .
+(39)
+ω = ωc
+ω = 2 ωc
+ω = ω-
+ω = ω-
+0
+1
+2
+3
+4
+-2
+-1
+0
+1
+2
+ω (rad s-1)
+δ (rad m-1)
+FIG. 1. Rotatory power (36) in terms of ω. Here, ωc = ωp (red
+line) and ωc = 2ωp (blue line), with the choice ωc = 1 rad s−1.
+Such a quantity is plotted in Fig. 2, which shows sin-
+gularity at the cyclotron frequency ωc.
+For ωc = ωp
+(red curve), the dichroism coefficient (39) is negative for
+ω < ωred
++ , positive for 2ωc < ω < ωred
+−
+and null for other
+frequencies. The case for ωc = ωp/2 (blue curve) differs
+in the fact that ωblue
++
+is greater than ωc, showing that
+(39) is now negative for ω < ωc.
+ω = ωc
+ω = 2 ωc
+ω = ω-
+ω = ω-
+ω = ω+
+ω = ω+
+0.0
+0.5
+1.0
+1.5
+2.0
+-2
+-1
+0
+1
+2
+ω (rad s-1)
+δd (rad m-1)
+FIG. 2. Dichroism coefficient (39) in terms of ω. Here, ωc =
+ωp (red line) and ωc = ωp/2 (blue line), with ωc = 1 rad s−1.
+IV.
+WAVE PROPAGATION IN CHIRAL
+ELECTRODYNAMICS
+Starting from the wave equation (24) and using the ex-
+pression of the cold plasma dielectric permittivity, given
+in Eq. (28), we obtain a linear homogeneous system,
+�
+�
+n2 − n2
+x − S
+iD − nxny + i (V0/ω) nz −nxnz − i (V0/ω) ny
+−iD − nxny − i (V0/ω) nz
+n2 − n2
+x − S
+−nynz + i (V0/ω) nx
+−nxnz + i (V0/ω) ny
+−nynz − i (V0/ω) nx
+n2 − n2
+z − P
+�
+�
+�
+�
+δEx
+δEy
+δEz
+�
+� = 0.
+(40)
+Let us consider, for simplicity, the case the refractive
+index is parallel to the magnetic field, n = nˆz, such that
+one obtains
+�
+�
+n2 − S
+iD + i (V0/ω) n
+0
+−iD − i (V0/ω) n
+n2 − S
+0
+0
+0
+−P
+�
+�
+�
+�
+δEx
+δEy
+δEz
+�
+� = 0,
+(41)
+for which det[Mij] = 0 provides the dispersion relations
+P
+�
+ω2 �
+n2 − S
+�2 − (ωD + nV0)2�
+= 0.
+(42)
+Longitudinal waves, n ∥ δE or δE = (0, 0, δEz), may
+emerge, when P = 0, with non propagating vibration
+at the plasma frequency, ω = ωp. For transverse waves,
+n ⊥ δE or δE = (δEx, δEy, 0), the dispersion relation
+(42) simplifies as
+�
+n2 − S
+�2 − (D + n (V0/ω))2 = 0,
+(43)
+also written as a fourth-order equation in n,
+n4 −
+�
+2S + (V0/ω)2�
+n2 − 2D (V0/ω) n +
+�
+S2 − D2�
+= 0.
+(44)
+Taking into account the relations (29), the dispersion re-
+lation (44) provides the following refractive indices:
+nR,M = − V0
+2ω ±
+�
+1 +
+� V0
+2ω
+�2
+−
+ω2p
+ω(ω − ωc),
+(45)
+nL,E = V0
+2ω ±
+�
+1 +
+� V0
+2ω
+�2
+−
+ω2p
+ω(ω + ωc),
+(46)
+In general, the indices nR, nL, nE, nM may be real,
+imaginary, or complex (presenting both pieces) at some
+frequency ranges. As well-known, the real part is asso-
+ciated with propagation, while the complex piece is con-
+cerned with absorption. Furthermore, these indices may
+
+6
+have positive or negative real pieces.
+The indices nL
+and nM are always positive and negative, respectively,
+the latter one being a negative refractive index. On the
+other hand, the indices nR and nE can be positive or
+negative, depending on the frequency zone examined, in
+such a way the associated modes can manifest negative
+refraction behavior (in a suitable frequency band).
+The propagating modes associated with the refractive
+indices in Eq. (45)) and Eq. (46) are obtained by insert-
+ing each one in Eq. (41) and carrying out the correspond-
+ing eigenvector (with a null eigenvalue). The emerging
+electric field are the ELCP and ERCP , given in Eq. (32),
+where nR, nM are associated with the RCP mode, and
+nL, nE are related to the LCP mode,
+nL, nE �→ ELCP =
+i
+√
+2
+�
+1
+i
+�
+,
+(47)
+nR, nM �→ ERCP =
+i
+√
+2
+�
+1
+−i
+�
+.
+(48)
+From the indices nR, nE, given by Eqs. (45) and (46),
+we obtain the same cutoff frequencies (33) of the standard
+case: in fact, ω− is related to the refractive index nR,
+and ω+ is associated with the refractive index nE. In
+contrast, the refractive indices nL and nM have no real
+root. The behavior of the refractive indices in Eqs. (45)
+and (46) will be examined in the following.
+A.
+About the index nR
+We initiate discussing some properties of the index nR.
+The behavior of nR in terms of the dimensionless param-
+eter ω/ωc is illustrated in Fig. 3, which displays the real
+imaginary pieces of the refractive index nR. We point
+out:
+(i) It takes on a finite value when ω → 0, given by
+nR (0) = 1
+V0
+�
+ω2
+p
+ωc
+�
+,
+(49)
+differing from the behavior of the usual magnetized
+plasma index n−, which provides n → ∞ near the
+origin.
+(ii) For 0 < ω < ωc, nR is positive since the square root
+in (45) is real, positive, and larger than the negative
+piece before it. Such a positivity also holds for the
+usual index n−. See the black line in this frequency
+zone in Fig. 3.
+(iii) For ω → ωc, nR → ∞, and there occurs a resonance
+at the cyclotron frequency.
+(iv) For ωc < ω < ωr, there appears a negative refrac-
+tive index zone with absorption, where Re[nR] < 0
+and Im[nR] ̸= 0, as shown in Fig. 3. The frequency
+ωr is the root of the radicand in Eq. (45),
+R− (ω) = 1 + V 2
+0
+4ω2 −
+ω2
+p
+ω (ω − ωc),
+(50)
+which yields a cubic equation in ω.
+(v) For ωr < ω < ω−, one finds a negative refractive
+index zone without absorption, that is, Re[nR] < 0
+and Im[nR] = 0.
+(vi) For ω > ω−, the quantity nR is always positive,
+corresponding to a propagating zone, with nR → 1
+in the high-frequency limit.
+ω = ω-
+ω = ωr
+ω = ωc
+0
+1
+2
+3
+4
+-2
+-1
+0
+1
+2
+3
+4
+ω/ωc
+n
+FIG. 3. Index of refraction nR in terms of the frequency ω.
+The dashed blue (black) line corresponds to the imaginary
+piece of nR (n−), while the solid blue (black) line represents
+the real piece of nR (n−).
+Here ωc = ωp, V0 = 2ωp, and
+ωc = 1 rad s−1.
+The frequency zone in which Im[nR] ̸= 0, that is,
+ωc < ω < ωr, corresponds to the absorption zone for
+the metamaterial (negative refractive index) RCP wave,
+as already mentioned before. The frequency ranges in
+which Im[nR] = 0 define the propagation zone for the
+RCP wave.
+B.
+About the index nL
+The index nL, given in Eq.
+(46), has no real root,
+presenting the following features:
+(i) For ω → 0, nL → +∞. Then, the presence of the
+term V0 turns the refractive index real and pos-
+itively divergent at the origin, differing from the
+usual index n+ behavior, see Eq. (31), which is
+complex and divergent, Im[n+] → ∞, at the ori-
+gin.
+
+7
+(ii) For ω > 0, it is necessary to analyze the radicand
+in Eq. (46),
+R+ (ω) = 1 + V 2
+0
+4ω2 −
+ω2
+p
+ω (ω + ωc),
+(51)
+since it can be positive or negative, which deter-
+mines the absence or presence of an absorption
+zone, respectively. Note that for ω > ω+ the term
+1 − ω2
+p/ω (ω + ωc) is greater than zero (ω+ is the
+root of such a term), such that R+ is positive.
+Therefore, the possibility of R+ being negative oc-
+curs only in the range 0 < ω < ω+, for which the
+term 1 − ω2
+p/ω (ω + ωc) is less than zero. Hence,
+this positivity for R+ is stated by the condition,
+V 2
+0
+4ω2 >
+�����1 −
+4ω2
+p
+ω(ω + ωc)
+�����
+ω<ω+
+,
+(52)
+for which R+ is always positive and the refractive
+index nL is real for any ω > 0. This corresponds
+to a propagating mode for the entire frequency do-
+main. The behavior of nL in terms of the dimen-
+sionless parameter ω/ωc, considering the condition
+(52), that is, R+ > 0, is shown in Fig. 4.
+(iii) On the other hand, for
+V 2
+0
+4ω2 <
+�����1 −
+4ω2
+p
+ω(ω + ωc)
+�����
+ω<ω+
+,
+(53)
+one has R+
+<
+0 and nL becomes complex,
+Im[nL] ̸= 0, determining the opening of an absorp-
+tion zone located within the interval ωi < ω < ωf,
+as shown in Fig. 5. The frequencies ωi and ωf are
+positive and real roots of R+, a cubic equation in
+the frequency.
+ω = ω+
+0
+1
+2
+3
+4
+-2
+0
+2
+4
+6
+ω/ωc
+n
+FIG. 4. Refractive index nL (blue lines) for the condition (52),
+R+ > 0. Refractive index n+ (black lines) of Eq. (31). The
+dashed (solid) lines correspond to the imaginary (real) pieces
+of nL and n+. Here ωc = ωp, V0 = 2ωp, and ωc = 1 rad s−1.
+ω = ω+
+ω = ωf
+ω = ωi
+0.0
+0.5
+1.0
+1.5
+2.0
+-1
+0
+1
+2
+3
+4
+ω/ωc
+n
+FIG. 5. Refractive index nL (blue lines) for the condition (53),
+R+ < 0. Refractive index n+ (black lines) of Eq. (31). The
+dashed (solid) lines correspond to the imaginary (real) pieces
+of nL and n+. Here ωc = ωp, V0 = 0.7ωp, and ωc = 1 rad s−1.
+C.
+About the index nE
+The quantity nE is a refractive index that only exists
+as a positive quantity due to the presence of the chiral
+Lorentz-violating term. In the case we set V0 = 0, the
+second relation in Eq. (46) yields Re[nE] < 0 (negative
+index of refraction). For V0 ̸= 0, the index nE presents a
+small positivity range, Re[nE] > 0, which provides prop-
+agation for the associated LCP wave. We present below
+some aspects of nE:
+(i) For ω → 0, the index nE tends to a finite value at
+origin,
+nE (0) = 1
+V0
+�
+ω2
+p
+ωc
+�
+,
+(54)
+which is inversely proportional to the magnitude of
+the chiral factor, V0.
+(ii) Since the radicand of nE is the same one of nL, see
+Eq. (46), it holds here the same procedure applied
+for nL. For values of V0 that satisfy the condition
+(52), R+ > 0, nE is always real, Im[nE] = 0, being
+positive within the interval 0 < ω < ω+, and nega-
+tive for ω > ω+, since
+�
+R+ > V0/2ω at this range.
+The real and imaginary parts of nE are represented
+in Fig. 6.
+(iii) Considering the condition (53), nE becomes com-
+plex and exhibits an absorption zone, Im[nE] ̸= 0,
+in the interval ωi < ω < ωf, with ωi, ωf < ω+, as
+shown in Fig. 7. Such a figure depicts the real and
+imaginary pieces of nE (under the condition (53)).
+
+8
+ω = ω+
+0.0
+0.5
+1.0
+1.5
+2.0
+-4
+-3
+-2
+-1
+0
+1
+2
+ω/ωc
+n
+FIG. 6.
+Red line: plot of the index nE for the condition
+(52), R+ > 0. Black line: plot of the index −n+ of Eq. (31).
+Dashed (solid) lines represent the imaginary (real) pieces of
+nE and −n+. Here, we have used ωc = ωp and V0 = 2ωp,
+with the choice ωc = 1 rad s−1.
+ω = ω+
+ω = ωf
+ω = ωi
+0.0
+0.5
+1.0
+1.5
+2.0
+-3
+-2
+-1
+0
+1
+2
+3
+ω/ωc
+n
+FIG. 7.
+Red line: plot of the index nE for the condition
+(53), R+ < 0. Black line: plot of the index −n+ of Eq. (31).
+Dashed (solid) lines represent the imaginary (real) pieces of
+nE and −n+. Here, we have set ωc = ωp, V0 = 0.7ωp, and
+ωc = 1 rad s−1.
+D.
+About the index nM
+The additional index nM, given in Eq. (45), is always
+negative (negative refraction) and has no real root. The
+behavior of nM in terms of the dimensionless parameter
+ω/ωc is shown in Fig. 8. We notice the following features:
+(i) For 0 < ω < ωc, nM is real and negative since
+the square root in (45) is real. This is the same
+behavior of the index −n+. See the black line in
+Fig. (8).
+(ii) For ω → ωc, nM → −∞, and there occurs a reso-
+nance at the cyclotron frequency.
+(iii) For ωc < ω < ωr, there appears an absorption zone
+for metamaterial, Re[nM] < 0 and Im[nM] ̸= 0,
+while the index −n+ is purely imaginary, Re[nM] =
+0 and Im[nM] ̸= 0, as shown in Fig. 8. The fre-
+quency ωr is the root of R−.
+(iv) For ω > ωr, the quantity nM is always negative,
+corresponding to a negative propagation zone, with
+nM → −1 in the high-frequency limit.
+ω = ωc
+ω = ω-
+ω = ωr
+0
+1
+2
+3
+4
+-6
+-4
+-2
+0
+ω/ωc
+n
+FIG. 8. Blue line: plot of the index nM. Black line: plot
+of the real piece of −n− of Eq. (31).
+Dashed (solid) lines
+represent the imaginary (real) pieces of nM and −n−. Here,
+we have used: ωc = ωp, V0 = 2ωp, and ωc = 1 rad s−1.
+E.
+Dispersion relations behavior
+The wave dispersion associated with each refractive in-
+dex is usually visualized in plots ω × k. In the following,
+we work with dimensionless plots, (ω/ωc) × (k/ωc).
+The dispersion relations associated with nR and nM
+are depicted in Fig. 9 for ωc = ωp.
+The propagation
+occurs for 0 < ω < ωc and ω > ω−, while absorption
+takes place in ωc < ω < ωr. The range ωr < ω < ω−
+corresponds to negative refraction propagation zone (k <
+0) for nR. The refractive index nM is negative for k < 0
+and ω > 0.
+Figure 10 depicts the dispersion relations related to nL
+and nE. The wave associated with nL propagates for all
+frequencies. For nE, the conventional propagation zone
+occurs in 0 < ω < ω+. For ω > ω+, there occurs a prop-
+agation zone with negative refraction. For the standard
+indices, ±n+, the absorption zone is 0 < ω < ω+.
+Furthermore, Fig. 11 shows the dispersion relations for
+nL and nE in the case there is a modified absorption zone
+for ωi < ω < ωf, while the free propagation occurs for
+0 < ω < ωi and ω > ωf. The frequencies ωi, ωf and
+ωr define the limits for unusual propagation zones. As
+already discussed, these frequencies are obtained from
+the radicands (50) and (51).
+
+9
+Modified absorption
+Usual absorption zone
+ω = ω-
+ω = ωc
+ω = ωr
+ω = k
+-4
+-2
+0
+2
+4
+0
+1
+2
+3
+4
+k/ωc
+ω
+ωc
+FIG. 9. Plot of the dispersion relations related to refractive
+indices nR (solid red line) and nM (solid blue line).
+The
+dashed black line corresponds to the indices of the usual case
+(±n−). The highlighted area in red (gray) indicates the ab-
+sorption zone for nR,M (±n−). Here, we have used ωc = ωp
+and V0 = 2ωp, with ωc = 1 rad s−1.
+Usual absorption zone
+ω = ω+
+ω = k
+-2
+-1
+0
+1
+2
+0.0
+0.5
+1.0
+1.5
+2.0
+k/ωc
+ω
+ωc
+FIG. 10. Plot of the dispersion relations related to refractive
+indices nL (solid red line) and nE (solid blue line). The dashed
+line corresponds to the indices ±n+ of the usual case. The
+highlighted gray area indicates the absorption zone for ±n+,
+where now also occurs propagation. Here, we have used ωc =
+ωp and V0 = ωp, with ωc = 1 rad s−1.
+Modified absorption zone
+ω = ω+
+ω = ωi
+ω = ωf
+ω = k
+-2
+-1
+0
+1
+2
+0.0
+0.5
+1.0
+1.5
+2.0
+k/ωc
+ω
+ωc
+FIG. 11. Plot of the dispersion relations related to refractive
+indices nL (solid red line) and nE (solid blue line). The dashed
+line corresponds to the usual case with indices ±n+.
+The
+highlighted areas in red (gray) indicate the absorption zone
+for nL,E (±n+). Here, we have used ωc = ωp and V0 = 0.7ωc,
+with ωc = 1 rad s−1.
+V.
+BIREFRINGENCE, ROTATORY POWER
+AND DICHROISM
+The phase velocity in terms of the refractive index n
+is defined (in natural units) as vphase = 1/n.
+Hence,
+the corresponding phase velocities, vR = 1/ (nR), vL =
+1/ (nL), vE = 1/ (nE), vM = 1/ (nM), can be defined
+with the indices nR, nL, nE, nM of Eqs. (45) and (46).
+Accordingly with the previous analysis of the refractive
+indices, in general, the RCP and LCP modes propagate
+at different phase velocities for each frequency value, gen-
+erating circular birefringence in the propagation band,
+expressed in terms of the rotatory power (35). On the
+other hand, in the absorption zones, there occurs dichro-
+ism, measured in terms of the coefficient of Eq. (38).
+A.
+Rotatory power
+In order to write the rotatory power (RP), we need to
+consider the refractive indices nL, nE, associated with
+the LCP wave, and the indices nR, nM, associated to
+the RCP wave. It allows, in principle, to determine four
+distinct RPs at the propagation zones, some of which we
+examine in this section.
+We start by writing the rotation power defined in terms
+of real pieces of the refractive indices nL and nR,
+δLR = −ω
+2 (Re[nL] − Re[nR]) ,
+(55)
+
+10
+or explicitly,
+δLR = −ω
+2 Re
+�
+V0/ω +
+�
+R+ −
+�
+R−
+�
+,
+(56)
+where R+ and R− are given in Eqs. (50) and (51). We
+find a positive frequency,
+ˆω =
+�
+ω2c + ω2p/2 − ω2p
+�
+4ω2c + V 2
+0
+2V0
+,
+(57)
+where the RP (56) undergoes a sign reversal.
+In Fig.
+12, we illustrate the behavior of RP for the condition
+(52). For the interval 0 < ω < ˆω, the RP is negative,
+and for ˆω < ω < ωc, it is positive. The RP reversion
+that occurs at ω = ˆω is not usual in cold plasmas theory.
+However, it is reported in graphene systems [84], rotat-
+ing plasmas [95], and bi-isotropic dielectrics supporting
+chiral magnetic current [96].
+For ω > ωc, the RP is
+always negative. Nevertheless, it is necessary to pay at-
+tention to the interval ωc < ω < ωr, where the refractive
+index nR has an imaginary piece and the RCP wave is
+absorbed. At ω = ωr, the real piece of nR undergoes a
+sharp change (see Fig. 3), which also appears in the RP
+profile of Fig. 12.
+ω = ω
+ω = ωc
+ω = ωr
+ω = ω-
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+-2
+-1
+0
+1
+2
+ω (rad s-1)
+δ (rad m-1)
+FIG. 12. The solid blue line represents the rotatory power
+(56) defined by the refractive index nL and nR, for the con-
+dition (52). The dashed black line corresponds to the usual
+rotatory power (36). Here, we have used ωc = ωp, V0 = ωp,
+and ωc = 1 rad s−1.
+We can safely claim that both modes associated with
+the nL and nR propagate for ω > ω−, range in which
+the RP magnitude decreases monotonically with ω, ap-
+proaching to its asymptotic value, −V0/2 (see Fig. 12).
+Assuming the limit where ω >> (ωp, ωc), we can write
+nL,R ≈ 1 ± V0
+2ω + V 2
+0
+8ω2 −
+ω2
+p
+2ω (ω ± ωc),
+(58)
+so that the rotatory power is
+δLR ≈ −V0
+2 − ω2
+pωc
+2ω2 .
+(59)
+Note that taking the limit V0 → 0, the usual Faraday
+effect RP (37) is recovered for the high-frequency regime.
+It is also interesting to point out that the Faraday effect
+disappears for a null magnetic field, ωc = 0. However,
+the birefringence still remains, due to the presence of the
+chiral term, which yields the following RP:
+δ ≈ −V0/2.
+(60)
+For the condition (53), the RP (56) also exhibits a sign
+reversal and a very similar profile to the one of Fig. 12,
+in such a way that it will not be depicted here.
+Considering now the refractive indices nE and nR, the
+rotatory power is:
+δER = −ω
+2 (Re[nE] − Re[nR]) ,
+(61)
+or,
+δER = −ω
+2 Re
+�
+V0/ω −
+�
+R+ −
+�
+R−
+�
+.
+(62)
+Recalling that the LCP wave associated with nE has a
+conventional free propagation for ω < ω+ and propaga-
+tion with negative refractive index (nE < 0) for ω > ω+
+(with ω+ < ωc), the RP magnitude is enhanced in the
+latter zone. This behavior is depicted in Fig. 13, which
+shows the RP (62) for nE given by the condition (52),
+R+ > 0. The RP is positive for ω < ωc and negative
+for ωc < ω < ω′′, becoming positive again for ω > ω′′,
+where ω′′ is the reversal frequency. For nE given by the
+condition condition (53), the RP is depicted in Fig. 14,
+revealing a small reversion at ω′′ < ωc. Note that the
+increasing RP with ω, depicted in Figs. 13 and 14, is due
+to the negative behavior of the index nE for ω > ω+.
+In the asymptotic limit, where ω >> (ωp, ωc), the RP
+(62) goes as
+δER ≈ ω − V0
+2 ,
+(63)
+presentig a predominant linear behavior in ω, as it ap-
+pears in Figs. 13 and 14. It is also worth mentioning
+that the limit V0 → 0, implying δ ≈ ω, does not stand
+for a valid result for usual magnetized plasma, since the
+RP (62) is not defined for achiral cold plasmas.
+B.
+Dichroism coefficients
+As well known, absorption depends on the magnitude
+of the imaginary parts of the refractive indices. When
+one mode is more absorbed than the other, there occurs
+dichroism. Considering the refractive indices nL and nR,
+the circular dichroism coefficient is
+δdLR = −ω
+2 (Im[nL] − Im[nR]) .
+(64)
+Considering the condition (52), only nR has imaginary
+part (localized in the interval ωc < ω < ω−), while nL is
+
+11
+-0.1
+0
+0.1
+ω
+δ
+ω = ω''
+ω = ω''
+ω = ωc
+0
+1
+2
+3
+4
+-4
+-2
+0
+2
+4
+ω (��� �-�)
+δ (��� �-�)
+FIG. 13.
+Solid blue lines: plot of the rotatory power (62)
+associated to the refractive indices nE and nR for the con-
+dition (52).
+The dashed line represents the usual rotatory
+power (36).
+Here, we have used ωc = ωp, V0 = ωp, and
+ωc = 1 rad s−1. The inset plot highlights the behavior of δ
+around ω = ω′′.
+- 1
+10
+0
+1
+10
+ω
+δ
+ω = ω''
+ω = ω''
+ω = ωc
+0
+1
+2
+3
+4
+-4
+-2
+0
+2
+4
+ω (��� �-�)
+δ (��� �-�)
+FIG. 14. Solid red lines: rotatory power (62) associated to
+the refractive indices nE and nR for the condition (53). The
+dashed line represents the usual rotatory power (36). Here,
+we have used ωc = ωp, V0 = 0.7ωp, and ωc = 1 rad s−1. The
+inset plot highlights the behavior of δ around ω = ω′′.
+real for ω > 0. In this case, the dichroism coefficient is
+given by
+δdLR =
+�
+�
+�
+�
+�
+0,
+for 0 < ω < ωc,
+�
+R−,
+for ωc < ω < ωr,
+0,
+for ω > ωr,
+(65)
+being non null only in the range ωc < ω < ω−, as prop-
+erly shown in Fig 15.
+Considering the condition (53), both nR and nL have
+non-null imaginary parts in the intervals ωc < ω < ωr
+and ωi < ω < ωf, respectively. The dichroism coefficient
+is null for 0 < ω < ωi, ωf < ω < ωc, and ω > ωr, being
+ω = ω+
+ω = ω-
+ωc = ωp
+ω = ωr
+0.0
+0.5
+1.0
+1.5
+2.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+ω (rad s-1)
+δd (rad m-1)
+FIG. 15. Plot of the dichroism coefficient (65)(red solid lines)
+associated to the refractive indices nL and nR, under the
+condition (52). The black dashed line represents the usual
+dichroism coefficient (39). Here ωc = ωp, V0 = (3/2) ωc, and
+ωc = 1 rad s−1.
+non-null only for
+δdLR =
+�
+− ω
+2
+�
+R+,
+for ωi < ω < ωf,
++ ω
+2
+�
+R−,
+for ωc < ω < ωr,
+(66)
+whose general behavior is exhibited in Fig. 16.
+ω = ωi
+ω = ωf
+ω = ω+
+ω = ω-
+ωc = ωp
+ω = ωr
+0.0
+0.5
+1.0
+1.5
+2.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+ω (rad s-1)
+δd (rad m-1)
+FIG. 16. Plot of the dichroism coefficient (66)(solid red lines)
+associated to the refractive indices nL and nR, under the con-
+dition (53). The dashed line represents the usual dichroism
+coefficient (39). Here, we have set ωc = ωp, V0 = 0.7ωp, and
+ωc = 1 rad s−1.
+For the refractive indices nE and nR, the circular
+dichroism coefficient is
+δdER = −ω
+2 (Im[nE] − Im[nR]) .
+(67)
+If we consider nE under the condition (52), the same
+behavior of Fig. 15 is obtained, since nE is always real,
+not contributing to the dichroism. On the other hand,
+regarding now the condition (53), both nR e nE have
+non-zero imaginary parts in the intervals ωc < ω < ωr
+
+12
+and ωi < ω < ωf, respectively). In this case, we have
+δdER =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0,
+for 0 < ω < ωi,
++ ω
+2
+�
+R+,
+for ωi < ω < ωf,
+0,
+for ωf < ω < ωc,
++ ω
+2
+�
+R−,
+for ωc < ω < ωr,
+0,
+for ω > ωr.
+(68)
+The general behavior of the dichroism coefficient (68)
+is illustrated in Fig. 17.
+ω = ωi
+ω = ωf
+ω = ω+
+ω = ω-
+ωc = ωp
+ω = ωr
+0.0
+0.5
+1.0
+1.5
+2.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+ω (rad s-1)
+δd (rad m-1)
+FIG. 17. Plot of the dichroism coefficients (68) associated to
+the refractive indices nE and nR (for the condition (53)). The
+dashed line represents the usual dichroism coefficient (39).
+Here, we have used ωc = ωp, V0 = 0.7ωp, and ωc = 1 rad s−1.
+VI.
+FINAL REMARKS
+In this work, we have examined the propagation of
+electromagnetic waves in a cold magnetized plasma in
+the context of the chiral MCFJ electrodynamics, describ-
+ing the implied optical effects as well. We have adopted
+a MCFJ timelike background vector in order to repre-
+sent the chirality factor that breaks the parity. Starting
+from the modified Maxwell equations and employing the
+usual methods, we obtained four modified refractive in-
+dices given by Eqs. (45) and (46), associated with cir-
+cularly polarized propagating modes. Such indices were
+analyzed in detail in the Secs. IV A – IV D, where some
+of them exhibited significant modifications, as the index
+nR, see Fig. 3. It presents a negative refraction behavior
+in the range ωc < ω < ω−, in which it occurs propagation
+with absorption for ωc < ω < ωr and free (metamaterial)
+propagation for ωr < ω < ω−. The usual counterpart in-
+dex presents only pure absorption in this range.
+Optical effects of this system, involving birefringence
+and dichroism, were discussed in Sec. V, considering the
+refractive index nL and nR and nE. In Sec.V A, the RP
+δLR was introduced, see Eq. (56), exhibiting sign rever-
+sion at ω = ˆω, for the conditions (52) and (53). The RP
+δER also exhibits sign change at ω = ω′′ > ωc for the con-
+dition (52), and ω = ω′′ < ωc under the condition (53),
+as shown in Figs. 13 and 14, respectively. Such a RP
+reversal is not usual in cold plasmas, being reported in
+graphene systems [84], rotating plasmas [95], Weyl met-
+als and semimetals with low electron density with chi-
+ral conductivity [92, 93], and bi-isotropic dielectrics with
+magnetic chiral conductivity [96]. Comparing our results
+with the rotating plasma scenario of Ref. [95], there ap-
+pear differences. In the rotating plasma, the RP under-
+goes reversal and decays as 1/ω2 for high frequencies. In
+the present case, the rotatory power tends to the asymp-
+totical value −V0, see Eq. (60), or increases with ω when
+it involves the negative refraction index, see Eq. (63).
+These distinct RP properties may provide a channel to
+optically characterize chiral cold plasmas.
+Besides the nonconventional effect of reversion, the RP
+can also be enhanced when it is defined in the negative
+refraction zone. Such an enhancement occurs for δER,
+given in Eq. (62), for ω > ω+ (zone in which nE is neg-
+ative), being a topic of interest in metamaterial plasmas
+[116–119].
+Dichroism was examined in Sec.V B, where
+the coefficients δdLR and δdER have been shown to be
+non-null only in the range ωc < ω < ωr, for the condi-
+tion (52) - see Figs. 15, and in the intervals ωc < ω < ωr,
+ωi < ω < ωf, for the condition (53), in accordance with
+Figs. 16 and 17.
+ACKNOWLEDGMENTS
+The authors express their gratitude to FAPEMA,
+CNPq, and CAPES (Brazilian research agencies) for
+their invaluable financial support. M.M.F. is supported
+by FAPEMA Universal/01187/18, CNPq/Produtividade
+311220/2019-3
+and
+CNPq/Universal/422527/2021-1.
+P.D.S.S
+is
+supported
+by
+FAPEMA
+BPD-12562/22.
+Furthermore, we are indebted to CAPES/Finance Code
+001 and FAPEMA/POS- GRAD-02575/21.
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+page_content='Cold plasma waves in the chiral Maxwell-Carroll-Field-Jackiw electrodynamics  Filipe S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Ribeiro  a,∗ Pedro D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Silva  a,† and Manoel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Ferreira Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  b‡  aPrograma de P´os-gradua¸c˜ao em F´ısica, Universidade Federal do Maranh˜ao,  Campus Universit´ario do Bacanga, S˜ao Lu´ıs (MA), 65080-805, Brazil and  bDepartamento de F´ısica, Universidade Federal do Maranh˜ao,  Campus Universit´ario do Bacanga, S˜ao Lu´ıs (MA), 65080-805, Brazil  In this work, we study the propagation and absorption of plasma waves in the chiral Maxwell-  Carroll-Field-Jackiw electrodynamics (MCJF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The Maxwell equations are rewritten for a cold,  uniform, and collisionless fluid plasma model, allowing us to determine the new refractive indices and  propagating modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The case of transversal propagation is examined considering a purely timelike  CFJ background that plays the role of the magnetic conductivity chiral parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We find four  distinct refractive indices associated with RCP and LCP waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For each index, the propagation  and absorption zones are illustrated for some specific parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The optical behavior is  investigated by means of the rotatory power (RP) and dichroism coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The existence of a  negative refraction zone enhances the rotatory power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It is also observed RP sign reversal, a feature  of rotating plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  PACS numbers: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='Cp, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='Jb, 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='+e, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='Lc  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  INTRODUCTION  The study of electromagnetic (EM) waves propagation  [1, 2] in cold magnetized plasma is based on magneto-  ionic theory [3–8], developed by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Appleton [9] and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Hartree [10] between 1929 and 1932 to describe the radio  waves propagation in the ionosphere, in the context of the  usual electrodynamics [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' EM waves in plasmas have  been studied in other scenarios recently, as in logarithmic  nonlinear electrodynamics [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The chiral magnetic effect (CME) is the macroscopic  generation of an electric current in the presence of a mag-  netic field, stemming from an asymmetry between the  number density of left- and right-handed chiral fermions  [13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It has been extensively investigated in several  distinct contexts, such as quark-gluon plasmas [18–20],  cosmology [21], neutron stars [22, 23], and electroweak  interactions [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The CME plays a very relevant role  in Weyl semimetals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' where it is usually connected to  the chiral anomaly associated with Weyl nodal points  [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' the absence of the Weyl nodes [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' anisotropic ef-  fects stemming from tilted Weyl cones [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' the CME  and anomalous transport in Weyl semimetals [28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' quan-  tum oscillations arising from the CME [29],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' computa-  tion of the electromagnetic fields produced by an elec-  tric charge near a topological Weyl semimetal with two  Weyl nodes [30],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' renormalization evaluations for Weyl  semimetals and Dirac materials [31],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' and solutions of ax-  ion electrodynamics [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The CME current can be classically described by the  axion Lagrangian [32–38],  L = −1  4F µνFµν + θ(E · B),  (1)  ∗ filipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='ribeiro@discente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='ufma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='br, filipe99ribeiro@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='com  † pedro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='dss@discente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='ufma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='br, pdiego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='10@hotmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='com  ‡ manojr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='ufma@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='com, manoel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='messias@ufma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='br  where θ is the axion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In this context, the Maxwell  equations are  ∇ · E = ρ − ∇θ · B,  (2)  ∇ × B − ∂tE = j + (∂tθ)B + ∇θ × E,  (3)  where the terms involving θ derivatives find association  with condensed matter effects [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Indeed, ∇θ · B repre-  sents an anomalous charge density, while ∇θ×B appears  in the anomalous Hall effect, and (∂tθ)B plays the role  of the chiral magnetic current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In the case the axion field  does not depend on the space coordinates, ∇θ = 0, the  Maxwell equations (2) and (3) read  ∇ · E = ρ,  ∇ × B − ∂tE = j + (∂tθ)B,  (4)  where (∂tθ)B, the chiral magnetic current, may also  be addressed as a term of Maxwell-Carroll-Field-Jackiw  (MCFJ) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' A classical electrodynamics scenario en-  dowed with a chiral magnetic current has been investi-  gated considering symmetric and antisymmetric conduc-  tivity [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The latter case has also been addressed in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The MCFJ model [41] is the CPT-odd part of the U (1)  gauge sector of the Standard Model Extension (SME)  [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It is described by the Lagrangian density  L = −1  4F µνFµν − 1  4ϵµναβ (kAF )µ AνFαβ − AµJµ,  (5)  with (kAF )µ being the 4-vector background which con-  trols the Lorentz violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' This theory has been investi-  gated in multiple respects [43], encompassing radiative  evaluations [44, 45], topological defects solutions [46],  supersymmetric generalizations [47], classical solutions,  quantum aspects and unitarity analysis [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It may also  be connected with the CME in the sense that it provides  a modified Amp`ere’s law,  ∇ × B − ∂E  ∂t = J + k0  AF B + kAF × E,  (6)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='02183v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='plasm-ph]  5 Jan 2023  ID2  containing the magnetic current, JB = k0  AF B, with the  component k0  AF playing the role of the magnetic conduc-  tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The SME photon sector is also composed of a CPT-  even term constituted of a rank 4 Lorentz-violating tensor  [49], whose components may be properly parametrized in  terms of dimensionless 3 × 3 matrices, κDE, κDB, κHE,  and κHB, which allow to write generalized constitutive  relations between the fields (D, E) and (H, B),  �  D  H  �  =  �  �  ϵ1 + κDE  κDB  κHE  µ−11 + κHB  �  �  �  E  B  �  ,  (7)  similar to the ones that hold in continuous medium elec-  trodynamics, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (8a) and (8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, D is the elec-  tric displacement, while H is the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' This  CPT-even electrodynamics was investigated in several  contexts, involving consistency aspects [50], finite tem-  perature and boundary effects [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Lorentz-violating  electrodynamics in continuous matter [52, 53] has been  a topic of interest in the latest years due to its potential  to describe interesting effects of the phenomenology of  new materials, such as Weyl semimetals [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' A classical  field theory description of wave propagation, refractive  indices, and optical effects in a continuous medium de-  scribed by the MCFJ electrodynamics (with usual consti-  tutive relations), including its Lorentz-violating higher-  order derivative version [55], was discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Chiral media are endowed with parity violation [57–  60], being described by parity-odd models, as bi-isotropic  [61] and bi-anisotropic electrodynamics [62–67], whose  constitutive relations read  D = ˆϵ E + ˆα B,  (8a)  H = ˆβ E + ˆζ B,  (8b)  and ˆϵ = [ϵij], ˆα = [αij], ˆβ = [βij], and ˆζ = [ζij] represent,  in principle, 3 × 3 complex matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The bi-isotropic  relations involve the diagonal isotropic tensors, ϵij = ϵδij,  αij = αδij, βij = βδij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  In chiral scenarios, LCP and  RCP waves travel at distinct phase velocities, implying  birefringence and optical rotation [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' This phenomenon  stems from the natural optical activity of the medium  or can be induced by the action of external fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=', Faraday effect [69–71]), and it is measured in terms  of the rotation angle per unit length or rotatory power  (RP) [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Magneto-optical effects are used to investigate  features of new materials, such as topological insulators  [73–79] and graphene compounds [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The RP is a probe to examine the optical behavior of  several distinct systems, for instance, crystals [81, 82],  organic compounds [57, 83], graphene phenomena at ter-  ahertz band [84], and gas of fast-spinning molecules [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The optical rotation may depend on the frequency (RP  dispersion) and undergo reversion (anomalous RP dis-  persion) [86–88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It also finds interesting applications in  chiral metamaterials [89–91], chiral semimetals [92, 93],  in the determination of the rotation direction of pulsars  [94], and in rotating plasmas, which constitutes a sce-  nario where RP sign reversal also takes place [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Re-  cently, RP reversal was also reported in a bi-isotropic  dielectric in the presence of chiral magnetic current [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Furthermore, in the presence of absorption, dichroism  is another useful tool for the optical characterization of  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It occurs when LCP and RCP light waves are  absorbed by the medium at different degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It has been  used to distinguish between Dirac and Weyl semimetals  [97], perform enantiomeric discrimination [98, 99], and  for developing graphene-based devices at terahertz fre-  quencies [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Another feature of chiral systems is the possible oc-  currence of negative refraction and negative refractive  index, which was first proposed by Veselago in 1968 [101]  and experimentally observed in 2000 [102, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Later,  other experiments confirmed the negative refraction by  using Snell’s law [104, 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  This unusual property  was achieved in constructed metamaterials with both  negative electric permittivity and magnetic permeability  [106, 107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The negative refractive index also appears in  quark-gluon plasmas [108, 109], magnetoelectric materi-  als [110], metasurfaces [111], chiral bi-anisotropic meta-  materials [112, 113], and new materials, such as Dirac  semimetals [114, 115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  In chiral plasmas described by  generalized bi-isotropic constitutive relations [116,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 117],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  the negative refractive index can occur within some fre-  quency band and is not necessarily associated with si-  multaneously negative electric permittivity and negative  magnetic permeability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' being attributed to the chirality  parameter introduced in the constitutive relations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Di = εijEj + iξcBi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Hi = µ−1Bi + iξcEi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (9)  where εij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' and ξc are the plasma electric permit-  tivity tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' the magnetic permeability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' and the con-  stant chirality parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Plasmas metamaterials have  been investigated as new media endowed with interesting  properties, such as negative refraction and nonlinearities  [118, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  In this work, we are interested in examining the  wave propagation in a magnetized cold plasma ruled  by the MCFJ model, a chiral route distinct from the  bi-isotropic/anisotropic electrodynamics of the relations  (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We carry out our analysis considering the timelike  Lorentz-violating background component, which plays  the role of chiral magnetic conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The refractive  indices are evaluated and optical effects, such as bire-  fringence and dichroism, are examined, which could be  useful to trace analogies with other material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  We also find that the chiral conductivity yields negative  refraction in specific frequency bands, enhancing the ro-  tatory power and dichroism signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  This paper is outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' II, we briefly  review some aspects of the MCFJ model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' III, the  main properties of propagation in usual cold magnetized  plasmas are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The dispersion relations and re-  fractive indices for cold plasmas in chiral electrodynamics  3  are addressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The optical effects are examined  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Finally, we summarize our results in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  BASICS ON MCFJ ELECTRODYNAMICS  The Carroll-Field-Jackiw model was proposed as a  gauge invariant CPT-odd electrodynamics constrained  by birefringence data of distant galaxies [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It was later  incorporated as the CPT-odd sector of the SME [42], and  it has been investigated in several respects [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In  matter, it is described by the following Lagrangian den-  sity1 [56]:  L = −1  4GµνFµν − 1  4ϵµναβ (kAF )µ AνFαβ − AµJµ, (10)  yielding the MCFJ equation of motion,  ∂ρGρκ + ϵβκµν (kAF )β Fµν = Jκ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (11)  Here, (kAF )µ =  �  k0  AF , kAF  �  is a constant 4-vector back-  ground responsible for the Lorentz violation, and  Fµν = ∂µAν − ∂νAµ,  Gµν = 1  2χµναβFαβ,  (12)  are the usual U(1) vacuum and continuous matter field  strength, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The 4-rank tensor, χµναβ, de-  scribes the medium constitutive tensor [120], whose com-  ponents provide the electric and magnetic responses of  the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Indeed, the electric permittivity and mag-  netic permeability tensor components are written as  ϵij ≡ χ0ij0 and µ−1  lk  ≡  1  4ϵijlχijmnϵmnk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  For isotropic polarization and magnetization, it holds  ϵij = ϵδij and µ−1  ij = µδij, providing the usual isotropic  constitutive relations,  D = ϵE,  H = µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (13)  A straightforward calculation from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (11) yields  ∇ · D = J0 − kAF · B,  (14)  ∇ × H − ∂D  ∂t = J + k0  AF B + kAF × E,  (15)  where Gi0 = Di and Gij = −ϵijkHk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The homogeneous  Maxwell equations are given by  ∇ · B = 0,  ∇ × E + ∂B  ∂t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (16)  By using a plane-wave ansatz for the electromagnetic  fields, the MCFJ equations (14)-(16) read:  1 We use natural units h = c = 1 and the Minkowski metric sig-  nature gµν = diag (1, −1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ik · D + kAF · B = J0,  (17a)  ik × H + iωD − k0  AF B − kAF × E = J,  (17b)  k · B = 0,  k × E − ωB = 0,  (17c)  where k is the wave vector and ω is the (angular) wave  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  In the presence of anisotropy, the permittivity and per-  meability are represented by rank 2 tensors, εij and µij,  which may also depend on the frequency (for a dispersive  medium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For an anisotropic medium, the constitutive  relations (13) are replaced by [1, 2],  Di = εij(ω)Ej,  Bi = µij(ω)Hj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (18)  For non-magnetic media with isotropic magnetic perme-  ability, it holds µij(ω) = µ0, where µ0 is the vacuum  permeability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Considering the constitutive relations (18),  the modified Amp`ere-Maxwell’s law, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (17b), and Fara-  day’s law, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (17c), in the absence of sources, we obtain  a modified wave equation for the electric field,  ki �  kjEj�  − k2Ei = −ω2µ0¯εij (ω) Ej,  (19)  where we define the extended permittivity tensor,  ¯εij(ω) = εij(ω) − ik0  AF  ω2 ϵikjkk − iϵikj  kk  AF  ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (20)  Using the definition to the refractive index, n = k/ω, the  modified wave equation becomes  MijEj = 0,  (21)  with Mij given by  Mij = n2δij −ninj − εij  ε0  − i  ω  �  V0ϵikjnk + ϵikjV k�  , (22)  in which ε0 is the vacuum electric permittivity, and  V0 = k0  AF /ε0,  V k = kk  AF /ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (23)  appear as the components of a redefined background,  V µ =  �  V0, V i�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The nontrivial solutions for the elec-  tric field require a vanishing determinant of the matrix  Mij, det Mij = 0, which provides the dispersion relations  that describe the wave propagation in the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  In this work, we will study plasma waves propagation  for a chiral (parity-odd) medium, which means restrain-  ing our investigation to the case of a purely timelike  Lorentz-violating background vector, (kAF )µ =  �  k0  AF , 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The latter also plays the role of chiral magnetic conduc-  tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In this scenario, the wave equation (21) becomes  �  n2δij − ninj − εij  ε0  − iV0  ω ϵikjnk  �  Ej = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (24)  4  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  THE USUAL MAGNETIZED COLD  PLASMA  In this work we will adopt the fluid theory approach  in the cold plasma limit [3–7]:  ∂n  ∂t + ∇ · (nu) = 0,  (25)  ∂u  ∂t + u · ∇u = q  m (E + u × B0) ,  (26)  where n is the electron number density, u is the electron  fluid velocity field, q and m are the electron charge and  mass, respectively, and B0 is the equilibrium magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For simplicity, the ions are supposed to be infinitely  massive, which is appropriate for high-frequency waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Furthermore, thermal and collisional effects are also dis-  regarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The linearized version of the magnetized cold  plasmas [7] consider fluctuations around average quanti-  ties, n0 and B0, which are constant in time and space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Thus, the plasma quantities read  n = n0 + δn,  (27a)  u = δu,  (27b)  E = δE  (27c)  B = B0 + δB,  (27d)  with δn, δu, δE and δB being first order plane wave  magnitude perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Following the usual procedure  [3–6], assuming B0 = B0ˆz, we write the corresponding  dielectric tensor,  εij(ω) = ε0  �  �  S  −iD 0  iD  S  0  0  0  P  �  � ,  (28)  where  S = 1−  ω2  p  (ω2 − ω2c), D =  ωcω2  p  ω (ω2 − ω2c), P = 1− ω2  p  ω2 , (29)  and  ωp = n0q2  mϵ0  ,  ωc = |q|B0  m  ,  (30)  are the plasma and cyclotron frequencies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  From the Maxwell theory, two distinct refractive in-  dices are obtained,  n± =  �  1 −  ω2p  ω (ω ± ωc),  (31)  which provide right-handed circularly polarized (RCP)  and left-handed circularly polarized (LCP) modes,  ELCP =  i  √  2  �  1  i  �  ,  ERCP =  i  √  2  �  1  −i  �  ,  (32)  for the propagating modes associated to n±, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  This is the standard result of wave propagation in the  usual magnetized cold plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We recall that a cutoff  happens whenever the refractive index, n, goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  On the other hand, a resonance occurs if n tends to infin-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' From the indices (31), we obtain the following cutoff  frequencies:  ω± = 1  2  ��  ω2c + 4ω2p ∓ ωc  �  ,  (33)  where ω± is related to n±, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  A very usual effect in magnetized plasmas is the circu-  lar birefringence2, which causes the rotation of the plane  of polarization of a linearly polarized wave that propa-  gates within the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Thus the linearly polarized  wave emerges from the medium with an electric field  whose polarization is rotated relative to its initial lin-  ear configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Such a phenomenon can be properly  explained by decomposing the initial wave into two cir-  cularly polarized waves (RCP and LCP) that travel with  different phase velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In this case, the rotation an-  gle of the electric field can be expressed as the difference  between the refractive indices associated with the RCP  and LCP waves [68, 72]:  θ = πL  λ0  (Re [nRCP ] − Re [nLCP ]) ,  (34)  where λ0 is the vacuum wavelength of the incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The rotation power δ = θ/L (phase difference per unit  length), is given as  δ = −ω  2 (Re [nLCP ] − Re [nRCP ]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (35)  In a cold magnetized plasma, the refractive indices (31)  provide the following rotatory power:  δ = −ω  2 Re  �  �  �  1 −  ω2p  ω (ω + ωc) −  �  1 −  ω2p  ω (ω − ωc)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (36)  The behavior of the RP (36) in terms of the frequency  ω is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' One notices that there is a di-  vergence at ωc, being positive for ω < ωc and negative  for ω > ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It tends to zero at the high-frequency limit  ω >> (ωp, ωc), where it decays as  δ ≈ −ω2  pωc  2ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (37)  Associated with the imaginary part of the refractive in-  dex, one can also examine dichroism, an optical effect  2 In plasmas, the birefringence is usually a consequence of the Fara-  day effect, occurring due to the presence of the external field  B0, which generates distinct phase velocities for the propagating  modes [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  5  that occurs when circularly polarized waves are absorbed  by the medium at different degrees [56, 58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Thus  dichroism coefficient refers to the difference in absorp-  tion of LCP and RCP waves, being given by:  δd = −ω  2 (Im[nLCP ] − Im[nRCP ]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (38)  which, for the refractive indices (31), implies  δd = −ω  2 Im  �  �  �  1 −  ω2p  ω (ω + ωc) −  �  1 −  ω2p  ω (ω − ωc)  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (39)  ω = ωc  ω = 2 ωc  ω = ω-  ω = ω-  0  1  2  3  4  2  1  0  1  2  ω (rad s-1)  δ (rad m-1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Rotatory power (36) in terms of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, ωc = ωp (red  line) and ωc = 2ωp (blue line), with the choice ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Such a quantity is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 2, which shows sin-  gularity at the cyclotron frequency ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  For ωc = ωp  (red curve), the dichroism coefficient (39) is negative for  ω < ωred  + , positive for 2ωc < ω < ωred  −  and null for other  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The case for ωc = ωp/2 (blue curve) differs  in the fact that ωblue  +  is greater than ωc, showing that  (39) is now negative for ω < ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ωc  ω = 2 ωc  ω = ω-  ω = ω-  ω = ω+  ω = ω+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  2  1  0  1  2  ω (rad s-1)  δd (rad m-1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Dichroism coefficient (39) in terms of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, ωc =  ωp (red line) and ωc = ωp/2 (blue line), with ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  WAVE PROPAGATION IN CHIRAL  ELECTRODYNAMICS  Starting from the wave equation (24) and using the ex-  pression of the cold plasma dielectric permittivity, given  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (28), we obtain a linear homogeneous system,  �  �  n2 − n2  x − S  iD − nxny + i (V0/ω) nz −nxnz − i (V0/ω) ny  −iD − nxny − i (V0/ω) nz  n2 − n2  x − S  −nynz + i (V0/ω) nx  −nxnz + i (V0/ω) ny  −nynz − i (V0/ω) nx  n2 − n2  z − P  �  �  �  �  δEx  δEy  δEz  �  � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (40)  Let us consider, for simplicity, the case the refractive  index is parallel to the magnetic field, n = nˆz, such that  one obtains  �  �  n2 − S  iD + i (V0/ω) n  0  −iD − i (V0/ω) n  n2 − S  0  0  0  −P  �  �  �  �  δEx  δEy  δEz  �  � = 0,  (41)  for which det[Mij] = 0 provides the dispersion relations  P  �  ω2 �  n2 − S  �2 − (ωD + nV0)2�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (42)  Longitudinal waves, n ∥ δE or δE = (0, 0, δEz), may  emerge, when P = 0, with non propagating vibration  at the plasma frequency, ω = ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For transverse waves,  n ⊥ δE or δE = (δEx, δEy, 0), the dispersion relation  (42) simplifies as  �  n2 − S  �2 − (D + n (V0/ω))2 = 0,  (43)  also written as a fourth-order equation in n,  n4 −  �  2S + (V0/ω)2�  n2 − 2D (V0/ω) n +  �  S2 − D2�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (44)  Taking into account the relations (29), the dispersion re-  lation (44) provides the following refractive indices:  nR,M = − V0  2ω ±  �  1 +  � V0  2ω  �2  −  ω2p  ω(ω − ωc),  (45)  nL,E = V0  2ω ±  �  1 +  � V0  2ω  �2  −  ω2p  ω(ω + ωc),  (46)  In general, the indices nR, nL, nE, nM may be real,  imaginary, or complex (presenting both pieces) at some  frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' As well-known, the real part is asso-  ciated with propagation, while the complex piece is con-  cerned with absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Furthermore, these indices may  6  have positive or negative real pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The indices nL  and nM are always positive and negative, respectively,  the latter one being a negative refractive index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' On the  other hand, the indices nR and nE can be positive or  negative, depending on the frequency zone examined, in  such a way the associated modes can manifest negative  refraction behavior (in a suitable frequency band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The propagating modes associated with the refractive  indices in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (45)) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (46) are obtained by insert-  ing each one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (41) and carrying out the correspond-  ing eigenvector (with a null eigenvalue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The emerging  electric field are the ELCP and ERCP , given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (32),  where nR, nM are associated with the RCP mode, and  nL, nE are related to the LCP mode,  nL, nE �→ ELCP =  i  √  2  �  1  i  �  ,  (47)  nR, nM �→ ERCP =  i  √  2  �  1  −i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (48)  From the indices nR, nE, given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (45) and (46),  we obtain the same cutoff frequencies (33) of the standard  case: in fact, ω− is related to the refractive index nR,  and ω+ is associated with the refractive index nE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In  contrast, the refractive indices nL and nM have no real  root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The behavior of the refractive indices in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (45)  and (46) will be examined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  About the index nR  We initiate discussing some properties of the index nR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The behavior of nR in terms of the dimensionless param-  eter ω/ωc is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 3, which displays the real  imaginary pieces of the refractive index nR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We point  out:  (i) It takes on a finite value when ω → 0, given by  nR (0) = 1  V0  �  ω2  p  ωc  �  ,  (49)  differing from the behavior of the usual magnetized  plasma index n−, which provides n → ∞ near the  origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (ii) For 0 < ω < ωc, nR is positive since the square root  in (45) is real, positive, and larger than the negative  piece before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Such a positivity also holds for the  usual index n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' See the black line in this frequency  zone in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (iii) For ω → ωc, nR → ∞, and there occurs a resonance  at the cyclotron frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (iv) For ωc < ω < ωr, there appears a negative refrac-  tive index zone with absorption, where Re[nR] < 0  and Im[nR] ̸= 0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The frequency  ωr is the root of the radicand in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (45),  R− (ω) = 1 + V 2  0  4ω2 −  ω2  p  ω (ω − ωc),  (50)  which yields a cubic equation in ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (v) For ωr < ω < ω−, one finds a negative refractive  index zone without absorption, that is, Re[nR] < 0  and Im[nR] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (vi) For ω > ω−, the quantity nR is always positive,  corresponding to a propagating zone, with nR → 1  in the high-frequency limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ω-  ω = ωr  ω = ωc  0  1  2  3  4  2  1  0  1  2  3  4  ω/ωc  n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Index of refraction nR in terms of the frequency ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The dashed blue (black) line corresponds to the imaginary  piece of nR (n−), while the solid blue (black) line represents  the real piece of nR (n−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Here ωc = ωp, V0 = 2ωp, and  ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The frequency zone in which Im[nR] ̸= 0, that is,  ωc < ω < ωr, corresponds to the absorption zone for  the metamaterial (negative refractive index) RCP wave,  as already mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The frequency ranges in  which Im[nR] = 0 define the propagation zone for the  RCP wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  About the index nL  The index nL, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (46), has no real root,  presenting the following features:  (i) For ω → 0, nL → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Then, the presence of the  term V0 turns the refractive index real and pos-  itively divergent at the origin, differing from the  usual index n+ behavior, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (31), which is  complex and divergent, Im[n+] → ∞, at the ori-  gin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  7  (ii) For ω > 0, it is necessary to analyze the radicand  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (46),  R+ (ω) = 1 + V 2  0  4ω2 −  ω2  p  ω (ω + ωc),  (51)  since it can be positive or negative, which deter-  mines the absence or presence of an absorption  zone, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Note that for ω > ω+ the term  1 − ω2  p/ω (ω + ωc) is greater than zero (ω+ is the  root of such a term), such that R+ is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Therefore, the possibility of R+ being negative oc-  curs only in the range 0 < ω < ω+, for which the  term 1 − ω2  p/ω (ω + ωc) is less than zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Hence,  this positivity for R+ is stated by the condition,  V 2  0  4ω2 >  �����1 −  4ω2  p  ω(ω + ωc)  �����  ω<ω+  ,  (52)  for which R+ is always positive and the refractive  index nL is real for any ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' This corresponds  to a propagating mode for the entire frequency do-  main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The behavior of nL in terms of the dimen-  sionless parameter ω/ωc, considering the condition  (52), that is, R+ > 0, is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (iii) On the other hand, for  V 2  0  4ω2 <  �����1 −  4ω2  p  ω(ω + ωc)  �����  ω<ω+  ,  (53)  one has R+  <  0 and nL becomes complex,  Im[nL] ̸= 0, determining the opening of an absorp-  tion zone located within the interval ωi < ω < ωf,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The frequencies ωi and ωf are  positive and real roots of R+, a cubic equation in  the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ω+  0  1  2  3  4  2  0  2  4  6  ω/ωc  n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Refractive index nL (blue lines) for the condition (52),  R+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Refractive index n+ (black lines) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The  dashed (solid) lines correspond to the imaginary (real) pieces  of nL and n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here ωc = ωp, V0 = 2ωp, and ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ω+  ω = ωf  ω = ωi  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1  0  1  2  3  4  ω/ωc  n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Refractive index nL (blue lines) for the condition (53),  R+ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Refractive index n+ (black lines) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The  dashed (solid) lines correspond to the imaginary (real) pieces  of nL and n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here ωc = ωp, V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='7ωp, and ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  About the index nE  The quantity nE is a refractive index that only exists  as a positive quantity due to the presence of the chiral  Lorentz-violating term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In the case we set V0 = 0, the  second relation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (46) yields Re[nE] < 0 (negative  index of refraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For V0 ̸= 0, the index nE presents a  small positivity range, Re[nE] > 0, which provides prop-  agation for the associated LCP wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We present below  some aspects of nE:  (i) For ω → 0, the index nE tends to a finite value at  origin,  nE (0) = 1  V0  �  ω2  p  ωc  �  ,  (54)  which is inversely proportional to the magnitude of  the chiral factor, V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (ii) Since the radicand of nE is the same one of nL, see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (46), it holds here the same procedure applied  for nL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For values of V0 that satisfy the condition  (52), R+ > 0, nE is always real, Im[nE] = 0, being  positive within the interval 0 < ω < ω+, and nega-  tive for ω > ω+, since  �  R+ > V0/2ω at this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The real and imaginary parts of nE are represented  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (iii) Considering the condition (53), nE becomes com-  plex and exhibits an absorption zone, Im[nE] ̸= 0,  in the interval ωi < ω < ωf, with ωi, ωf < ω+, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Such a figure depicts the real and  imaginary pieces of nE (under the condition (53)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  8  ω = ω+  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  4  3  2  1  0  1  2  ω/ωc  n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Red line: plot of the index nE for the condition  (52), R+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Black line: plot of the index −n+ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Dashed (solid) lines represent the imaginary (real) pieces of  nE and −n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, we have used ωc = ωp and V0 = 2ωp,  with the choice ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ω+  ω = ωf  ω = ωi  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  3  2  1  0  1  2  3  ω/ωc  n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Red line: plot of the index nE for the condition  (53), R+ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Black line: plot of the index −n+ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Dashed (solid) lines represent the imaginary (real) pieces of  nE and −n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, we have set ωc = ωp, V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='7ωp, and  ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  About the index nM  The additional index nM, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (45), is always  negative (negative refraction) and has no real root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The  behavior of nM in terms of the dimensionless parameter  ω/ωc is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We notice the following features:  (i) For 0 < ω < ωc, nM is real and negative since  the square root in (45) is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' This is the same  behavior of the index −n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' See the black line in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (ii) For ω → ωc, nM → −∞, and there occurs a reso-  nance at the cyclotron frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (iii) For ωc < ω < ωr, there appears an absorption zone  for metamaterial, Re[nM] < 0 and Im[nM] ̸= 0,  while the index −n+ is purely imaginary, Re[nM] =  0 and Im[nM] ̸= 0, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The fre-  quency ωr is the root of R−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (iv) For ω > ωr, the quantity nM is always negative,  corresponding to a negative propagation zone, with  nM → −1 in the high-frequency limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ωc  ω = ω-  ω = ωr  0  1  2  3  4  6  4  2  0  ω/ωc  n  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Blue line: plot of the index nM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Black line: plot  of the real piece of −n− of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Dashed (solid) lines  represent the imaginary (real) pieces of nM and −n−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here,  we have used: ωc = ωp, V0 = 2ωp, and ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Dispersion relations behavior  The wave dispersion associated with each refractive in-  dex is usually visualized in plots ω × k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In the following,  we work with dimensionless plots, (ω/ωc) × (k/ωc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The dispersion relations associated with nR and nM  are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 9 for ωc = ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The propagation  occurs for 0 < ω < ωc and ω > ω−, while absorption  takes place in ωc < ω < ωr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The range ωr < ω < ω−  corresponds to negative refraction propagation zone (k <  0) for nR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The refractive index nM is negative for k < 0  and ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Figure 10 depicts the dispersion relations related to nL  and nE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The wave associated with nL propagates for all  frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For nE, the conventional propagation zone  occurs in 0 < ω < ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For ω > ω+, there occurs a prop-  agation zone with negative refraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For the standard  indices, ±n+, the absorption zone is 0 < ω < ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 11 shows the dispersion relations for  nL and nE in the case there is a modified absorption zone  for ωi < ω < ωf, while the free propagation occurs for  0 < ω < ωi and ω > ωf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The frequencies ωi, ωf and  ωr define the limits for unusual propagation zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' As  already discussed, these frequencies are obtained from  the radicands (50) and (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  9  Modified absorption  Usual absorption zone  ω = ω-  ω = ωc  ω = ωr  ω = k  4  2  0  2  4  0  1  2  3  4  k/ωc  ω  ωc  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Plot of the dispersion relations related to refractive  indices nR (solid red line) and nM (solid blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The  dashed black line corresponds to the indices of the usual case  (±n−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The highlighted area in red (gray) indicates the ab-  sorption zone for nR,M (±n−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, we have used ωc = ωp  and V0 = 2ωp, with ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Usual absorption zone  ω = ω+  ω = k  2  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  k/ωc  ω  ωc  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Plot of the dispersion relations related to refractive  indices nL (solid red line) and nE (solid blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The dashed  line corresponds to the indices ±n+ of the usual case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The  highlighted gray area indicates the absorption zone for ±n+,  where now also occurs propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, we have used ωc =  ωp and V0 = ωp, with ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Modified absorption zone  ω = ω+  ω = ωi  ω = ωf  ω = k  2  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  k/ωc  ω  ωc  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Plot of the dispersion relations related to refractive  indices nL (solid red line) and nE (solid blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The dashed  line corresponds to the usual case with indices ±n+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The  highlighted areas in red (gray) indicate the absorption zone  for nL,E (±n+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, we have used ωc = ωp and V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='7ωc,  with ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  BIREFRINGENCE, ROTATORY POWER  AND DICHROISM  The phase velocity in terms of the refractive index n  is defined (in natural units) as vphase = 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Hence,  the corresponding phase velocities, vR = 1/ (nR), vL =  1/ (nL), vE = 1/ (nE), vM = 1/ (nM), can be defined  with the indices nR, nL, nE, nM of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (45) and (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Accordingly with the previous analysis of the refractive  indices, in general, the RCP and LCP modes propagate  at different phase velocities for each frequency value, gen-  erating circular birefringence in the propagation band,  expressed in terms of the rotatory power (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' On the  other hand, in the absorption zones, there occurs dichro-  ism, measured in terms of the coefficient of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Rotatory power  In order to write the rotatory power (RP), we need to  consider the refractive indices nL, nE, associated with  the LCP wave, and the indices nR, nM, associated to  the RCP wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It allows, in principle, to determine four  distinct RPs at the propagation zones, some of which we  examine in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  We start by writing the rotation power defined in terms  of real pieces of the refractive indices nL and nR,  δLR = −ω  2 (Re[nL] − Re[nR]) ,  (55)  10  or explicitly,  δLR = −ω  2 Re  �  V0/ω +  �  R+ −  �  R−  �  ,  (56)  where R+ and R− are given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (50) and (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We  find a positive frequency,  ˆω =  �  ω2c + ω2p/2 − ω2p  �  4ω2c + V 2  0  2V0  ,  (57)  where the RP (56) undergoes a sign reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  12, we illustrate the behavior of RP for the condition  (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For the interval 0 < ω < ˆω, the RP is negative,  and for ˆω < ω < ωc, it is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The RP reversion  that occurs at ω = ˆω is not usual in cold plasmas theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  However, it is reported in graphene systems [84], rotat-  ing plasmas [95], and bi-isotropic dielectrics supporting  chiral magnetic current [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  For ω > ωc, the RP is  always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Nevertheless, it is necessary to pay at-  tention to the interval ωc < ω < ωr, where the refractive  index nR has an imaginary piece and the RCP wave is  absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' At ω = ωr, the real piece of nR undergoes a  sharp change (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 3), which also appears in the RP  profile of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ω\uf111  ω = ωc  ω = ωr  ω = ω-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  2  1  0  1  2  ω (rad s-1)  δ (rad m-1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The solid blue line represents the rotatory power  (56) defined by the refractive index nL and nR, for the con-  dition (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The dashed black line corresponds to the usual  rotatory power (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, we have used ωc = ωp, V0 = ωp,  and ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  We can safely claim that both modes associated with  the nL and nR propagate for ω > ω−, range in which  the RP magnitude decreases monotonically with ω, ap-  proaching to its asymptotic value, −V0/2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Assuming the limit where ω >> (ωp, ωc), we can write  nL,R ≈ 1 ± V0  2ω + V 2  0  8ω2 −  ω2  p  2ω (ω ± ωc),  (58)  so that the rotatory power is  δLR ≈ −V0  2 − ω2  pωc  2ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (59)  Note that taking the limit V0 → 0, the usual Faraday  effect RP (37) is recovered for the high-frequency regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  It is also interesting to point out that the Faraday effect  disappears for a null magnetic field, ωc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' However,  the birefringence still remains, due to the presence of the  chiral term, which yields the following RP:  δ ≈ −V0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (60)  For the condition (53), the RP (56) also exhibits a sign  reversal and a very similar profile to the one of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 12,  in such a way that it will not be depicted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Considering now the refractive indices nE and nR, the  rotatory power is:  δER = −ω  2 (Re[nE] − Re[nR]) ,  (61)  or,  δER = −ω  2 Re  �  V0/ω −  �  R+ −  �  R−  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (62)  Recalling that the LCP wave associated with nE has a  conventional free propagation for ω < ω+ and propaga-  tion with negative refractive index (nE < 0) for ω > ω+  (with ω+ < ωc), the RP magnitude is enhanced in the  latter zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' This behavior is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 13, which  shows the RP (62) for nE given by the condition (52),  R+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The RP is positive for ω < ωc and negative  for ωc < ω < ω′′, becoming positive again for ω > ω′′,  where ω′′ is the reversal frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' For nE given by the  condition condition (53), the RP is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 14,  revealing a small reversion at ω′′ < ωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Note that the  increasing RP with ω, depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 13 and 14, is due  to the negative behavior of the index nE for ω > ω+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  In the asymptotic limit, where ω >> (ωp, ωc), the RP  (62) goes as  δER ≈ ω − V0  2 ,  (63)  presentig a predominant linear behavior in ω, as it ap-  pears in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It is also worth mentioning  that the limit V0 → 0, implying δ ≈ ω, does not stand  for a valid result for usual magnetized plasma, since the  RP (62) is not defined for achiral cold plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Dichroism coefficients  As well known, absorption depends on the magnitude  of the imaginary parts of the refractive indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' When  one mode is more absorbed than the other, there occurs  dichroism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Considering the refractive indices nL and nR,  the circular dichroism coefficient is  δdLR = −ω  2 (Im[nL] − Im[nR]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (64)  Considering the condition (52), only nR has imaginary  part (localized in the interval ωc < ω < ω−), while nL is  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content="1  ω  δ  ω = ω''  ω = ω''  ω = ωc  0  1  2  3  4  4  2  0  2  4  ω (��� �-�)  δ (��� �-�)  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Solid blue lines: plot of the rotatory power (62)  associated to the refractive indices nE and nR for the con-  dition (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  The dashed line represents the usual rotatory  power (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Here, we have used ωc = ωp, V0 = ωp, and  ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The inset plot highlights the behavior of δ  around ω = ω′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content="  1  10  0  1  10  ω  δ  ω = ω''  ω = ω''  ω = ωc  0  1  2  3  4  4  2  0  2  4  ω (��� �-�)  δ (��� �-�)  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Solid red lines: rotatory power (62) associated to  the refractive indices nE and nR for the condition (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The  dashed line represents the usual rotatory power (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here,  we have used ωc = ωp, V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='7ωp, and ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The  inset plot highlights the behavior of δ around ω = ω′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  real for ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In this case, the dichroism coefficient is  given by  δdLR =  �  �  �  �  �  0,  for 0 < ω < ωc,  �  R−,  for ωc < ω < ωr,  0,  for ω > ωr,  (65)  being non null only in the range ωc < ω < ω−, as prop-  erly shown in Fig 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Considering the condition (53), both nR and nL have  non-null imaginary parts in the intervals ωc < ω < ωr  and ωi < ω < ωf, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The dichroism coefficient  is null for 0 < ω < ωi, ωf < ω < ωc, and ω > ωr, being  ω = ω+  ω = ω-  ωc = ωp  ω = ωr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  ω (rad s-1)  δd (rad m-1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Plot of the dichroism coefficient (65)(red solid lines)  associated to the refractive indices nL and nR, under the  condition (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The black dashed line represents the usual  dichroism coefficient (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here ωc = ωp, V0 = (3/2) ωc, and  ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  non-null only for  δdLR =  �  − ω  2  �  R+,  for ωi < ω < ωf,  + ω  2  �  R−,  for ωc < ω < ωr,  (66)  whose general behavior is exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ωi  ω = ωf  ω = ω+  ω = ω-  ωc = ωp  ω = ωr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  ω (rad s-1)  δd (rad m-1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Plot of the dichroism coefficient (66)(solid red lines)  associated to the refractive indices nL and nR, under the con-  dition (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The dashed line represents the usual dichroism  coefficient (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Here, we have set ωc = ωp, V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='7ωp, and  ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  For the refractive indices nE and nR, the circular  dichroism coefficient is  δdER = −ω  2 (Im[nE] − Im[nR]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (67)  If we consider nE under the condition (52), the same  behavior of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 15 is obtained, since nE is always real,  not contributing to the dichroism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' On the other hand,  regarding now the condition (53), both nR e nE have  non-zero imaginary parts in the intervals ωc < ω < ωr  12  and ωi < ω < ωf, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In this case, we have  δdER =  �  �  �  �  �  �  �  �  �  �  �  �  �  0,  for 0 < ω < ωi,  + ω  2  �  R+,  for ωi < ω < ωf,  0,  for ωf < ω < ωc,  + ω  2  �  R−,  for ωc < ω < ωr,  0,  for ω > ωr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  (68)  The general behavior of the dichroism coefficient (68)  is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ω = ωi  ω = ωf  ω = ω+  ω = ω-  ωc = ωp  ω = ωr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='0  ω (rad s-1)  δd (rad m-1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Plot of the dichroism coefficients (68) associated to  the refractive indices nE and nR (for the condition (53)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The  dashed line represents the usual dichroism coefficient (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Here, we have used ωc = ωp, V0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='7ωp, and ωc = 1 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  FINAL REMARKS  In this work, we have examined the propagation of  electromagnetic waves in a cold magnetized plasma in  the context of the chiral MCFJ electrodynamics, describ-  ing the implied optical effects as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' We have adopted  a MCFJ timelike background vector in order to repre-  sent the chirality factor that breaks the parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Starting  from the modified Maxwell equations and employing the  usual methods, we obtained four modified refractive in-  dices given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (45) and (46), associated with cir-  cularly polarized propagating modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Such indices were  analyzed in detail in the Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' IV A – IV D, where some  of them exhibited significant modifications, as the index  nR, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' It presents a negative refraction behavior  in the range ωc < ω < ω−, in which it occurs propagation  with absorption for ωc < ω < ωr and free (metamaterial)  propagation for ωr < ω < ω−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The usual counterpart in-  dex presents only pure absorption in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Optical effects of this system, involving birefringence  and dichroism, were discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' V, considering the  refractive index nL and nR and nE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='V A, the RP  δLR was introduced, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (56), exhibiting sign rever-  sion at ω = ˆω, for the conditions (52) and (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' The RP  δER also exhibits sign change at ω = ω′′ > ωc for the con-  dition (52), and ω = ω′′ < ωc under the condition (53),  as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 13 and 14, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Such a RP  reversal is not usual in cold plasmas, being reported in  graphene systems [84], rotating plasmas [95], Weyl met-  als and semimetals with low electron density with chi-  ral conductivity [92, 93], and bi-isotropic dielectrics with  magnetic chiral conductivity [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Comparing our results  with the rotating plasma scenario of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' [95], there ap-  pear differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In the rotating plasma, the RP under-  goes reversal and decays as 1/ω2 for high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' In  the present case, the rotatory power tends to the asymp-  totical value −V0, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (60), or increases with ω when  it involves the negative refraction index, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  These distinct RP properties may provide a channel to  optically characterize chiral cold plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Besides the nonconventional effect of reversion, the RP  can also be enhanced when it is defined in the negative  refraction zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Such an enhancement occurs for δER,  given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' (62), for ω > ω+ (zone in which nE is neg-  ative), being a topic of interest in metamaterial plasmas  [116–119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Dichroism was examined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='V B, where  the coefficients δdLR and δdER have been shown to be  non-null only in the range ωc < ω < ωr, for the condi-  tion (52) - see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 15, and in the intervals ωc < ω < ωr,  ωi < ω < ωf, for the condition (53), in accordance with  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 16 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors express their gratitude to FAPEMA,  CNPq, and CAPES (Brazilian research agencies) for  their invaluable financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' is supported  by FAPEMA Universal/01187/18, CNPq/Produtividade  311220/2019-3  and  CNPq/Universal/422527/2021-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='S  is  supported  by  FAPEMA  BPD-12562/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Furthermore, we are indebted to CAPES/Finance Code  001 and FAPEMA/POS- GRAD-02575/21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Appleton, Wireless Studies of the Iono-  sphere, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' Electr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' Hartree, The propagation of electromagnetic  waves in a stratified medium, Mathematical Proceed-  ings of the Cambridge Philosophical Society, 25:97–120,  1929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' Ratcliff, The formation of the ionosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Kharzeev, The chiral magnetic effect and anomaly-  induced transport, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' 75, 133  (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Kharzeev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' Voloshin, and  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content='  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE0T4oBgHgl3EQfPwB8/content/2301.02183v1.pdf'}
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+VS-Net: Multiscale Spatiotemporal Features for
+Lightweight Video Salient Document Detection
+1st Hemraj Singh
+Dept. of Computer Sc. & Engg.
+National Institute of Technology
+Warangal, Telangana, India
+hs720079@student.nitw.ac.in
+2nd Mridula Verma
+Institute for Development and Research
+in Banking Technology
+Hyderabad, Telangana, India
+vmridula@idrbt.ac.in
+3rd Ramalingaswamy Cheruku
+Dept. of Computer Sc. & Engg.
+National Institute of Technology
+Warangal, Telangana, India
+rmlswamy@nitw.ac.in
+Abstract—Video Salient Document Detection (VSDD) is an es-
+sential task of practical computer vision, which aims to highlight
+visually salient document regions in video frames. Previous tech-
+niques for VSDD focus on learning features without considering
+the cooperation among and across the appearance and motion
+cues and thus fail to perform in practical scenarios. Moreover,
+most of the previous techniques demand high computational
+resources, which limits the usage of such systems in resource-
+constrained settings. To handle these issues, we propose VS-Net,
+which captures multi-scale spatiotemporal information with the
+help of dilated depth-wise separable convolution and Approxi-
+mation Rank Pooling. VS-Net extracts the key features locally
+from each frame across embedding sub-spaces and forwards
+the features between adjacent and parallel nodes, enhancing
+model performance globally. Our model generates saliency maps
+considering both the background and foreground simultaneously,
+making it perform better in challenging scenarios. The immense
+experiments regulated on the benchmark MIDV-500 dataset show
+that the VS-Net model outperforms state-of-the-art approaches
+in both time and robustness measures.
+Index Terms—Separable convolution, Approximation Rank
+Pooling, Variational Autoencoder, Multi-scale features
+I. INTRODUCTION
+Video Salient Document Detection (VSDD) is an essential
+task in several real-world applications, such as video document
+recognition [1], video document compression [2], video doc-
+ument captioning [3] and many more. In real-life scenarios,
+a number of challenges appear due to an unconstrained envi-
+ronment (as shown in Fig.1). Current state-of-the-art (SOTA)
+models [4], [1] utilize non-selective attentional resources in the
+dynamic scenes. They employ limited static features and thus
+face difficulties in detecting the intended object in multiple
+real-world scenarios.
+Most of the existing VSDD models [6], [7], [8] extract
+the spatial features separately using a computationally costly
+process and then integrate them to generate a spatial saliency
+map. Later, they use a different method to extract the refined
+spatial-temporal features. Segregating these two steps reduces
+the quality of the generated frame. This segregation also fails
+to capture the longer-term motion arrangement, which links
+with some actions.
+Supported by Ministry of Electronics and Information Technology (MeiTy),
+Government of India and IIT Bhilai Innovation and Technology Foundation
+(IBITF) under the project entitled ”Blockchain and Machine Learning Pow-
+ered Unified Video KYC Framework”
+(a) Complex Scene
+(b) Keyboard Scene
+(c) Partial Scene
+(d) Noise
+(e) Motion Blur
+(f) Illumination
+Fig. 1. Challenging scenarios from MIDV-500 [5] dataset.
+One of the most popular model for video analytics is U-
+Net [9], which extends the temporal dimension for substituting
+2D filters with 3D filters and produces a little benediction of
+annotated videos to help the 3D convolution layers. Sebastian
+et al. [3] proposed DeepDeSRT, an end-to-end system for
+table understanding in document images and detecting PDF
+documents. But it failed in the video datasets due to poor
+handling of temporal features. To solve these problems, Recur-
+rent Neural Networks (RNNs) [7] used memory cells for the
+long-term pattern, which parses the video frames sequentially
+and encodes the information. The LSTMs use convolutional
+neural networks [10] and output in the form of action labels
+or video specifications. The Autoencoder LSTM model [4] is
+proposed to use either an instant or the next frame for accurate
+reconstruction. Tenet model in [11] acquired salient object de-
+tection metrics and performed unsupervised training on CNN.
+Sheshkus et al. [2] proposed HoughEncoder neural network
+architecture and performed Fast Hough Transform to calculate
+low-level features for the image semantic segmentation task,
+however, failed to resolve challenges due to an unconstrained
+environment. In [12] a spatiotemporal conditional random
+field is proposed to establish the relationships between local
+and global context regions, but the method failed to extract
+the high-level features. Wujie Zhou et al. [13] designed a
+convolution residual module to send equally distributed feature
+maps between the encoder and the decoder but failed due to
+long-range skip connections. The recent method [14] proposed
+Vnet to optimize the skip connection but failed to combine
+arXiv:2301.04447v1  [cs.CV]  11 Jan 2023
+
+D
+Mustermann
+Erika
+3
+12.08.64
+4a.
+22.01.15
+MUSTER
+Berlln
+Musterhausen
+4o,Landratsent
+21.01.30
+B072RRE2155
+am see%
+5
+ni
+FUHRERSCHEIN BUNDESRERUBLK DEUTSCHLAND
+D
+Mustermann
+MUSTER
+Erika
+1.15
+3.
+12.08.64
+Berlin
+4a.22.01.15
+4c.Landratsamt
+Musterhausen
+am See
+4b.21.01.30
+B072RRE2155
+A
+9.CM/B/LP
+D
+C01X0006H
+MUSTERMANN
+ERIKA
+DEUTSCH
+12.08.1964
+BERLIN
+01.11.2007
+31.10.2017
+D6H1D<<6408125F1710319<<<<<<<<<<<<<0FUHRERSCMEINEARE究
+ETIKS
+12.06.54
+22.01:15
+Mueterpnuoen
+21-01.39
+B672RRE2159
+PA/B1Hustermann
+Erlka
+MUSTERmultilevel feature information.
+We propose a novel VS-Net model to overcome these
+problems, which utilizes the separable convolution in the
+combination of the variational encoder to extract the key
+features from each frame across embedding sub-spaces and
+forward the features between adjacent and parallel nodes.
+Our model extracts the spatiotemporal features locally and
+makes better predictions globally. The main contributions of
+our paper are given below: noitemsep, nolistsep
+• We design a novel spatiotemporal-based VS-Net model
+with separable convolutions in variational autoencoder ar-
+chitecture (VAE) [15], which reduces the skip-connection
+between two nodes and generates the generalized latent
+space vector.
+• We utilize the Approximation Rank Pooling (ARP) [16],
+which takes input features from separable convolutions
+intermediate layers to train the VS-Net model. It pro-
+vides low-rank approximation features to preserve their
+temporal locality.
+• We have conducted experiments with MIDV-500 [5]
+dataset and demonstrated that VS-Net performs better in
+terms of both efficiency and time.
+II. PROPOSED METHODOLOGY
+A. VS-Net Architecture
+Based on prior knowledge, spatial and temporal-based
+methods can capture better location information and preserve
+location boundaries than pixel-wise CNN methods. Therefore,
+we design a novel spatial and temporal-based VS-Net model
+with separable convolutions [17] in variational auto-encoder
+architecture (VAE) [15], which reduces the skip-connection
+between two nodes and generates the generalized latent space
+vector (shown in fig.2).
+Given a sequence of input frames (Sn|n = 1, 2, 3, · · ·, N),
+and corresponding ground-truth maps (Gn|n = 1, 2, 3, · · ·, N)
+are first passed into the VS-Net model to extract the spatial and
+temporal features, which uses pretrained weights of ResNet50
+[7]. Our proposed model has two branches with different
+purposes. The first is the down-sampling operation performing
+top to bottom, extracting the spatial and temporal features from
+each node and reducing the feature vectors’ dimension. The
+second is upsampling operation from bottom to top, which
+decodes the spatial-temporal latent space and enhances the
+feature vectors. At last, we combine feature vectors from
+previous nodes and parallel nodes.
+During the down-sampling, we perform separable convolu-
+tion operation with 3 × 3 filters on input frames to extract
+spatial and temporal features, which have rich spatial and
+temporal information. Then max-pooling operation with 2 × 2
+filters succeeded by a ReLU activation operation performs to
+downgrade the dimension of the features vectors and generate
+the latent spatial and temporal map.
+Sd ∼ Down(Sn) = SConv(S1, S2, S3, S4, · · ·, Sn),
+(1)
+where SConv is a separable convolution operation.
+Before the up-sampling, we perform a convolution operation
+using 1 × 1 filters with learnable weight θ and applied the
+ReLU activation function to reduce the dimension of the latent
+features vectors and apply a dropout operation to dropout the
+50% neurons for generating the latent space of spatial and
+temporal features vectors.
+Sl = Dropout(ReLU(Conv(Sd, θ))).
+(2)
+During the up-sampling operation from bottom to top, we
+decode the latent space vectors using separable convolution
+layers with 3× 3 filters, succeeded by up-sampling layers and
+ReLU activation operation to reconstruct the dimension of the
+features vectors.
+Sup ∼ UP(Sl) = SConv(S1, S2, S3, S4, · · ·, Sn),
+(3)
+where SConv is a separable convolution layers with 3×3 filters.
+We extract the spatial and temporal features from latent space
+during the upsampling. The extracted spatial and temporal
+features of each parallel node and adjacent node both are
+concatenated, i.e.,
+Sc = Conc(Sup, Sd),
+(4)
+where Conc is concatenation operation. Then we perform
+separable convolution layers with 3 × 3 filters succeeded by
+the ReLU activation function to enhance the quality of the
+spatial-temporal feature vectors and reconstruct the original
+dimension of latent features vectors. Further, we apply the
+Sigmoid function using the convolution layer with 1×1 filters
+to simplify the spatial and temporal features, i.e.,
+Sm = Sigmoid(Sc).
+(5)
+The network has approximately 3.5 million trainable pa-
+rameters. We notice that each layer of the VS-Net generates
+a feature map with a spatial structure in places of the video
+frames. Max Pooling layers use to increase the feature’s map
+generation speed, which updates the weight matrix of the
+backbone models during the feature extraction from every
+separable convolution layer. During the bottom-up extraction
+of features from high to low resolution, upsampling operations
+with 2×2 filters are used to distribute the latent feature space
+and combine them with the previous layer and parallel layer’s
+nodes’ features.
+We used Approximation Rank Pooling (ARP) [16], which
+takes input features from the intermediate layers of a VS-
+Net, trains on sub-sequences, and generates the output of
+a subspace. ARP not only gives low-rank approximation
+features, but it also conserves temporal order. The low-rank
+approximation differentiated and captured important character-
+istics of the data, which summarizes the document’s position
+and orientation. Further, a quadratic ranking function captured
+the temporal order, which handles non-linear dependencies of
+the input features. Generally, the temporal order deals with the
+protuberances of the input channels onto the substance.
+Due to the low-rank approximation, the down-sampling
+generates the generalized latent space vector. The sampling of
+
+Input Frame
+Saliency Map
+UpSampling
+Down Sampling
+SConv1
+SConv2
+SConv+MaxPool+ReLU
+SConv+MaxPool+ReLU
+SConv+MaxPool+ReLU
+SConv+MaxPool+ 
+ReLU
+SConv6
+SConv7
+SConv8
+SConv9
+SConv10
+SConv14
+SConv15
+Fig. 2. Architecture of the proposed VS-Net.
+mean and variance gives the efficient latent distribution of the
+VS-Net architecture. Based on the latent Gaussian distribution,
+the latent vector is generalized. The down-sampling and up-
+sampling are performed based on the variational encoder and
+decoder. For handling the over-fitting of the proposed model,
+the latent space of the down-sampling is normalized with the
+help of convolution layers and passed to up-sampling layers.
+After concatenating all the spatial and temporal features, we
+applied a convolution layer with 1×1 filters at the last node to
+generate feature vectors. At last, the previous node and parallel
+node features are aggregated and provide the saliency map.
+TABLE I
+PERFORMANCE COMPARISON OF THE SOTA AND PROPOSED MODEL
+(VS-NET) IN TERMS OF BCE+IOU LOSS(%), AND TESTING SPEED (FPS)
+Input frame
+Model
+Output frame
+Loss (BCE+IoU)
+Testing speed (FPS)
+RNN+LSTM [7]
+0.032
+12.34
+U-Net [9]
+0.028
+10.45
+FCNN [18]
+0.038
+12.36
+HE [2]
+0.041
+14.45
+STCRF [12]
+0.039
+13.54
+AED [13]
+0.035
+12.89
+CNN+LSTM [10]
+0.036
+12.63
+RCNN [19]
+0.025
+11.54
+VS-Net
+0.021
+8.36
+B. Loss function
+During training, we use input frames Sk with the corre-
+sponding ground-truth Gk at frame t. The binary cross-entropy
+loss Lbce [20] is used to calculate the dissimilarity of the
+output and target, which is given as follows,
+lbce(Sk, Gk) = −
+�
+i,j
+[Gk(i, j)log(Sk(i, j))+
+(1 − Gk(i, j))log(1 − Sk(i, j))]
+(6)
+where (i,j) represents a coordinate of the frames. The IoU
+loss [21] is computed as:
+lIoU(Sk, Gk) = 1 −
+�w
+i=1
+�h
+j=1 Sk(i, j)Gk(i, j)
+�w
+i=1
+�h
+j=1[Sk(i, j) + Gk(i, j) − Sk(i, j)Gk(i, j)]
+(7)
+The total loss function is derived as,
+Totalloss = lbce(Sk, Gk) + αlIoU(Sk, Gk),
+(8)
+where α is the weighting parameter for the IoU. For the final
+saliency map prediction, we utilize Sk and Gk since it shows
+that our experiments better utilized spatial-temporal cues.
+III. EXPERIMENTAL SETUP AND RESULT ANALYSIS
+A. MIDV-500 Dataset
+We have considered the MIDV-500 dataset [5], which con-
+tains video clips of 50 different identity documents (seventeen
+Id cards, fourteen passports, six identity documents, and
+thirteen driving licenses) of various countries. It has eight
+different variations of background and foreground scenes. The
+attributes of the dataset are TS (Table Scene), TA (Table
+Action), KS (Keyboard Scene), KA (Keyboard Action), HS
+(Hand Scene), HA (Hand Action), PS (Partial Scene), PA
+(Partial Action), CS (Complex Scene), CA (Complex Action).
+Thus a total of 500 videos are generated (50 documents× 5
+desperation×2 devices). Each video has a duration of three
+seconds, which is split into ten frames per second and the
+corresponding annotation. The dataset contains examples of
+multiple challenging scenarios, such as complex scenes, small
+
+FUHRERSCHEIN BUNDESREPUBLKDEUTSCHLAND
+D
+Mustermann
+Erika
+12.08.64
+MUSTER
+4a.22.01.15
+Berlin
+Musterhausen
+4c.Landratsant
+21.0:.30
+B072RRE2155
+am See
+M/B/LFUHRERSCHEIN BUNDESREPUBLKDEUTSCHLAND
+D
+Mustermann
+Erika
+12.08.64
+MUSTER
+4a.22.01.15
+Berlin
+Musterhausen
+4c.Landratsant
+21.0:.30
+B072RRE2155
+am See
+M/B/Lobjects with different variations of frames, appearances of
+background and foreground, cluttered background, etc. (a
+few examples are shown in Fig: 1). The ground truth is
+prepared for each extracted video frame with various document
+locations in JSON format. It has 48 photo patches, 40 signature
+patches, and 546 text patches. The patches convert Cyrillic,
+Greek, Chinese, Japanese, Arabic, and Persian with singular
+Latin characters.
+B. Optimization and Propagation
+For preparing the VS-Net model, the Adaptive Moment Esti-
+mation (ADAM) optimizer is used to facilitate the computation
+of learning rates of each parameter using the first and second
+moment of the gradient. The saliency map optimization is cast
+as a “label propagation” problem, where uncertain labels are
+propagated based on background and foreground seeds. Fur-
+ther, we perform shift and rotation invariance robustly to the
+deformation of gray value variations. These random variations
+of deformation are sampled from the Gaussian distribution.
+Drop-out layers are used at the bottom stop and the previous
+level to normalize the latent space vectors. The proposed VS-
+Net model optimizes the loss of saliency maps from top-to-
+bottom and bottom-to-top and propagative seeds sequentially
+interact. The sequential seeding procedure optimizes feature
+map loss and improves the robustness to construct a saliency
+map.
+C. Evaluation Metrics
+We used Intersection Over Union (IoU) loss as an evaluation
+metric, which is defined as the intersection over the predicted
+boundary box (bbox), and the actual bbox and divided with
+their union. A prediction considers True Positive if Inter-
+section over union (IoU)>threshold, and False Positive If
+IoU<threshold [21].
+IoU = Area
+of
+Overlap
+Area
+of
+Union
+(9)
+The smaller the IoU value, the better the performance. In our
+experiments, the threshold is set as 0.5. The comparison results
+in terms of accuracy are shown in Tab. II).
+TABLE II
+COMPARISON RESULTS WITH SOTA MODELS AND PROPOSED MODEL ON
+MIDV-500 DATASET
+Comparison of document detection accuracy (%)
+Model
+TS,
+KS,
+HS,
+PS,
+CS
+TA
+KA
+HA
+PA
+CA
+U-Net (2019) [9]
+96.44
+97.43
+97.12
+96.64
+97.40
+RNN+LSTM (2021) [7]
+97.12
+96.33
+97.56
+95.56
+96.59
+FCNN (2017)[18]
+97.78
+98.39
+98.89
+96.43
+97.43
+HE (2020) [2]
+96.86
+97.58
+97.54
+96.32
+96.45
+STCRF (2018) [12]
+97.43
+96.98
+97.56
+96.43
+97.98
+AED (2020)[13]
+96.97
+98.54
+98.74
+96.78
+97.69
+CNN+LSTM (2019)[10]
+97.56
+98.54
+98.74
+96.78
+97.69
+RCNN (2019)[19]
+98.43
+98.69
+97.78
+97.19
+98.56
+VS-Net
+99.25
+99.67
+99.62
+99.45
+99.74
+D. Implementation Details
+1) Training Setup: The experiments are accomplished on
+Intel Xeon(R) CPU E5-2640 v4 @ 2.40GHz X40 processor
+with 32GB RAM. We used Tensorflow-gpu2.0, Keras as the
+backend, and Python3.6 accelerated by NVIDIA RTX Graphic
+card (Quadro P5000 / PCIe /SSE2). Input frames are of size
+256×256. The ADAM optimizer, weight decay of 0.006, batch
+size of 128, and learning rate of 0.001 are used to train the
+model, and BCE +IoU loss function is used to calculate the
+loss of the VS-Net model. The running time comparison is
+given in Tab. III.
+2) Testing Setup and Runtime: We resize the frame 256 ×
+256 to feed into the corresponding branch for testing. The
+distribution of the dataset is in the ratio of 3:7. The average
+testing speed of our model is 9.46 fps which is less than the
+existing models. Additionally, we do not perform any pre-
+/post-processing [19]. For validating purpose, the 5-fold cross-
+validation are used to check the overfitting (see Fig.3).
+Fig. 3.
+5-fold cross-validation comparison result of VS-Net and SOTA
+models.
+Ablation Study - The qualitative evaluation shows that
+our VS-Net model gives the best results on MIDV-500 [5]
+datasets but is an extremely lightweight setting that has fewer
+parameters and FLOPs (see in Table. IV).
+Speed Comparison The speed performance is calculated
+on a 64-bit Linux Ubuntu-18.04 operating system with Intel
+Core i7-4590 CPU @ 3.3 GHz. It has 32GB RAM and 1 TB
+Hard disk. The average speed is computed on all frame images
+of MIDV-500 datasets and does not include I/O file time. The
+parallel processing of multiple images is not allowed. Priorly,
+SOTA models require high computational powerful GPU sys-
+tem. For a fair comparison, the recent unsupervised models
+compare the speed on a normal CPU and the speed of deep
+learning methods reported on GPU. The speed comparison is
+given in Table III. We show the trade-off between loss and
+testing speed in Table I. Further, the average time cost of
+each step is examined. VS-Net consumes 256 ms and SOTA
+models 270 ms, respectively, while saliency prediction and
+global optimization use 8.36 ms and 10.0 ms.
+IV. CONCLUSION AND FUTURE WORK
+In this paper, we proposed a novel efficient and fast VS-Net
+model that fully leverages the spatiotemporal features to detect
+
+100
+99
+98
+Accuracy
+97
+96
+95
+94
+HE
+AED
+STCRF
+VS-Net
+RNN+LSTM
+CNN+LSTMTABLE III
+RUNTIME COMPARISON OF SOTA MODELS AND VS-NET MODEL
+U-Net[9]
+RNN+LSTM[7]
+FCNN[18]
+HE [2]
+STCRF[12]
+AED[13]
+CNN+LSTM[10]
+RCNN[19]
+VS-Net
+Runtime (s)
+188
+191
+190
+198
+200
+199
+196
+189
+175
+Step (ms)
+265
+266
+273
+278
+275
+268
+267
+260
+256
+TABLE IV
+ABLATION STUDY WITH RESNET-50 AS BACKBONE.
+# Param(M)
+FLOPS(G)
+Runtime (s)
+Steps(ms)
+Accuracy
+5.42
+10.2
+200
+285
+97.98
+3.54
+7.3
+185
+275
+98.85
+1.20
+3.7
+175
+265
+99.75
+0.95
+2.5
+165
+255
+96.45
+the video identity document. The proposed model used sep-
+arable convolutions with variational autoencoder architecture,
+which performs downsampling for extracting the features and
+upsampling operation for decoding the latent space features.
+The Approximation Rank Pooling (ARP) is used low-rank
+approximation on frames to preserve the temporal locality. The
+bottom-end the generalized latent space is generated. Further,
+These features are combined to generate in fuse spatial and
+temporal characteristics. We validated each module of the
+proposed model with the help of extensive experiments, which
+can be considered as the unified solution advancing VSDD.
+REFERENCES
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+
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+page_content='VS-Net: Multiscale Spatiotemporal Features for  Lightweight Video Salient Document Detection  1st Hemraj Singh  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' of Computer Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' & Engg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  National Institute of Technology  Warangal, Telangana, India  hs720079@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='nitw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='in  2nd Mridula Verma  Institute for Development and Research  in Banking Technology  Hyderabad, Telangana, India  vmridula@idrbt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='in  3rd Ramalingaswamy Cheruku  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' of Computer Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' & Engg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  National Institute of Technology  Warangal, Telangana, India  rmlswamy@nitw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='in  Abstract—Video Salient Document Detection (VSDD) is an es-  sential task of practical computer vision, which aims to highlight  visually salient document regions in video frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Previous tech-  niques for VSDD focus on learning features without considering  the cooperation among and across the appearance and motion  cues and thus fail to perform in practical scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Moreover,  most of the previous techniques demand high computational  resources, which limits the usage of such systems in resource-  constrained settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' To handle these issues, we propose VS-Net,  which captures multi-scale spatiotemporal information with the  help of dilated depth-wise separable convolution and Approxi-  mation Rank Pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' VS-Net extracts the key features locally  from each frame across embedding sub-spaces and forwards  the features between adjacent and parallel nodes, enhancing  model performance globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Our model generates saliency maps  considering both the background and foreground simultaneously,  making it perform better in challenging scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The immense  experiments regulated on the benchmark MIDV-500 dataset show  that the VS-Net model outperforms state-of-the-art approaches  in both time and robustness measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Index Terms—Separable convolution, Approximation Rank  Pooling, Variational Autoencoder, Multi-scale features  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' INTRODUCTION  Video Salient Document Detection (VSDD) is an essential  task in several real-world applications, such as video document  recognition [1], video document compression [2], video doc-  ument captioning [3] and many more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' In real-life scenarios,  a number of challenges appear due to an unconstrained envi-  ronment (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Current state-of-the-art (SOTA)  models [4], [1] utilize non-selective attentional resources in the  dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' They employ limited static features and thus  face difficulties in detecting the intended object in multiple  real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Most of the existing VSDD models [6], [7], [8] extract  the spatial features separately using a computationally costly  process and then integrate them to generate a spatial saliency  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Later, they use a different method to extract the refined  spatial-temporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Segregating these two steps reduces  the quality of the generated frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' This segregation also fails  to capture the longer-term motion arrangement, which links  with some actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Supported by Ministry of Electronics and Information Technology (MeiTy),  Government of India and IIT Bhilai Innovation and Technology Foundation  (IBITF) under the project entitled ”Blockchain and Machine Learning Pow-  ered Unified Video KYC Framework”  (a) Complex Scene  (b) Keyboard Scene  (c) Partial Scene  (d) Noise  (e) Motion Blur  (f) Illumination  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Challenging scenarios from MIDV-500 [5] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  One of the most popular model for video analytics is U-  Net [9], which extends the temporal dimension for substituting  2D filters with 3D filters and produces a little benediction of  annotated videos to help the 3D convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Sebastian  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' [3] proposed DeepDeSRT, an end-to-end system for  table understanding in document images and detecting PDF  documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' But it failed in the video datasets due to poor  handling of temporal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' To solve these problems, Recur-  rent Neural Networks (RNNs) [7] used memory cells for the  long-term pattern, which parses the video frames sequentially  and encodes the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The LSTMs use convolutional  neural networks [10] and output in the form of action labels  or video specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The Autoencoder LSTM model [4] is  proposed to use either an instant or the next frame for accurate  reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Tenet model in [11] acquired salient object de-  tection metrics and performed unsupervised training on CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Sheshkus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' [2] proposed HoughEncoder neural network  architecture and performed Fast Hough Transform to calculate  low-level features for the image semantic segmentation task,  however, failed to resolve challenges due to an unconstrained  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' In [12] a spatiotemporal conditional random  field is proposed to establish the relationships between local  and global context regions, but the method failed to extract  the high-level features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Wujie Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' [13] designed a  convolution residual module to send equally distributed feature  maps between the encoder and the decoder but failed due to  long-range skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The recent method [14] proposed  Vnet to optimize the skip connection but failed to combine  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='04447v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='CV]  11 Jan 2023  D  Mustermann  Erika  3  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='64  4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='15  MUSTER  Berlln  Musterhausen  4o,Landratsent  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='30  B072RRE2155  am see%  5  ni  FUHRERSCHEIN BUNDESRERUBLK DEUTSCHLAND  D  Mustermann  MUSTER  Erika  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='64  Berlin  4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+page_content='Landratsamt  Musterhausen  am See  4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='30  B072RRE2155  A  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='CM/B/LP  D  C01X0006H  MUSTERMANN  ERIKA  DEUTSCH  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='1964  BERLIN  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='2007  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='2017  D6H1D<<6408125F1710319<<<<<<<<<<<<<0FUHRERSCMEINEARE究  ETIKS  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='54  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='01:15  Mueterpnuoen  21-01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='39  B672RRE2159  PA/B1Hustermann  Erlka  MUSTERmultilevel feature information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  We propose a novel VS-Net model to overcome these  problems, which utilizes the separable convolution in the  combination of the variational encoder to extract the key  features from each frame across embedding sub-spaces and  forward the features between adjacent and parallel nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Our model extracts the spatiotemporal features locally and  makes better predictions globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The main contributions of  our paper are given below: noitemsep, nolistsep  We design a novel spatiotemporal-based VS-Net model  with separable convolutions in variational autoencoder ar-  chitecture (VAE) [15], which reduces the skip-connection  between two nodes and generates the generalized latent  space vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  We utilize the Approximation Rank Pooling (ARP) [16],  which takes input features from separable convolutions  intermediate layers to train the VS-Net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' It pro-  vides low-rank approximation features to preserve their  temporal locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  We have conducted experiments with MIDV-500 [5]  dataset and demonstrated that VS-Net performs better in  terms of both efficiency and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' PROPOSED METHODOLOGY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' VS-Net Architecture  Based on prior knowledge, spatial and temporal-based  methods can capture better location information and preserve  location boundaries than pixel-wise CNN methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Therefore,  we design a novel spatial and temporal-based VS-Net model  with separable convolutions [17] in variational auto-encoder  architecture (VAE) [15], which reduces the skip-connection  between two nodes and generates the generalized latent space  vector (shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Given a sequence of input frames (Sn|n = 1, 2, 3, · · ·, N),  and corresponding ground-truth maps (Gn|n = 1, 2, 3, · · ·, N)  are first passed into the VS-Net model to extract the spatial and  temporal features, which uses pretrained weights of ResNet50  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Our proposed model has two branches with different  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The first is the down-sampling operation performing  top to bottom, extracting the spatial and temporal features from  each node and reducing the feature vectors’ dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The  second is upsampling operation from bottom to top, which  decodes the spatial-temporal latent space and enhances the  feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' At last, we combine feature vectors from  previous nodes and parallel nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  During the down-sampling, we perform separable convolu-  tion operation with 3 × 3 filters on input frames to extract  spatial and temporal features, which have rich spatial and  temporal information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Then max-pooling operation with 2 × 2  filters succeeded by a ReLU activation operation performs to  downgrade the dimension of the features vectors and generate  the latent spatial and temporal map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Sd ∼ Down(Sn) = SConv(S1, S2, S3, S4, · · ·, Sn),  (1)  where SConv is a separable convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Before the up-sampling, we perform a convolution operation  using 1 × 1 filters with learnable weight θ and applied the  ReLU activation function to reduce the dimension of the latent  features vectors and apply a dropout operation to dropout the  50% neurons for generating the latent space of spatial and  temporal features vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Sl = Dropout(ReLU(Conv(Sd, θ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  (2)  During the up-sampling operation from bottom to top, we  decode the latent space vectors using separable convolution  layers with 3× 3 filters, succeeded by up-sampling layers and  ReLU activation operation to reconstruct the dimension of the  features vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Sup ∼ UP(Sl) = SConv(S1, S2, S3, S4, · · ·, Sn),  (3)  where SConv is a separable convolution layers with 3×3 filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  We extract the spatial and temporal features from latent space  during the upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The extracted spatial and temporal  features of each parallel node and adjacent node both are  concatenated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=',  Sc = Conc(Sup, Sd),  (4)  where Conc is concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Then we perform  separable convolution layers with 3 × 3 filters succeeded by  the ReLU activation function to enhance the quality of the  spatial-temporal feature vectors and reconstruct the original  dimension of latent features vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Further, we apply the  Sigmoid function using the convolution layer with 1×1 filters  to simplify the spatial and temporal features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=',  Sm = Sigmoid(Sc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  (5)  The network has approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='5 million trainable pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' We notice that each layer of the VS-Net generates  a feature map with a spatial structure in places of the video  frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Max Pooling layers use to increase the feature’s map  generation speed, which updates the weight matrix of the  backbone models during the feature extraction from every  separable convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' During the bottom-up extraction  of features from high to low resolution, upsampling operations  with 2×2 filters are used to distribute the latent feature space  and combine them with the previous layer and parallel layer’s  nodes’ features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  We used Approximation Rank Pooling (ARP) [16], which  takes input features from the intermediate layers of a VS-  Net, trains on sub-sequences, and generates the output of  a subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' ARP not only gives low-rank approximation  features, but it also conserves temporal order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The low-rank  approximation differentiated and captured important character-  istics of the data, which summarizes the document’s position  and orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Further, a quadratic ranking function captured  the temporal order, which handles non-linear dependencies of  the input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Generally, the temporal order deals with the  protuberances of the input channels onto the substance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Due to the low-rank approximation, the down-sampling  generates the generalized latent space vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The sampling of  Input Frame  Saliency Map  UpSampling  Down Sampling  SConv1  SConv2  SConv+MaxPool+ReLU  SConv+MaxPool+ReLU  SConv+MaxPool+ReLU  SConv+MaxPool+  ReLU  SConv6  SConv7  SConv8  SConv9  SConv10  SConv14  SConv15  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Architecture of the proposed VS-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  mean and variance gives the efficient latent distribution of the  VS-Net architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Based on the latent Gaussian distribution,  the latent vector is generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The down-sampling and up-  sampling are performed based on the variational encoder and  decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' For handling the over-fitting of the proposed model,  the latent space of the down-sampling is normalized with the  help of convolution layers and passed to up-sampling layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  After concatenating all the spatial and temporal features, we  applied a convolution layer with 1×1 filters at the last node to  generate feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' At last, the previous node and parallel  node features are aggregated and provide the saliency map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  TABLE I  PERFORMANCE COMPARISON OF THE SOTA AND PROPOSED MODEL  (VS-NET) IN TERMS OF BCE+IOU LOSS(%), AND TESTING SPEED (FPS)  Input frame  Model  Output frame  Loss (BCE+IoU)  Testing speed (FPS)  RNN+LSTM [7]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='032  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='34  U-Net [9]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='028  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='45  FCNN [18]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='038  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='36  HE [2]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='041  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='45  STCRF [12]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='039  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='54  AED [13]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='035  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='89  CNN+LSTM [10]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='036  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='63  RCNN [19]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='025  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='54  VS-Net  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='021  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='36  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Loss function  During training, we use input frames Sk with the corre-  sponding ground-truth Gk at frame t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The binary cross-entropy  loss Lbce [20] is used to calculate the dissimilarity of the  output and target, which is given as follows,  lbce(Sk, Gk) = −  �  i,j  [Gk(i, j)log(Sk(i, j))+  (1 − Gk(i, j))log(1 − Sk(i, j))]  (6)  where (i,j) represents a coordinate of the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The IoU  loss [21] is computed as:  lIoU(Sk, Gk) = 1 −  �w  i=1  �h  j=1 Sk(i, j)Gk(i, j)  �w  i=1  �h  j=1[Sk(i, j) + Gk(i, j) − Sk(i, j)Gk(i, j)]  (7)  The total loss function is derived as,  Totalloss = lbce(Sk, Gk) + αlIoU(Sk, Gk),  (8)  where α is the weighting parameter for the IoU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' For the final  saliency map prediction, we utilize Sk and Gk since it shows  that our experiments better utilized spatial-temporal cues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' EXPERIMENTAL SETUP AND RESULT ANALYSIS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' MIDV-500 Dataset  We have considered the MIDV-500 dataset [5], which con-  tains video clips of 50 different identity documents (seventeen  Id cards, fourteen passports, six identity documents, and  thirteen driving licenses) of various countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' It has eight  different variations of background and foreground scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The  attributes of the dataset are TS (Table Scene), TA (Table  Action), KS (Keyboard Scene), KA (Keyboard Action), HS  (Hand Scene), HA (Hand Action), PS (Partial Scene), PA  (Partial Action), CS (Complex Scene), CA (Complex Action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Thus a total of 500 videos are generated (50 documents× 5  desperation×2 devices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Each video has a duration of three  seconds, which is split into ten frames per second and the  corresponding annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The dataset contains examples of  multiple challenging scenarios, such as complex scenes, small  FUHRERSCHEIN BUNDESREPUBLKDEUTSCHLAND  D  Mustermann  Erika  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='64  MUSTER  4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='15  Berlin  Musterhausen  4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='Landratsant  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='0:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='30  B072RRE2155  am See  M/B/LFUHRERSCHEIN BUNDESREPUBLKDEUTSCHLAND  D  Mustermann  Erika  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='64  MUSTER  4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='15  Berlin  Musterhausen  4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='Landratsant  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='0:.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='30  B072RRE2155  am See  M/B/Lobjects with different variations of frames, appearances of  background and foreground, cluttered background, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' (a  few examples are shown in Fig: 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The ground truth is  prepared for each extracted video frame with various document  locations in JSON format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' It has 48 photo patches, 40 signature  patches, and 546 text patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The patches convert Cyrillic,  Greek, Chinese, Japanese, Arabic, and Persian with singular  Latin characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Optimization and Propagation  For preparing the VS-Net model, the Adaptive Moment Esti-  mation (ADAM) optimizer is used to facilitate the computation  of learning rates of each parameter using the first and second  moment of the gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The saliency map optimization is cast  as a “label propagation” problem, where uncertain labels are  propagated based on background and foreground seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Fur-  ther, we perform shift and rotation invariance robustly to the  deformation of gray value variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' These random variations  of deformation are sampled from the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Drop-out layers are used at the bottom stop and the previous  level to normalize the latent space vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The proposed VS-  Net model optimizes the loss of saliency maps from top-to-  bottom and bottom-to-top and propagative seeds sequentially  interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The sequential seeding procedure optimizes feature  map loss and improves the robustness to construct a saliency  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Evaluation Metrics  We used Intersection Over Union (IoU) loss as an evaluation  metric, which is defined as the intersection over the predicted  boundary box (bbox), and the actual bbox and divided with  their union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' A prediction considers True Positive if Inter-  section over union (IoU)>threshold, and False Positive If  IoU<threshold [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  IoU = Area  of  Overlap  Area  of  Union  (9)  The smaller the IoU value, the better the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' In our  experiments, the threshold is set as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The comparison results  in terms of accuracy are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  TABLE II  COMPARISON RESULTS WITH SOTA MODELS AND PROPOSED MODEL ON  MIDV-500 DATASET  Comparison of document detection accuracy (%)  Model  TS,  KS,  HS,  PS,  CS  TA  KA  HA  PA  CA  U-Net (2019) [9]  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='44  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='43  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='12  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='64  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='40  RNN+LSTM (2021) [7]  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='12  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='33  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='56  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='56  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='59  FCNN (2017)[18]  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='78  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='39  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='89  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='43  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='43  HE (2020) [2]  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='86  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='58  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='54  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='32  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='45  STCRF (2018) [12]  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='43  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='98  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='56  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='43  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='98  AED (2020)[13]  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='97  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='54  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='74  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='78  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='69  CNN+LSTM (2019)[10]  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='56  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='54  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='74  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='78  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='69  RCNN (2019)[19]  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='43  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='69  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='78  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='19  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='56  VS-Net  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='25  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='67  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='62  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='45  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='74  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Implementation Details  1) Training Setup: The experiments are accomplished on  Intel Xeon(R) CPU E5-2640 v4 @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='40GHz X40 processor  with 32GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' We used Tensorflow-gpu2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='0, Keras as the  backend, and Python3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='6 accelerated by NVIDIA RTX Graphic  card (Quadro P5000 / PCIe /SSE2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Input frames are of size  256×256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The ADAM optimizer, weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='006, batch  size of 128, and learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='001 are used to train the  model, and BCE +IoU loss function is used to calculate the  loss of the VS-Net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The running time comparison is  given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  2) Testing Setup and Runtime: We resize the frame 256 ×  256 to feed into the corresponding branch for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The  distribution of the dataset is in the ratio of 3:7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The average  testing speed of our model is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='46 fps which is less than the  existing models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Additionally, we do not perform any pre-  /post-processing [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' For validating purpose, the 5-fold cross-  validation are used to check the overfitting (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  5-fold cross-validation comparison result of VS-Net and SOTA  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Ablation Study - The qualitative evaluation shows that  our VS-Net model gives the best results on MIDV-500 [5]  datasets but is an extremely lightweight setting that has fewer  parameters and FLOPs (see in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  Speed Comparison The speed performance is calculated  on a 64-bit Linux Ubuntu-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='04 operating system with Intel  Core i7-4590 CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='3 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' It has 32GB RAM and 1 TB  Hard disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The average speed is computed on all frame images  of MIDV-500 datasets and does not include I/O file time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The  parallel processing of multiple images is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Priorly,  SOTA models require high computational powerful GPU sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' For a fair comparison, the recent unsupervised models  compare the speed on a normal CPU and the speed of deep  learning methods reported on GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The speed comparison is  given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' We show the trade-off between loss and  testing speed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Further, the average time cost of  each step is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' VS-Net consumes 256 ms and SOTA  models 270 ms, respectively, while saliency prediction and  global optimization use 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='36 ms and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='0 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' we proposed a novel efficient and fast VS-Net  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='model that fully leverages the spatiotemporal features to detect  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='99  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='98  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='Accuracy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='97  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='96  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='95  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='94  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='HE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='AED  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='STCRF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='VS-Net  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='RNN+LSTM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='CNN+LSTMTABLE III  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='RUNTIME COMPARISON OF SOTA MODELS AND VS-NET MODEL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='U-Net[9]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='RNN+LSTM[7]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='FCNN[18]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='HE [2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='STCRF[12]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='AED[13]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='CNN+LSTM[10]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='RCNN[19]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='VS-Net  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='Runtime (s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='188  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='191  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='190  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='198  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='199  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='196  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='189  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='175  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='Step (ms)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='265  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='266  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='273  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='278  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='275  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='268  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='267  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='260  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='256  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='TABLE IV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='ABLATION STUDY WITH RESNET-50 AS BACKBONE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  # Param(M)  FLOPS(G)  Runtime (s)  Steps(ms)  Accuracy  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='42  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='2  200  285  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='98  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='54  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='3  185  275  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='7  175  265  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='95  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='5  165  255  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='45  the video identity document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The proposed model used sep-  arable convolutions with variational autoencoder architecture,  which performs downsampling for extracting the features and  upsampling operation for decoding the latent space features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  The Approximation Rank Pooling (ARP) is used low-rank  approximation on frames to preserve the temporal locality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' The  bottom-end the generalized latent space is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Further,  These features are combined to generate in fuse spatial and  temporal characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' We validated each module of the  proposed model with the help of extensive experiments, which  can be considered as the unified solution advancing VSDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  REFERENCES  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Burie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Forn´es, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Santosh, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+page_content=' Luqman, “Deep learning  for graphics recognition: document understanding and beyond,” pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+page_content=' Nikolaev, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+page_content='  IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+page_content='  IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Wang, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Liu, “Content-aware rate control  scheme for hevc based on static and dynamic saliency detection,”  Neurocomputing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' 411, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' 393–405, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content='  [5] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Arlazarov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Bulatov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+page_content=' Ahmed, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Tarlow, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' Batra, “Optimizing expected intersection-  over-union with candidate-constrained crfs,” in Proceedings of the IEEE  international conference on computer vision, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
+page_content=' 1850–1858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BtE3T4oBgHgl3EQfUAqQ/content/2301.04447v1.pdf'}
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+1
+Universal Multimodal Representation for
+Language Understanding
+Zhuosheng Zhang#, Kehai Chen, Rui Wang#, Masao Utiyama, Eiichiro Sumita, Zuchao Li, Hai Zhao*
+Abstract—Representation learning is the foundation of natural language processing (NLP). This work presents new methods to employ
+visual information as assistant signals to general NLP tasks. For each sentence, we first retrieve a flexible number of images either
+from a light topic-image lookup table extracted over the existing sentence-image pairs or a shared cross-modal embedding space that
+is pre-trained on out-of-shelf text-image pairs. Then, the text and images are encoded by a Transformer encoder and convolutional
+neural network, respectively. The two sequences of representations are further fused by an attention layer for the interaction of the two
+modalities. In this study, the retrieval process is controllable and flexible. The universal visual representation overcomes the lack of large-
+scale bilingual sentence-image pairs. Our method can be easily applied to text-only tasks without manually annotated multimodal parallel
+corpora. We apply the proposed method to a wide range of natural language generation and understanding tasks, including neural
+machine translation, natural language inference, and semantic similarity. Experimental results show that our method is generally effective
+for different tasks and languages. Analysis indicates that the visual signals enrich textual representations of content words, provide fine-
+grained grounding information about the relationship between concepts and events, and potentially conduce to disambiguation.
+Index Terms—Artificial Intelligence, Natural Language Understanding, Vision-Language Modeling, Multimodal Machine Translation.
+!
+1
+INTRODUCTION
+L
+EARNING contextualized representations of human lan-
+guages is one of the major themes in natural language
+processing (NLP), which is also fundamental to training
+machines to understand human languages and handle ad-
+vanced tasks, such as machine translation, question an-
+swering, and human-computer conversations. Text repre-
+sentation learning has evolved from standard distributed
+representations [1, 2] to contextualized language represen-
+tation from deep pre-trained representation models (PRMs)
+[3, 4, 5, 6]. Despite the success of PRMs, NLP models
+commonly model the world knowledge (e.g, commonsense,
+rules, events, assertions extracted from raw texts) solely
+from textual features without grounding of the outside
+world, such as visual conception [7]. Languages are abstract
+•
+Z. Zhang, R. Wang, Z. Li, H. Zhao are with the Department of
+Computer Science and Engineering, Shanghai Jiao Tong University,
+China and also with Key Laboratory of Shanghai Education Com-
+mission for Intelligent Interaction and Cognitive Engineering, Shang-
+hai Jiao Tong University, China. K. Chen is with the School of
+Computer Science and Technology, Harbin Institute of Technology,
+Shenzhen, China. M. Utiyama and E. Sumita are with the Na-
+tional Institute of Information and Communications Technology (NICT),
+Japan. E-mail: {zhangzs, charlee}@sjtu.edu.cn; chenkehai@hit.edu.cn;
+{mutiyama, eiichiro.sumita}@nict.go.jp; wangrui.nlp@gmail.com; zhao-
+hai@cs.sjtu.edu.cn.
+•
+H. Zhao is supported by Key Projects of National Natural Science
+Foundation of China (U1836222 and 61733011). R. Wang is supported
+by National Natural Science Foundation of China (No. 6217020129),
+Shanghai Pujiang Program (No. 21PJ1406800), Shanghai Municipal
+Science and Technology Major Project (No. 2021SHZDZX0102), Beijing
+Academy of Artificial Intelligence (BAAI) (No. 4), CCF-Baidu Open
+Fund (No. CCF-BAIDU OF2022018). K. Chen is supported by National
+Natural Science Foundation of China (No. 62276077) and Shenzhen
+College Stability Support Plan (No. GXWD20220811170358002 and
+GXWD20220817123150002).
+•
+Z. Zhang and R. Wang contribute equally to this work. Part of this work
+was finished when Z. Zhang and Z. Li visited NICT, and R. Wang was
+with NICT. Corresponding author: Hai Zhao.
+and rather difficult for the brain to retain, whereas visuals
+are concrete and, as such, more easily remembered [8, 9].
+Adopting multimodality would be essential for better back-
+ground perception. Therefore, a trend of research has been
+motivated to apply non-linguistic modalities to language
+representations [7, 10, 11, 12, 13, 14].
+Most of previous works focus on joint modeling im-
+ages and texts, involving vision-language (VL) pre-training
+[15, 16, 17, 18, 19, 20] and multimodal (MM) application
+tasks [14, 21, 22, 23, 24]. However, these studies rely on
+large-scale text-image annotations as the paired input and
+thus are confined to VL or MM tasks, such as image cap-
+tioning and visual question answering. It is natural to boost
+the performance on VL and MM tasks as the concerned
+datasets are human-labeled with high quality. However, the
+essential challenge lies with the real-world scenario as there
+is no such high-quality annotated text-image aligned corpus
+for text-only NLP applications. Therefore, it is critical to
+investigate a general method to take advantage of visual
+information in a wide range of mono-modal (e.g., text-only)
+tasks. In addition, it is still not clear the role of images
+in language representation, as well as how to apply the
+multimodality in the standard NLP scenario.
+Taking multimodal machine translation (MMT) as an
+example, the starting point is to leverage visual information
+to improve the quality of the translation from the source
+to the target languages. However, the effectiveness heavily
+relies on the availability of bilingual parallel sentence pairs
+with manual image annotations, which hinders the image
+applicability to neural machine translation (NMT). As a re-
+sult, the visual information is only applied to the translation
+task over specific multimodal datasets [25, 26, 27, 28, 29], in-
+stead of general text-only NMT [30, 31, 32] and low-resource
+text-only NMT [33, 34, 35, 36]. In addition, because of the
+high cost of annotation, the content of one bilingual parallel
+Copyright © 20XX IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media,
+including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists,
+or reuse of any copyrighted component of this work in other works.
+arXiv:2301.03344v1  [cs.CL]  9 Jan 2023
+
+2
+sentence pair is paired with a single image, which is weak in
+capturing the diversity of visual information. Therefore, the
+current study of introducing visual information falls into a
+bottleneck in the multimodal NMT and is not feasible for
+text-only NMT and low-resource NMT.
+Our previous work [37] finds that using monolingual
+corpora with image annotations can overcome the lack of
+large-scale bilingual sentence-image pairs, thereby extend-
+ing image applicability in NMT, with performance gains.
+The method of using a lookup table is task-agnostic. This
+work stimulates our further thinking, and we are interested
+in answering the three major aspects of questions:
+(i) Global Multimodality: Can we apply the multi-
+modality to standard text-only NLP tasks to enhance the
+language representations (Section 5.4), e.g., natural lan-
+guage generation (Section 5.1.1) and natural language un-
+derstanding (Section 5.1.2)?
+(ii) Interpretability: Why does the universal representa-
+tion method work (Section 6.2)? How does multimodality
+improve language representation, and what is the network
+learned(Section 6.2-6.6)?
+(iii) Quality: How to control the quality of visual-text
+alignment to reduce noise (Section 6.9)?
+In this paper, we present a universal visual represen-
+tation (UVR) method relying only on a seed set of task-
+independent annotations, instead of the existing approach
+that depends on large-scale task-specific image-text anno-
+tation, thus breaking the bottleneck of using visual infor-
+mation in standard text-only NLP tasks. For each sentence,
+we retrieve diverse images from either a light topic-image
+lookup table or pre-trained shared text-visual embedding
+space that is pre-trained on a large-scale of text-image
+pairs, to connect both the mono-modal paths of text and
+image embeddings. The text and images are encoded by
+Transformer language model (LM) and a pre-trained convo-
+lutional neural network (CNN), respectively. A simple and
+effective attention layer is then designed to fuse the two
+sequences of representations.
+Our approach can be easily applied to text-only tasks
+without manually annotated multimodal parallel corpora.
+Therefore, our method is universal in terms of the task re-
+quirements, in contrast to the recent vision-language models
+that require large-scale and expensive annotation datasets
+for each downstream task. The proposed method is eval-
+uated on 14 NLP benchmark datasets involving natural
+language inference (NLI), semantic similarity, text classifica-
+tion, and machine translation. The experiments and analysis
+verify the effectiveness of the proposed method. To summa-
+rize, our contributions are primarily three-fold:
+(i) This work studies the universal visual representa-
+tion for language representation in a broader view of the
+natural language processing scenario. Besides neural ma-
+chine translation, this work leverages visual information
+as assistant signals for general NLP tasks, with the focus
+on investigating the global multimodality for general NLP,
+interpretability of effectiveness, and quality control of using
+universal visual representation.
+(ii) For the technical side, this work proposes new meth-
+ods of semantic sentence-image matching from a shared
+cross-modal space to give more accurately paired images
+as topic information. We also present a new multimodal
+representation framework and systematically study the two
+main instances, including the model with the original TF-
+IDF topic-image lookup table and the newly proposed one
+from the retrieval from cross-modal retrieval.
+(iii) Experiments are extended to 14 representative NLP
+tasks, which show the effectiveness of the proposed method.
+A series of in-depth analyses indicate that the visual signals
+enrich textual representations of content words, provide
+fine-grained grounding information about the relationship
+between concepts and events, and potentially conduce to
+disambiguation.
+2
+BACKGROUND
+2.1
+Vision-Language Integration
+This study is related to that of VL methods (VLMs). Re-
+cently, there has been a great deal of interest in integrating
+image presentations in pre-trained Transformer architec-
+tures [15, 16, 17, 18, 19, 20, 38]. The common strategy is
+to take a Transformer model [32], such as BERT, as the back-
+bone and learn aligned representations of visual and lan-
+guage in a pre-training manner inspired by the masked lan-
+guage modeling mechanism in pre-trained language models
+[5]. These studies require the annotation of task-dependent
+sentence-image pairs, which are limited to VL tasks, such as
+image captioning and visual question answering.
+Two studies [22, 37] are closely related to general image-
+enhanced LM. Glyce [22] proposes incorporating glyph
+vectors for Chinese character representations. However, it
+can only be used for Chinese and only involves single
+image enhancement. Regarding the technical part, previous
+methods only benefit from one image per sentence. We
+propose taking advantage of a group of similar images using
+a filtering mechanism to form a more fine-grained visual-
+aware context. Our early version of this work [37] proposes
+using multiple images for NMT, based on a text-image
+lookup table trained over a sentence-image pair corpus.
+However, the number of images is fixed because of the lack
+of similarity measurement in the simple lookup method,
+which possibly makes the resulting model suffer from the
+noise of irrelevant images. This work is improved from the
+perspectives of motivation and technique. It is motivated
+by cross-modal semantic retrieval in the shared embedding
+space. It adopts a neural matching method with a similarity
+threshold to control the expected matching degree flexibly,
+which is generally applicable to a broader range of NLP
+tasks. In addition, we conduct an in-depth analysis to inves-
+tigate how the visual modality helps text representation.
+2.2
+Visual-Semantic Embeddings
+Another research line is language grounding for images
+whose major topic is multimodality and cross-modality be-
+tween images and text. The major focus is to bridge the gap
+between text and images through building visual-semantic
+embeddings [39, 40, 41, 42, 43].
+Prior studies have verified that representations of images
+and text can be jointly leveraged to build visual-semantic
+embeddings in a shared representation space [39, 40, 41, 44].
+To this end, a popular approach is to connect both the mono-
+modal text and image encoding paths using fully connected
+
+3
+layers [45, 46]. The shared deep embedding can be used
+for cross-modal retrieval; thus, it can associate sentence
+text with associated images. Partly inspired by this line of
+research, we are motivated to incorporate visual awareness
+into sentence modeling by retrieving a group of images for
+a given sentence.
+One of the first techniques to align two views of hetero-
+geneous data is the canonical correlation analysis method
+[47], in which linear projections defined on both sides are
+optimized to maximize the cross-correlation. Recent studies
+have followed the two-path architecture [45, 46], in which
+the encoder consists of a joint embedding of textual and
+image representations extracted from both the images and
+corresponding caption. Notably, Engilberge et al. [46] adopts
+RNN to encode sentence embeddings in the same space
+with extracted image representations from CNN. Portaz
+et al. [48] enhances cross-modal retrieval using multilingual
+text. Inspired by the previous success of visual-semantic
+embeddings, we apply neural image retrieval from the joint
+space to fetch a group of associated images.
+3
+UNIVERSAL REPRESENTATION FRAMEWORK
+This section overviews our universal representation frame-
+work. Given a sentence, we first fetch a group of matched
+images from our retrieval methods (details of our retrieval
+methods will be given in the next section). The text and
+images are encoded, respectively, by the text feature extrac-
+tor and image feature extractor. Then the two sequences of
+representations are integrated using multi-head attention to
+form a joint representation, which is passed to downstream
+task-specific layers. Figure 1 overviews the whole multi-
+modal representation model.
+3.1
+Encoding Layer
+3.1.1
+Text Encoder
+We pair each sentence with the top matched m images
+according to the retrieval method above. Following [5], the
+sentence is fed into the multi-layer Transformer encoder [32]
+to learn the text representation H ∈ Rn×d where n and d are
+the input text length and dimension of hidden states for the
+text representation.
+Let X = {x1, . . . , xn} be the input sentence in length
+n. We feed the sequence to a PRM encoder (e.g., BERT
+[5]). In the encoder, the input sequence is firstly mapped
+to embeddings. Then, the embeddings are passed to multi-
+head attention layers [32] to obtain the contextualized rep-
+resentations, which is defined as
+H = FFN(MultiHead(K, Q, V )),
+(1)
+where K, Q, V are packed from the input sequence repre-
+sentation X. As the common practice, we set K = Q = V
+in the implementation. MultiHead is short for multi-head
+attention.
+3.1.2
+Image Encoder
+Similar to the standard way of retrieving word embeddings,
+the image embeddings are fetched from a lookup table ∈
+Rnm×dm that contains the image features encoded by a pre-
+trained ResNet [49],1 where nm is the number of the total
+number of unique images +1 and dm is the dimension of the
+image features.2 The first row of the lookup table is filled by
+all-zero vectors, which will be used when no image is paired
+for the sentence.
+After the feature lookup process, we obtain the image
+embeddings E ∈ Rm×dm for the m input images. Then, the
+embeddings are passed to a feedforward layer, to produce
+the image representation M ∈ Rm×d with the same hidden
+dimension as H:
+M = FFN(ResNet(E)).
+(2)
+There may exist cases when no word in the sentence can
+be found in the topic-image lookup table. When there is no
+paired image retrieved, we use the first-row all-zero vectors
+of the image lookup table as the “blank features” in the
+intuition to tell the model to ignore them.
+3.2
+Multimodal Integration Layer
+We connect the visual and text modalities by calculating the
+attention between image and text features:
+α = softmax(H(WgM + bg)⊤),
+(3)
+H′ = αM,
+(4)
+where Wg and bg are parameters to learn. α ∈ Rn×m
+denotes the weights assigned to the different hidden states
+in the sentence and the image sequences. H′ ∈ Rn×d is the
+weighted sum of all the hidden states and it represents how
+the sentence can be aligned to each hidden state in the image
+representation.
+The retrieval process may possibly introduce noise of
+irrelevant images. To alleviate the influence, we use a neural
+gating mechanism for information filtering. In detail, we
+compute λ ∈ [0, 1]n×d to weight the expected importance
+of image representation for each source word:
+λ = sigmoid(WλH′ + UλH),
+(5)
+where Wλ and Uλ are model parameters. We then fuse
+H and H′ and pass the resulting representation to layer
+normalization and learn an effective source representation:
+ˆH = LayerNorm(H + λH′).
+(6)
+The text and image representations are jointly encoded
+as ˆH, which is fed to the task-specific layers for downstream
+decoding or predictions depending on task settings.
+3.3
+Task-specific Layer
+In this section, we show how the joint representation ˆH is
+used for downstream tasks by considering NMT and NLU
+tasks as examples, generally following the standard proce-
+dure of the concerned tasks. For NMT, ˆH is directly fed to
+the decoder to learn a dependent-time context vector to pre-
+dict the target translation. For other tasks, ˆH is directly fed
+1. Note that this is the standard lookup table in embedding imple-
+mentations, which is not our topic-image lookup table.
+2. We used the maxpooling layer of ResNet, which is in the size of
+Rnm×2400.
+
+4
+Add & Norm
+Feed Forward
+Add & Norm
+Multi-head
+Attention
+Add & Norm
+Multi-head
+Attention
+Feed Forward
+ResNet
+Text 
+Embedding
+Sentence
+Text Encoder
+Image
+Corpus
+Image Retrieval
+Pooling
+Linear
+Downstream Adaption
+(NLU, NMT, etc)
+Image Encoder
+Integration
+L×
+Fig. 1. Overview of the universal representation framework. Given a sentence as input, a group of related images will be retrieved by our image
+retrieval methods. The text and images are encoded by the text feature extractor and image feature extractor, respectively. Then the two sequences
+of representations are integrated using multi-head attention to form a joint representation in the same shape as the original text sequence
+representation. Finally, the joint representation is passed to downstream task-specific layers to give predictions.
+to a feed-forward layer to make the prediction, which fol-
+lows the same downstream procedure as the Transformer-
+based LMs, like BERT [5] and RoBERTa [50]. Specifically,
+for sentence-pair tasks, we maintain the pairwise input as
+that in LMs and separate the encoded text representation
+into two individual sentence representations {H1, H2}, ac-
+cording to the positions. The two text representations are
+integrated with the corresponding image representations
+respectively {M 1, M 2}, and then the resulting sequences
+are concatenated for prediction, ˆH = ˆH1 ◦ ˆH2.
+4
+IMAGE RETRIEVAL METHODS
+In this section, we describe our two visual retrieval models
+used for image retrieval given sentence text:
+(i) UVR-TILT: retrieval by topic-image lookup table;
+(ii) UVR-CMRM: retrieval from cross-modal embed-
+ding.
+4.1
+Model-I: Retrieval by Topic-Image Lookup Table
+4.1.1
+Topic-image Lookup Table Conversion
+In this section, we will introduce the proposed universal
+visual representation method. Our basic intuition is to trans-
+form the existing sentence-image pairs into a topic-image
+lookup table,3 which assumes the topic words in a sentence
+should be relevant to the paired image. The procedure can
+be seen as the inverted index where a topic word is mapped
+to a list of images. Consequently, a sentence can possess a
+group of images by retrieving the topic-image lookup table.
+To focus on the major part of the sentence and suppress
+the noise such as stopwords and low-frequency words, we
+design a filtering method to extract the “topic” words of
+the sentence through the term frequency-inverse document
+3. We use the training set of the Multi30K dataset to build the topic-
+image lookup table.
+Algorithm 1 Topic-image Lookup Table Conversion
+Require: Input sentences, S = {X1, X2, . . . XI} and paired
+images E = {e1, e2, . . . , eI}
+Ensure: Topic-image lookup table Q where each word is asso-
+ciated with a group of images
+1: Obtain the TF-IDF dictionary F = TF-IDF(S)
+2: Transform sentence-image pair to topic-image lookup table
+Q = LookUp(S, E, F)
+3: procedure TF-IDF(S)
+4:
+for each sentence in S do
+5:
+Filter stop-words in the sentence
+6:
+Calculate the TF-IDF weight for each word
+7:
+end for
+8:
+return TF-IDF dictionary F
+9: end procedure
+10: procedure LOOKUP(S, E, F)
+11:
+for For each pair {Xi, ei} ∈ zip{S, E} do
+12:
+Rank and pick out the top-w “topic” words in the
+sentence according to the TF-IDF score in the dictionary F,
+and each sentence is reformed as T = {t1, t2, . . . , tw}
+13:
+for For each word tjin T do
+14:
+if ei not in Q[tj] then
+15:
+Add ej to the corresponding image set Q[tj]
+for word tj
+16:
+end if
+17:
+end for
+18:
+end for
+19:
+return Topic-image lookup table Q
+20: end procedure
+frequency (TF-IDF),4 inspired by [51]. Specifically, given an
+original input sentence X = {x1, x2, . . . , xI} of length I
+and its paired image e, X is first filtered by a stopword list,5
+and then the sentence is treated as a document g. We then
+4. We describe our methods by regarding the processing unit as word
+though this method can also be applied to a subword-based sentence
+for which the subword is considered to be the processing unit.
+5. https://github.com/stopwords-iso/stopwords-en.
+
+UBS
+SOONERS
+MEGA
+OMEGA
+0
+SOON5
+   dog is playing in the snow
+dog (1,512)
+playing (1,531)
+snow (439)
+(a) a black dog and a spotted dog are fighting
+(b) a dog is running in the snow
+(c) a dog is playing with a hose
+(d) a family playing on a tractor on a beautiful day
+(e) two people working on removing snow from a roof 
+(f) a black dog and a white dog are standing on snow 
+corpus (29,000)
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+sentence-image pairs
+topic-image lookup table
+associated images for input sentence
+tokenize, filtering
+word-image transform
+sampling
+ranking
+Fig. 2. Illustration of the TILT method. We first transform the existing sentence-image pairs from seed small-scale sentence-image datasets into a
+topic-image lookup table. For a given sentence, we extract its topic words and the associated images will be retrieved from the lookup table.
+compute TF-IDF TIi,j for each word xi in g,
+TIi,j =
+oi,j
+�
+k ok,j
+× log
+|G|
+1 + |j : xi ∈ g|,
+(7)
+where oi,j represents the number of occurrences of the word
+xi in the input sentence g, |G| the total number of source
+language sentences in the training data, and |j : xi ∈ g|
+the number of source sentences including word xi in the
+training data. We then select the top-w high TF-IDF words as
+the new image description T = {t1, t2, . . . , tw} for the input
+sentence. After the preprocessing, each filtered sentence T
+is paired with an image e, and each word ti ∈ T is regarded
+as the topic word for image e. After processing the whole
+corpus (i.e., Multi30K), we form a topic-image lookup table
+Q as described in Algorithm 1, in which each topic word ti
+would be paired with dozens of images.
+4.1.2
+Image Retrieval
+For the input sentence, we first obtain its topic words
+according to the text preprocessing method described above.
+Then we retrieve the associated images for each topic word
+from the lookup table Q and group all the retrieved images
+together to form an image list G. We observe that an image
+might be associated with multiple topic words so that it
+would occur multiple times in the list G. Thus, we sort the
+images according to the frequency of occurrences in G to
+maintain the same total number of images for each sentence
+at m.
+Figure 2 illustrates the retrieval process. In the left block,
+we show six examples of sentence-image pairs in which the
+topic words are in boldface. Then we process the corpus
+using the topic-image transformation method demonstrated
+above and obtain the topic-image lookup table. For example,
+the word dog is associated with 1,512 images. For an input
+source sentence, we obtain the topic words (in boldface)
+using the same preprocessing. Then we retrieve the corre-
+sponding images from the lookup table for each topic word.
+Now we have a list of images, and some images appear
+multiple times as they have various topics (like the boxed
+image in Figure 2). So we sort the retrieved image list by the
+count of occurrence to pick out the top-m images that cover
+the most topics of the sentence.
+At test time, the process of getting images is done using
+the image lookup table built by the training set, so we do
+not need to use the images from the validation and test sets
+in Multi30K dataset.6 Intuitively, we do not strictly require
+the manual alignment of the word (or concept) and image
+but rely on the co-occurrence of the topic word and image,
+which is simpler and more general. In this way, we call our
+method universal visual retrieval.
+4.2
+Model-II: Retrieval from Cross-modal Embedding
+Following Engilberge et al. [46], we train a semantic-visual
+embedding on a text-image corpus, which is then used for
+image retrieval. The semantic-visual embedding architec-
+ture comprises two paths to encode the text and images into
+vectors. Based on our preliminary experiments, we maintain
+the same settings in Engilberge et al. [46] by using the simple
+recurrent unit as text encoder, and the fully convolutional
+residual ResNet-152 [52] with Weldon pooling [53] as image
+encoder for our cross-modal retrieval model.
+During training, each text X is paired with (i) a posi-
+tive image Y that is paired with the text and (ii) a hard
+negative Z, which is selected as the image that has the
+6. The lookup table can be easily adapted to a wide range of other
+NLP tasks even without any paired image, and therefore opens our
+proposed model to generalization.
+
+6
+highest similarity to the text while not being associated
+with it. Triplet loss [54, 55, 56] is used to enable the images
+to converge correctly to improve the performance of the
+proposed method:
+loss(X, Y, Z) = max(0, γ − E(X) · E(Y ) + E(X) · E(Z)),
+(8)
+where E(X), E(Y ), and E(Z) are the embeddings of X, Y ,
+and Z, respectively. γ is the minimum margin between the
+similarity of the correct caption and the unrelated caption.
+The loss function enables the sentence X to be closer to the
+corresponding image Y than the unrelated image Z. During
+the prediction time, the relationship between the text and
+images is calculated using the cosine similarity.
+For general use, it is reasonable that some sentences,
+such as social constructs or metaphorical usage, are not
+paired with images after retrieval and have a low similarity
+score. In these cases, visual information might not be help-
+ful. To measure how similar the retrieved images should be,
+we set a threshold δ to choose the top-ranked images for
+each sentence.
+5
+EXPERIMENTS
+5.1
+Task Settings
+Our evaluation is performed on the widely-used natural
+language generation and understanding tasks involving 14
+NLP benchmark datasets that involve machine translation,
+natural language inference (NLI), semantic similarity, and
+text classification. Part of the NLU tasks is available from
+the GLUE benchmark [57], which is a collection of nine NLU
+tasks.
+5.1.1
+Neural Machine Translation
+Five widely-used translation tasks are used for model eval-
+uation, including WMT’16 English-to-Romanian (En-Ro),
+WMT’14 English-to-German (En-De), WMT’14 English-to-
+French (En-Fr), and Multi30K dataset for WMT’16 and
+WMT’17, which are standard corpora for NMT and MMT
+evaluation.
+(i) For the En-Ro task, we experiment with the officially
+provided parallel corpus: Europarl v7 and SETIMES2 from
+WMT’16 with 0.6M sentence pairs. We use newsdev2016 as
+the validation set and newstest2016 as the test set.
+(ii) The En-De task has 4.43M bilingual sentence pairs
+of the WMT14 dataset used as training data, including
+Common Crawl, News Commentary, and Europarl v7. The
+newstest2013 and newstest2014 datasets are used as the vali-
+dation set and test set, respectively.
+(iii) The En-Fr task has 36M bilingual sentence pairs
+from the WMT14 dataset used as training data. Newstest12
+and newstest13 are combined for validation and newstest14
+is used as the test set, following the setting of [31].
+(iv) Multi30K dataset contains 29K English→{German,
+French} parallel sentence pairs with visual annotations.
+The 1,014 English→{German, French} sentence pairs with
+visual annotations serve as the validation set. For WMT’16
+and WMT’17 tasks, we have two test sets, test2016 and
+test2017, with 1,000 pairs for each.
+5.1.2
+Natural Language Understanding
+The NLU task involves natural language inference, semantic
+similarity, and classification subtasks.
+Natural Language Inference involves reading a pair of sen-
+tences and assessing the relationship between their mean-
+ings, such as entailment, neutral, and contradiction. We
+evaluate the proposed method on four diverse datasets:
+SNLI [58], MNLI [59], QNLI [60], and RTE [61].
+Semantic Similarity aims to predict whether two sentences
+are semantically equivalent. Three datasets are used: Mi-
+crosoft Paraphrase Corpus (MRPC) [62], Quora Question
+Pairs (QQP) dataset [63], and Semantic Textual Similarity
+benchmark (STS-B) [64].
+Classification CoLA [65] is used to predict whether an
+English sentence is linguistically acceptable. SST-2 [66] pro-
+vides a dataset for sentiment classification that needs to
+determine whether the sentiment of a sentence extracted
+from movie reviews is positive or negative.
+5.2
+Retrieval Setup
+This part describes the implementation of the image re-
+trieval by the topic-image lookup table (TILT) and cross-
+modal retrieval model (CMRM):
+TILT: We segment the sentences using the same BPE
+vocabulary as that for each source language. We select top-
+8 (w = 8) high TF-IDF words, and the default number of
+images m is set to 5.7 The detailed case study is shown in
+Section 6.9. Image features are extracted from the averaged
+pooled features of a pre-trained ResNet50 CNN [49]. The
+dimension of the feature maps is V ∈ R2048.
+CMRM: The cross-modal retrieval model is trained on
+the MS-COCO dataset [67], which contains 123,287 images
+with five English captions per image. It is split into 82,783
+training images, 5,000 validation images, and 5,000 test
+images. We use the Karpathy split [40] that forms 113,287
+training, 5,000 validation and 5,000 test images. The model
+is implemented following the same settings as Engilberge
+et al. [46], and produces state-of-the-art results (94.0% R@10)
+for cross-modal retrieval. To ensure that each task can enjoy
+enough images, we set the similarity threshold δ to 0.4 and
+rank the paired images according to the similarity score. The
+maximum number of retrieved images m for each sentence
+is set to eight according to our preliminary experiments.
+Multi30K and COCO datasets are used as the candidate
+seed image retrieval corpus for our downstream tasks.
+5.3
+Model Implementation
+Since our task involves text generation and understanding,
+we have two kinds of baselines, the NLG model for transla-
+tion and the NLU model for the other tasks.
+5.3.1
+NLG Model
+Our baseline for NLG is encoder-decoder Transformer [32].
+We use six layers for the encoder and the decoder. The
+number of dimensions of all input and output layers is set to
+512 and 1024 for base and big models. For MMT experiments
+7. In some cases when there is no paired image retrieved, we use
+the first-row all-zero vectors of the image lookup table as the “blank
+features”.
+
+7
+TABLE 1
+Results for the NMT tasks. “++/+” after the BLEU score indicates that the proposed method (base: 5-8; large:9-12) was significantly better than the
+corresponding baseline Transformer (base or big) at significance level p <0.01/0.05.
+#
+Model
+En→Ro
+En→De
+En→Fr
+BLEU
+#Param
+BLEU
+#Param
+BLEU
+#Param
+Text-only Transformer
+1
+Transformer-Base
+32.66
+61.54M
+27.31
+63.44M
+38.52
+63.83M
+2
+Transformer-Big
+33.85
+207.02M
+28.45
+210.88M
+41.10
+211.66M
+Our MMT systems
+3
+UVR-TILTMulti30K
+33.78++
+63.04M
+28.14++
+64.94M
+39.64++
+65.33M
+4
+UVR-TILTCOCO
+34.08++
+63.04M
+27.79+
+64.94M
+39.84++
+65.33M
+5
+UVR-CMRMMulti30K
+34.38++
+63.04M
+27.82+
+64.94M
+39.76++
+65.33M
+6
+UVR-CMRMCOCO
+34.40++
+63.04M
+27.86+
+64.94M
+40.24++
+65.33M
+7
+UVR-TILTMulti30K
+34.46+
+211.02M
+29.14++
+214.89M
+41.83+
+215.66M
+8
+UVR-TILTCOCO
+34.51+
+211.02M
+29.18++
+214.89M
+41.76+
+215.66M
+9
+UVR-CMRMMulti30K
+34.60++
+211.02M
+28.96+
+64.94M
+41.79+
+215.66M
+10
+UVR-CMRMCOCO
+34.62++
+211.02M
+29.21++
+64.94M
+41.82+
+215.66M
+TABLE 2
+Results (BLEU) from the test2016 and test2017 for the MMT task. “++/+” after the BLEU score indicates that the proposed method (base: 5-8;
+large: 9-12) was significantly better than the corresponding baseline Transformer (base or big) at significance level p <0.01/0.05.
+#
+Model
+En-De
+En-Fr
+Test2016
+Test2017
+#Param
+Test2016
+Test2017
+#Param
+Text-only Transformer
+1
+Transformer-Base
+35.59
+26.31
+49.15M
+57.88
+48.55
+49.07M
+2
+Transformer-Big
+36.86
+27.62
+186.38M
+56.97
+48.17
+186.23M
+Standard MMT systems
+3
+MMT-Base
+35.09
+27.10
+50.72M
+57.40
+48.02
+50.65M
+4
+MMT-Big
+35.60
+28.02
+190.58M
+57.87
+49.63
+190.43M
+Our MMT systems
+5
+UVR-TILTMulti30K
+35.72
+26.87+
+50.72M
+58.32+
+48.69
+50.65M
+6
+UVR-TILTCOCO
+35.67
+26.89+
+50.72M
+58.21+
+48.73
+50.65M
+7
+UVR-CMRMMulti30K
+36.38+
+27.34++
+50.72M
+58.53+
+49.28+
+50.65M
+8
+UVR-CMRMCOCO
+35.78
+26.92+
+50.72M
+58.46+
+48.58
+50.65M
+9
+UVR-TILTMulti30K
+37.02
+28.63++
+190.58M
+57.53+
+48.46
+190.43M
+10
+UVR-TILTCOCO
+36.94
+28.69++
+190.58M
+57.62+
+48.39
+190.43M
+11
+UVR-CMRMMulti30K
+37.16
+28.82++
+190.58M
+58.37++
+48.77+
+190.43M
+12
+UVR-CMRMCOCO
+37.28+
+28.71++
+190.58M
+57.60+
+48.42
+190.43M
+on the Multi30K dataset, we also use the tiny setting where
+the dimension of the input and output layer is 128. The inner
+feed-forward neural network layer is set to 2048. The heads
+of all multi-head modules are set to eight in both the encoder
+and decoder layers. For the Multi30K dataset, we further
+evaluate a multimodal baseline (denoted as MMT) where
+each source sentence was paired with an original image.
+The other settings were the same as our proposed model.
+The byte pair encoding algorithm is adopted to segment
+sentences into subword sequences, with the vocabulary size
+set to 40,000. In each training batch, a set of sentence pairs
+contains approximately 4096×4 source tokens and 4096×4
+target tokens. During training, the value of label smoothing
+is set to 0.1, and the attention dropout and residual dropout
+rates are p = 0.1. We used Adam optimizer [74] to tune the
+parameters of the model. The learning rate is varied under
+a warm-up strategy with 8,000 steps. For evaluation, we
+validate the model with an interval of 1,000 batches on the
+validation set. For the Multi30K dataset, we train the model
+up to 10,000 steps, and the training will be early-stopped
+if the validation set BLEU score does not improve for ten
+epochs. For the En-De, En-Ro, and En-Fr tasks, following
+the training of 200,000 batches, the model with the highest
+BLEU score of the validation set is selected to evaluate
+the test sets. During the decoding, the beam size is set to
+five. Multi-bleu.perl is used to compute case-sensitive 4-
+gram BLEU scores for all test sets.8 We follow the model
+configurations of [32] to train big models for WMT En-Ro,
+En-De, and En-Fr translation tasks. The experiments of NLG
+are conducted with fairseq [75].9
+For the statistical tests, we perform the paired bootstrap
+resampling test [76] to measure the reliability of the con-
+clusion that our system is better than the baseline. Our im-
+8. https://github.com/moses-smt/mosesdecoder/tree/
+RELEASE-4.0/scripts/generic/multi-bleu.perl.
+9. https://github.com/pytorch/fairseq.
+
+8
+TABLE 3
+Comparison with public methods on the Multi30K MMT dataset. The results of existing methods are from [68]. “++/+” after the BLEU score
+indicates that the proposed method was significantly better than the corresponding baseline Transformer (tiny) at significance level p <0.01/0.05.
+#
+Model
+En-De
+En-Fr
+Test2016
+Test2017
+#Param
+Test2016
+Test2017
+#Param
+Text-only Transformer
+1
+Transformer-Tiny
+40.38
+32.86
+2.6M
+61.00
+52.42
+2.6M
+Existing MMT systems
+2
+GMNMT [69]
+39.8
+32.2
+4.0M
+60.9
+53.9
+-
+3
+DCCN [70]
+39.7
+31.0
+17.1M
+61.2
+54.3
+16.9M
+Our MMT systems
+4
+UVR-TILTTiny
+41.27++
+33.62++
+2.9M
+61.60+
+54.83++
+2.9M
+5
+UVR-CMRMTiny
+40.94+
+33.11+
+2.9M
+61.50+
+53.64++
+2.9M
+TABLE 4
+Test results on the GLUE benchmark. The best results are marked in boldface.
+#
+Model
+Classification
+Semantic Similarity
+Language Inference
+Average
+CoLA
+SST-2
+MRPC
+STS-B
+QQP
+MNLI
+QNLI
+RTE
+SNLI
+Public Systems
+1
+BERT [5]
+60.5
+94.9
+85.4
+87.6
+89.3
+86.7
+92.7
+70.1
+-
+83.4
+2
+MT-DNN [71]
+62.5
+95.6
+88.2
+89.5
+89.6
+86.7
+93.1
+81.4
+91.6
+86.4
+3
+BERT + Voken-cls [72]
+-
+92.2
+-
+-
+88.6
+82.6
+88.6
+-
+-
+-
+4
+UniT [73]
+-
+91.5
+-
+-
+88.4
+79.8
+88.0
+-
+-
+-
+-
+Our Systems
+5
+Baseline (BERTWWM)
+63.6
+93.6
+87.0
+90.2
+88.8
+87.2
+93.9
+77.3
+91.6
+85.9
+6
+UVR-TILTMulti30K
+62.5
+94.7
+87.7
+89.8
+89.4
+87.2
+94.1
+84.5
+91.7
+86.8
+7
+UVR-TILTCOCO
+62.8
+94.9
+87.4
+90.2
+89.7
+86.9
+94.0
+83.6
+91.7
+86.8
+8
+UVR-CMRMMulti30K
+63.0
+94.3
+87.8
+90.2
+89.6
+87.3
+93.8
+83.8
+91.6
+86.8
+9
+UVR-CMRMCOCO
+63.2
+94.6
+87.9
+90.3
+89.8
+87.4
+94.2
+83.9
+91.7
+87.0
+plementation is based on the public toolkit.10 Two thousand
+bootstrap samples are used for each significance test.
+5.3.2
+NLU Model
+For the NLU tasks, the baseline is encoder-only BERT [5].11
+We use the whole-word-mask (WWM) version of the pre-
+trained weights due to its more stable, reproducible, and
+slightly better performance than the original large version
+[5]. The initial learning rate is set in the range {2e-5, 3e-5}
+with a warm-up rate of 0.1 and L2 weight decay of 0.01.
+The batch size is selected from {16, 24, 32}. The maximum
+number of epochs is set in the range [2, 5]. Texts are
+tokenized using SentencePiece,12 with a maximum length
+of 128.
+5.4
+Main Results
+Tables 1-4 show the results for the 14 NMT, MMT, and NLU
+tasks, respectively. According to the results, we have the
+following observations:
+(i) According to the machine translation results in Tables
+1-2, the proposed UVR methods significantly outperform
+the baselines according to the statistical test, demonstrating
+10. https://github.com/neubig/util-scripts/blob/master/
+paired-bootstrap.py
+11. https://github.com/huggingface/transformers.
+12. https://github.com/google/sentencepiece.
+the effectiveness of modeling visual information for text-
+only NMT. In particular, the superiority is observed in
+the translation tasks of three language pairs with different
+training data scales, verifying that the proposed approach is
+a universal method for improving translation performance.
+(ii) Our method introduces only 1.5M and 4.0M param-
+eters for the base and big Transformers, respectively. The
+number is less than 3% of the baseline parameters as we use
+the fixed image embeddings from the pre-trained ResNet
+feature extractor. Besides, the training time is basically the
+same as the baseline model (Section 6.10).
+(iii) Results in Tables 2-3 show that our model can
+generally outperform the Transformer baseline in multi-
+modal settings that could benefit from the gold sentence-
+image annotations. Compared with the results in text-only
+NMT, we find that the image enhancement sometimes gives
+marginal contribution, which is consistent with the findings
+in previous work [77, 78, 79]. The most plausible reason
+might be that the sentences in Multi30K are quite simple,
+short, and repetitive, so that the source text itself is sufficient
+to perform the translation [10, 79]. We also see that the big
+models sometimes show inferior results. The possible reason
+is that the dataset is too small to effectively train such big
+models, which easily suffer from over-fitting issues. The
+hypothesis is also supported by our superior results with
+the tiny model setting in Table 3. The observation verifies
+our assumption of the current bottleneck of MMT due to
+
+9
+0.4
+0.5
+Percentage
+D-value
+Coverage
+CoLA
+SST-2
+MRPC
+STS-B
+QQP
+MNLI
+QNLI
+RTE
+SNLI
+0
+5
+10
+Accuracy
+Fig. 3. Accuracy difference between our method and baseline compared
+with the coverage percentage of tokens that can be paired with images
+in each dataset.
+TABLE 5
+Validation results on GLUE datasets in different sizes: small datasets
+with less than 10k examples (RTE and STS-B), and a large dataset
+with more than 10k examples (QNLI). The MT-DNN results are
+reproduced using the released weights [71].
+Model
+RTE
+STS-B
+QNLI
+MT-DNNbase
+78.94±0.83
+88.14±0.40
+90.32±0.12
+w/ UVR-TILT
+81.59±0.63
+90.16±0.07
+91.31±0.19
+MT-DNNlarge
+78.70±1.65
+90.16±0.17
+92.04±0.27
+w/ UVR-TILT
+82.19±1.37
+91.32±0.12
+92.78±0.12
+the limitation of Multi30K and shows the necessity of our
+new methodology of transferring multimodality into more
+standard and mature text-only NMT tasks.
+(iv) Table 4 shows our method is generally helpful for a
+wide range of NLU tasks in the GLUE benchmark, which
+verifies the effectiveness of modeling visual information
+for language understanding. We are interested in whether
+public methods, such as MT-DNN, can be further enhanced
+by our method, we apply our UVR-TILT method to the MT-
+DNN model based on the same implementation in BERT.
+According to the results in Table 5, both the base and
+large models are enhanced, and we observe consistent gains
+in different datasets, especially the small datasets. For the
+results in Tables 4-5, we notice a few inferior or marginally
+better performances in the CoLA, MNLI, and QNLI tasks.
+We calculate the accuracy difference (D-value) between our
+method and baseline compared with the coverage percent-
+age of tokens that can be paired with images in each dataset
+using UVR-TILTMulti30K. From Figure 3, we see that those
+datasets are commonly paired with a relatively small num-
+ber of images so that the visual signals can enhance only a
+small proportion of token representations. In addition, some
+datasets, i.e., ColA, mainly require linguistic knowledge for
+solving the tasks, so introducing visual modality might not
+benefit such tasks, which corresponds to the common short-
+coming of vision injection for language tasks. For MNLI and
+QNLI, the possible reason for the marginal improvements
+would be that both of the datasets are quite large. Still, the
+task is relatively simple, so the model might well solve the
+tasks directly via the text representations. In this scenario,
+the visual features might only provide the regularization
+effect to improve the model robustness [80, 81, 82].
+0
+5
+10
+15
+20
+epoch
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+value
+Fig. 4. Illustration of the gate values λ with the UVR-TILT method on
+Multi30K En-De Test2016.
+(v) For the two retrieval methods, we observe that UVR-
+CMRM is slightly better than UVR-TILT in general, and
+the results of the two methods are pretty close for the
+MMT task. We find a performance tradeoff between them:
+there might be more accurate similarity calculation after
+cross-modal pre-training than the direct topic extraction.
+However, controlling the proper similarity threshold would
+be a heuristic for each dataset. In contrast, the advantage of
+UVR-TILT is the simple preprocessing, which only requires
+TF-IDF-based topic extraction and matching.
+(vi) We also compare the results using different seed cor-
+pora, i.e., Multi30K and COCO. For the MMT task in Table
+2, using Multi30k is basically similar to or slightly better
+than COCO in general, as the task could enjoy the images
+in the same domain. For the out-of-domain evaluations in
+Table 1 and Table 4, we obverse that using COCO generally
+achieves better results because the size of COCO images
+is three times that of Multi30K, which could provide more
+diverse image features.
+6
+ANALYSIS
+This section presents our exploration on the role of visual
+contexts, which involves two aspects, when the visual con-
+text helps and how the visual context helps language repre-
+sentations. In the following analysis part, we use Multi30K
+for UVR-TILT and COCO for UVR-CMRM by default.
+6.1
+Dynamics of the visual information
+To explore the role of visual context in the training process,
+we illustrate the gate values λ (defined in Eq. 5) in Figure
+4 where a larger value indicates more dependence on the
+visual context in the fusion process. We observe that the
+model relies on the visual context in the early stages, and
+the role of visual information changes dynamically across
+training. When the training starts, the model accommodates
+the visual information to a large extent (λ ≥ 0.6), indicating
+that the model tends to trust the visual context, which could
+provide useful information in the early stages. With more
+knowledge captured as the training continues, the contribu-
+tions of the visual contexts appeal to decay. In the following
+parts, we will further discuss the specific effectiveness of
+visual representations in language modeling.
+
+10
+A girl in a purple tutu dances in the yard.
+A little girl is walking over a path of numbers.
+A girl jumping rope on a sidewalk near a parking garage.
+A young girl washes an automobile.
+Fig. 5. Examples of sentences that share the same retrieved images, in which the common topic is about “girl”. Sentences with similar topics tend to
+be paired with similar or even the same images, and vice versa. This means that images may provide topic information, which benefits the modeling
+of similar sentences.
+6.2
+Pairwise relationship across modalities
+The benefits of the universal representation method could
+be two folds: (i) the content connection of the sentences and
+images; (ii) the topic-aware co-occurrence of similar images
+and sentences. According to Distributional Hypothesis [83],
+which states that words that occur in similar contexts tend to
+have similar meanings, we are inspired to extend the concept
+in the multimodal world, the sentences with similar meanings
+would be likely to pair with similar even the same images. There-
+fore, the consistent images (with a related topic) could play
+the role of topic or type hints for similar sentence modeling.
+After using our image retrieval method, sentences with
+similar topics tend to be paired with similar or even the
+same images, and vice versa. Figure 5 shows examples in
+which the common topic is about “girl”. This means that
+images may provide topic information, which benefits the
+modeling of similar sentences. Thus, aside from the image
+embeddings’ inner meaning (vectors), there is a mapping
+relationship between the sentence and images after the
+retrieval process.
+For the image embeddings, as described in Section 3.1.2,
+we adopt the embedding lookup to fetch the embedding
+features for each image, which is very similar to the way of
+using word embedding by treating each image as a “word”.
+The weights of the embedding features are derived from
+the average pooled output of ResNet, where each image is
+represented as a 2400-d vector. For all the 29,000 images
+(e.g., using Multi30K), we have an embedding layer with
+size (29000, 2400). The “content” of the image can be seen
+as embedding initialization. The pre-initialized embedding
+weights might yield slight improvement gains. However,
+the neural network can also be effectively trained with
+random initialization [84, 85]. In contrast, whether to use
+the embedded feature is more critical. In other words, the
+mapping relationship of the sentences and images in image
+embedding would be essential, i.e., similar sentences (with
+the same topic words) tend to map the same or similar
+image after the word-image lookup process.
+To verify the hypotheses, we conduct the following
+ablations: we replace the ResNet50 feature extractor in our
+UVR-TILT model with (1) ResNet101 and (2) ResNet152;
+additionally, we compare the results with the following
+operations: (3) Shuffle: shuffle the image features but retain
+TABLE 6
+Ablation for the image embedding operation on En-Ro. The scores are
+reported by means and standard deviations for three random seeds.
+Method
+BLEU Score
+Baseline
+32.75±0.10
+UVR-TILT
+33.72±0.08
+w/ (1) Res101
+33.65±0.06
+w/ (2) Res152
+33.82±0.07
+w/ (3) Shuffle
+33.40±0.14
+w/ (4) Random Init
+33.08±0.17
+w/ (5) Random Mapping
+32.05±0.14
+the lookup table; (4) Random Init: randomly initialize the
+image embedding but keep the lookup table; (5) Random
+Mapping: randomly retrieve unrelated images.
+Table 6 shows the ablation results. The BLEU scores
+of models 1-4 are close to the proposed UVR method,
+and those ablated methods still outperform the baseline,
+indicating that using image features generally yields better
+performance than the baseline. In addition, either replacing
+the trained image features (model 4) or disturbing the
+mapping information (model 5) leads to a performance
+drop (↓0.64/↓1.67, respectively), which indicates that both
+the image features and mapping information are contribut-
+ing factors. Compared with image features, the mapping
+information has a larger impact, which verifies our prior
+hypothesis that the consistent images with a related topic
+could play the role of topic or type hints for similar sentence
+modeling in the whole training process. With the mapping
+information, the same images will be assigned to the same
+context. During training, the image features will be learned
+just like word embeddings. Therefore, it does not mean that
+the image features are not very helpful, but the mapping
+information in the lookup table reduces the dependence on
+the trained image features.
+From the view of an individual image, image content
+(embedding) has an effect. If we maintain the pairwise
+relationship between the sentence and image, the result is
+still higher than the baseline, even with shuffled or random
+image embeddings. This indicates that the pairwise rela-
+tionship is a vital contributor. From the macro perspective
+of sentence-image co-occurrence, image information plays
+the role of topic information, where similar sentences tend
+
+11
+a) baseline method without visual features
+b) our method with visual features
+Fig. 6. Visualization of (a) attention weights of the input tokens with regards to the probed token “lock” across different layers (Y-axis) using the
+BERT baseline; (b) image-to-word attention from our model. The illustration shows that the images provide fine-grained grounding information about
+the relationship between concepts and events, e.g., “lock”, “door”, “fancy”, “hotel”.
+TABLE 7
+Results of MMT with incomplete source texts by removing visually
+grounded tokens in the Multi30K dataset. The scores are reported by
+means and standard deviations for three random seeds.
+Model
+En-De
+En-Fr
+Test2016
+Test2017
+Test2016
+Test2017
+Baseline
+10.94±0.21
+7.75±0.24
+18.61±0.16
+15.01±0.15
+UVR-TILT
+12.80±0.18
+9.15±0.19
+19.60±0.17
+15.77±0.20
+to pair with similar images. The observation corresponds
+to the distributional hypothesis. This explains the potential
+effects of the pairwise relationship.
+This finding may potentially facilitate future research
+because most existing studies focus on the content of the
+individual image itself. We highlight the pairwise relation-
+ship across modalities as a different research line to bridge
+the gap between language and image modeling.
+6.3
+Handling incomplete source texts
+Content words are naturally related to specific content, such
+as car, room, play. We collect a list of tokens that have
+more than ten occurrences in the Multi30K training set
+after removing all stop words following [72]. We remove
+those tokens in the source sentence, which occupy 42.87%
+of tokens in the token dictionary. Table 7 shows the results
+of the baseline model and our model with the incomplete
+source texts. We observe that our model achieves noticeable
+gains over the baseline. The results verify that the visual
+representation can reduce the gap of the missing informa-
+tion from the content words in the source texts.
+6.4
+Knowledge grounding with the visual context
+To gain an insight into the process of multimodal integra-
+tion by our model, we analyze the attention distributions
+(α in Eq.3) at the multimodal integration layer. Figure 6
+shows the attention distributions of (a) the baseline and
+(b) our model 13 for an example randomly selected from
+13. Our model is the UVR-TILT trained on the CoLA dataset.
+TABLE 8
+Results (BLEU score) of the multimodal disambiguation experiments
+on WAT’19 English to Hindi dataset. The scores are reported by means
+and standard deviations for three random seeds.
+Model
+Validation
+Test
+Challenge
+Baseline
+47.04±0.27
+39.33±0.24
+20.52±0.14
+UVR-CMRM
+47.49±0.06
+39.81±0.12
+21.62±0.08
+our GLUE validation sets, “You should always lock your door,
+no matter how fancy the hotel might be.” For comparison,
+the baseline is implemented following Abnar and Zuidema
+[86]. Concretely, we collect the attention weights of all the
+input tokens with regards to a targeted token “lock” across
+different layers (i.e., {2, 4, . . . , 24}). As the baseline uses the
+representation of the last layer for prediction, we focus on
+the attention distributions of the last layer.
+Compared with the baseline that only captures partial
+relations in the last layer, e.g., with a lack of relationship
+among {“lock”, “door”, “hotel”}, our model provides more
+fine-grained connections. In detail, two generic patterns are
+observed in these examples.
+(i) the images appear to match the concepts and actions
+with the texts, in other words, the images tend to provide
+fine-grained grounding information about the relationship
+between concepts and events, e.g., {”lock”, ”door”, ”fancy”,
+”hotel”}.
+(ii) our model can resist irrelevant information from
+noisy images. For example, the second and third images
+yield low attention scores for the texts.
+6.5
+Disambiguation
+A natural intuition of using visual clues for text represen-
+tation is the advantage of alleviating the ambiguation of
+language. To evaluate the model performance for disam-
+biguation, we use a dataset from the HVG [88], which serves
+as a part of the WAT’19 Multimodal Translation Task.14 The
+dataset consists of a total of 31525 randomly selected images
+14. http://lotus.kuee.kyoto-u.ac.jp/WAT/WAT2019/index.html
+
+1.0
+always
+lock
+0.8
+Joop
+0.6
+0.4
+how
+fancy
+ 0.2
+might
+be
+0.0Toilet
+f1忆
+2
+0.175
+0.150
+8
+0.125
+4
+0.100
+2
+0
+0.075
+8
+0.050
+9
+4
+0.025
+2
+2
+matt12
+a) baseline method without visual features
+b) our method with visual features
+Fig. 7. Visualization of (a) attention weights of the input tokens with regards to the ambiguous token “apple” across different layers (Y-axis) using the
+BERT baseline; b) image-to-word attention from our model. The illustration shows that the images bridge the connection between “apple”, “watch”,
+“time”, helping disambiguate the meaning of “apple”.
+TABLE 9
+Selected eight probing tasks [87] to study what syntactic and semantic properties are captured by the encoders.
+Probing Tasks
+Content
+Syntactic
+TrDep
+Checking whether an encoder infers the hierarchical structure of sentence
+ToCo
+Sentences should be classified in terms of the sequence of top constituents immediately below the
+sentence node
+BShif
+Testing whether two consecutive tokens within the sentence have been inverted
+Semantic
+Tense
+Asking for the tense of the main clause verb
+SubN
+Focusing on the number of the main clause’s subject
+ObjN
+Testing for the number of the direct object of the main clause
+SoMo
+Some sentences are modified by replacing a random noun or verb with another one and the classifier
+should tell whether a sentence has been modified
+CoIn
+Containing sentences made of two coordinate clauses
+TABLE 10
+Classification accuracy on eight probing tasks of evaluating linguistics embedded in the encoder outputs.
+Model
+Syntactic
+Semantic
+TrDep
+ToCo
+BShif
+Tense
+SubN
+ObjN
+SoMo
+CoIn
+Baseline
+28.34
+58.33
+76.34
+80.66
+72.02
+68.57
+64.42
+67.51
+UVR-CMRM
+28.53
+58.64
+77.72
+80.97
+73.79
+69.66
+65.44
+67.23
+from Visual Genome [89] and a parallel image caption
+corpus in English-Hindi for selected image segments. The
+training part consists of 29K English and Hindi short cap-
+tions of rectangular areas in photos of various scenes, and
+it is complemented by three evaluation subsets: validation,
+test, and challenge test set (Challenge). The challenge test set
+is created by searching for (particularly) ambiguous English
+words based on the embedding similarity and manually
+selecting those where the image helps to resolve the am-
+biguity. We do not use the images but follow the same
+settings as the experiments on Multi30K. As the results
+shown in Table 8, we observe that our UVR model works
+effectively on the challenge disambiguation set, indicating
+that the visual information induced by retrieved images
+allows disambiguation of translation.
+Figure 7 shows a heatmap of the attention visualization
+on an ambiguous sentence, “apple watch keeps telling me
+the time”. In Figure 7(a), the baseline model fails to capture
+the relationship between “apple” with “time”. In Figure
+7(b), the retrieved images bridge the connection among
+“apple”, “watch” and “time”, which helps disambiguate the
+meaning of “apple”.
+6.6
+Linguistic Analysis
+We are interested in what knowledge is learned in the
+universal representations. In this section, we select eight
+widely-used language probing tasks [87] (see Table 9) to
+study what kind of syntactic and semantic properties are
+captured by the encoders. Specifically, we use the encoders
+of the baseline BERT-based SNLI model,15 and our UVR-
+CMRM visual representation model to generate the sentence
+representations of input, which are used to carry out the
+above eight probing tasks. The results are as shown in Table
+10.
+15. We select the SNLI model because NLI models show good
+generalization capacity for language representation [90, 91], which is
+supposed to be a strong test-bed for the evaluation.
+
+1.0
+apple
+watch
+ 0.8
+0.6
+telling
+0.4
+the
+ 0.2
+time
+0.0AppleWatchEdition
+WATCHEDITIONWATCHWATCH7AppStore&iTunes忆
+0.25
+2
+0.20
+4
+ 0.15
+2
+0
+ 0.10
+8
+6
+0.05
+4
+2
+me
+the
+telling13
+0
+1
+3
+5
+7
+9
+15 20 30
+33
+33.5
+Number of images per sentence
+BLEU
+Fig. 8. Influence of the number of images on the BLEU score.
+TABLE 11
+Experiments on different source languages using the Multi30K dataset.
+The baseline is Transformer-Tiny.
+Model
+De-En
+Fr-En
+Test2016
+Test2017
+Test2016
+Test2017
+Baseline
+42.88
+40.57
+54.60
+48.45
+UVR-TILT
+43.10
+40.07
+54.92
+49.10
+Concerning semantic properties, our model gains the
+most significant improvement on the SubN and ObjN tasks.
+The result indicates that visual information helps NMT to
+identify and represent the subject and object information,
+which is consistent with our hypotheses.
+6.7
+Effectiveness across languages
+Table 11 shows the experiment results on different source
+languages. We observe that our method is applicable when
+the source texts are in other languages such as German
+and French. Our proposed methods are supposed to be
+independent of languages because the calculation for image
+retrieval only relies on the light lookup table, which can be
+extracted from an off-the-shelf seed corpus that is available
+for many languages.
+6.8
+Joint Training and Fine-tuning
+Since the multimodal and text-only machine translation
+tasks can benefit from the visual modality after retrieving
+images from the seed corpus, it is possible to bridge both
+tasks to train an even more powerful model. The connec-
+tions between the texts and images inside the Multi30K
+datasets could be strong indications to bridge the gap be-
+tween text and image modalities.
+Therefore, we train a unified model based on UVR-
+CMRM by using the joint En-De datasets of Multi30K and
+WMT’14 (Joint Model), and respectively train the Multi30K
+(Fine-tuned Multi30K) and WMT’14 (Fine-tuned WMT) mod-
+els by initializing the trainable model parameters using the
+joint model. According to the results shown in Table 12, we
+summarize the following observations:
+(i) Two-stage training (joint training and fine-tuning) can
+boost the performance on the two concerned datasets, which
+indicates that the highly relevant Multi30K dataset can play
+the role of the seed data for training a text-only NMT model
+using our topic-image lookup table.
+0
+50
+100
+Percentage
+BLEU
+Data Size
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+33.5
+34
+34.5
+35
+Percentage
+BLEU
+Fig. 9. BLEU score for different similarity thresholds.
+TABLE 12
+Results of joint training and fine-tuning.
+Model
+Multi30K Task
+WMT Task
+Joint training baseline
+37.28
+27.68
++ Fine-tuned Multi30K
+-
+27.96
++ Fine-tuned WMT
+43.13
+-
+(ii) The major gain is achieved by fine-tuning the smaller
+dataset, showing that the large-scale text-only dataset with
+our lookup table can also provide valuable complementary
+information for training a multimodal model on a much
+smaller dataset. The result further verifies the effectiveness
+of our method in low-resource settings.
+6.9
+Parameter Sensitivity Analysis
+In this section, we analyze our model sensitivity against
+parameter settings, including similarity threshold, number
+of images, and gating weight.
+Influence of the similarity threshold. We investigate the
+influence of the similarity threshold δ that is set to filter the
+top-ranked images for each sentence in UVR-CMRM. Figure
+9 shows the performance for thresholds in [0.0, 0.1, 0.2, 0.3,
+0.4, 0.5, 0.6] on the En-Ro test set. We observe that setting
+the threshold around 0.4 can yield a good balance of data
+size and BLEU score. It is reasonable that the best thresholds
+vary for different datasets because of the domain divergence
+of the image corpus for pre-training.
+Influence of the number of images. To evaluate the
+influence of the number of paired images m for UVR-TILT,
+we constrain m in {0, 1, 3, 5, 7, 9, 15, 20, 30} for experiments
+on the En-Ro test set, as shown in Figure 8. When m = 0,
+the model is the baseline NMT model, whose BLEU score
+is lower than all the models with images. As the number
+of images increases, the BLEU score also increases at the
+beginning (from 32.66 to 33.78) and then slightly decreases
+when m exceeds 5. The reason might be that too many
+images for a sentence would have a higher chance of noise.
+Therefore, we set m = 5 in our models.
+Influence of gating weight λ. In our model, the weight
+λ of the gated aggregation method is learned automatically
+to measure the importance of the visual information. We
+compare by manually setting the weight λ to scalar values
+in {0.1, 0.3, 0.5, 0.7, 0.9} for experiments of UVR-TILT on the
+En-Ro test set. Figure 10 shows that all models with manual
+
+14
+0.1
+0.3
+0.5
+0.7
+0.9
+33
+33.5
+Weight λ
+BLEU
+Manual weight
+Transformer-Base
+Ours
+Fig. 10. Quantitative study of the gating weight λ.
+λ outperform the baseline Transformer-base, indicating the
+effectiveness of image information. In contrast, they are in-
+ferior to the performance of our model. This means that the
+degree of dependency for image information varies for each
+source sentence, indicating the necessity of automatically
+learning the gating weights of image representations.
+6.10
+Computation Efficiency
+There are mainly two extra computation costs using our
+method, including (i) obtaining image data for sentences
+and (ii) learning image representations, which are negligible
+compared with training an NMT model. The time of retriev-
+ing image data for MT sentences for the En-Ro dataset is
+less than 1 minute using GPU. The lookup table is formed
+as the mapping of the token (only topic words) index to
+the image id. Then, the retrieval method is applied as the
+tensor indexing from the sentence token indices (only topic
+words) to image ids, which is the same as the procedure of
+word embedding. The retrieved image ids are then sorted
+by frequency. Learning image representations takes about
+2 minutes for all the 29,000 images in Multi30K using 6G
+GPU memory for feature extraction and eight CPU threads
+for transforming images. The extracted features are formed
+as the “image embedding layer” in the size of (29000, 2400)
+for quick access in the neural network.
+7
+CONCLUSIONS
+This work investigates a flexible framework to incorporate
+visual information into sentence modeling by image re-
+trieval from a light lookup table and learned cross-modal
+embedding space. Extensive empirical experiments on 14
+benchmark datasets verify the effectiveness of the proposed
+method. A series of case studies are conducted to evaluate
+visual benefits and influence factors. Our method is general
+and can be easily implemented in existing deep-learning
+NLP systems for different languages. Through the proposed
+retrieval methods, we can provide a group of images that
+disclose a diversity of implicit topics that might be entailed
+in sentences, yielding better context grounding with fine-
+grained information. We show that our method enriches
+the representation of content words, provide fine-grained
+grounding information about the relationship between con-
+cepts and events, and potentially enhances the accuracy
+of disambiguation. Besides incorporating images to build
+the pairwise relationship across modalities, it is potential
+to incorporate various extra knowledge as alignment topic
+information in the future, such as audio, not only images.
+ACKNOWLEDGEMENT
+Part of this study has been published as “Neural Machine
+Translation with Universal Visual Representation” [37] in
+the Eighth International Conference on Learning Represen-
+tations (ICLR 2020). The extension includes three sides: (i)
+general tasks: this work studies the universal visual rep-
+resentation for language representation in a broader view
+of the natural language processing scenario, with experi-
+ments on 14 representative NLP tasks; (ii) new method: this
+work investigates new methods of semantic sentence-image
+matching from a shared cross-modal space, to give more
+accurately paired images as topic information; (iii) in-depth
+analysis to interpret the benefits from the visual modality.
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+Zhuosheng Zhang received his Bachelor’s de-
+gree in internet of things from Wuhan University
+in 2016, his M.S. degree in computer science
+from Shanghai Jiao Tong University in 2020. He
+is working towards his Ph.D. degree in computer
+science with the Center for Brain-like Computing
+and Machine Intelligence of Shanghai Jiao Tong
+University. He was an internship research fellow
+at NICT from 2019-2020. His research interests
+include natural language processing, machine
+reading comprehension, dialogue systems, and
+machine translation.
+Kehai Chen is an Assistant Professor at Harbin
+Institute of Technology (Shenzhen) since 2022.
+Before that, he was a researcher at Japan Na-
+tional Institute of Information and Communica-
+tions Technology (NICT) from 2018 to 2021. He
+received the Ph.D. degree in computer science
+from Harbin Institute of Technology in 2018. His
+research interests include machine translation
+and natural language processing.
+Rui Wang is an associate professor at Shanghai
+Jiao Tong University since 2021. Before that, he
+was a researcher (tenured in 2020) at Japan
+National Institute of Information and Communi-
+cations Technology (NICT) from 2016 to 2020.
+He received his B.S. degree from Harbin Institute
+of Technology in 2009, his M.S. degree from the
+Chinese Academy of Sciences in 2012, and his
+Ph.D. degree from Shanghai Jiao Tong Univer-
+sity in 2016, all of which are in computer science.
+He was a joint Ph.D. at Centre Nationnal de
+la Recherche Scientifique, France in 2014. His research interests are
+machine translation and natural language processing.
+Masao Utiyama is a research manager of the
+National Institute of Information and Communi-
+cations Technology, Japan. He completed his
+doctoral dissertation at the University of Tsukuba
+in 1997. His main research field is machine
+translation.
+Eiichiro Sumita received the Bachelor and Mas-
+ter degree in computer science from The Univer-
+sity of Electro-Communications, Japan in 1980
+and 1982 and the Ph.D degree in Engineering
+from Kyoto University, Japan in 1999. He is cur-
+rently Director of Multilingual Translation Labora-
+tory of National Institute of Information and Com-
+munication Technology from 2006. He worked
+at Advanced Telecomunications Research Insti-
+tute International from 1992 to 2009 and IBM
+Research-Tokyo from 1980 to 1991. His re-
+search interests include machine translation and e-Learning.
+Zuchao Li received the B.S. degree from Wuhan
+University, Wuhan, China, in 2017. Since 2017,
+he has been a Ph.D. student with the Center for
+Brain-like Computing and Machine Intelligence
+of Shanghai Jiao Tong University, Shanghai,
+China. His research focuses on natural language
+processing, especially syntactic and semantic
+parsing.
+Hai Zhao received the BEng degree in sensor
+and instrument engineering, and the MPhil de-
+gree in control theory and engineering from Yan-
+shan University in 1999 and 2000, respectively,
+and the PhD degree in computer science from
+Shanghai Jiao Tong University, China in 2005.
+He is currently a full professor at department
+of computer science and engineering, Shanghai
+Jiao Tong University after he joined the university
+in 2009. He was a research fellow at the City
+University of Hong Kong from 2006 to 2009, a
+visiting scholar in Microsoft Research Asia in 2011, a visiting expert in
+NICT, Japan in 2012. He is an ACM professional member, and served as
+area co-chair in ACL 2017 on Tagging, Chunking, Syntax and Parsing,
+(senior) area chairs in ACL 2018, 2019 on Phonology, Morphology and
+Word Segmentation. His research interests include natural language
+processing and related machine learning, data mining and artificial
+intelligence.
+
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+page_content='1  Universal Multimodal Representation for  Language Understanding  Zhuosheng Zhang#, Kehai Chen, Rui Wang#, Masao Utiyama, Eiichiro Sumita, Zuchao Li, Hai Zhao*  Abstract—Representation learning is the foundation of natural language processing (NLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This work presents new methods to employ  visual information as assistant signals to general NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For each sentence, we first retrieve a flexible number of images either  from a light topic-image lookup table extracted over the existing sentence-image pairs or a shared cross-modal embedding space that  is pre-trained on out-of-shelf text-image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then, the text and images are encoded by a Transformer encoder and convolutional  neural network, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The two sequences of representations are further fused by an attention layer for the interaction of the two  modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In this study, the retrieval process is controllable and flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The universal visual representation overcomes the lack of large-  scale bilingual sentence-image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Our method can be easily applied to text-only tasks without manually annotated multimodal parallel  corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We apply the proposed method to a wide range of natural language generation and understanding tasks, including neural  machine translation, natural language inference, and semantic similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Experimental results show that our method is generally effective  for different tasks and languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Analysis indicates that the visual signals enrich textual representations of content words, provide fine-  grained grounding information about the relationship between concepts and events, and potentially conduce to disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Index Terms—Artificial Intelligence, Natural Language Understanding, Vision-Language Modeling, Multimodal Machine Translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  1  INTRODUCTION  L  EARNING contextualized representations of human lan-  guages is one of the major themes in natural language  processing (NLP), which is also fundamental to training  machines to understand human languages and handle ad-  vanced tasks, such as machine translation, question an-  swering, and human-computer conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Text repre-  sentation learning has evolved from standard distributed  representations [1, 2] to contextualized language represen-  tation from deep pre-trained representation models (PRMs)  [3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Despite the success of PRMs, NLP models  commonly model the world knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g, commonsense,  rules, events, assertions extracted from raw texts) solely  from textual features without grounding of the outside  world, such as visual conception [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Languages are abstract Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Zhao are with the Department of  Computer Science and Engineering, Shanghai Jiao Tong University,  China and also with Key Laboratory of Shanghai Education Com-  mission for Intelligent Interaction and Cognitive Engineering, Shang-  hai Jiao Tong University, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Chen is with the School of  Computer Science and Technology, Harbin Institute of Technology,  Shenzhen, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Utiyama and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Sumita are with the Na-  tional Institute of Information and Communications Technology (NICT),  Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' E-mail: {zhangzs, charlee}@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content=' chenkehai@hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  {mutiyama, eiichiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='sumita}@nict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='jp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' wangrui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='nlp@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' zhao-  hai@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Zhao is supported by Key Projects of National Natural Science  Foundation of China (U1836222 and 61733011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Wang is supported  by National Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 6217020129),  Shanghai Pujiang Program (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 21PJ1406800), Shanghai Municipal  Science and Technology Major Project (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 2021SHZDZX0102), Beijing  Academy of Artificial Intelligence (BAAI) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 4), CCF-Baidu Open  Fund (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' CCF-BAIDU OF2022018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Chen is supported by National  Natural Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 62276077) and Shenzhen  College Stability Support Plan (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' GXWD20220811170358002 and  GXWD20220817123150002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Zhang and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Wang contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Part of this work  was finished when Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Zhang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Li visited NICT, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Wang was  with NICT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Corresponding author: Hai Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  and rather difficult for the brain to retain, whereas visuals  are concrete and, as such, more easily remembered [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Adopting multimodality would be essential for better back-  ground perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Therefore, a trend of research has been  motivated to apply non-linguistic modalities to language  representations [7, 10, 11, 12, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Most of previous works focus on joint modeling im-  ages and texts, involving vision-language (VL) pre-training  [15, 16, 17, 18, 19, 20] and multimodal (MM) application  tasks [14, 21, 22, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' However, these studies rely on  large-scale text-image annotations as the paired input and  thus are confined to VL or MM tasks, such as image cap-  tioning and visual question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' It is natural to boost  the performance on VL and MM tasks as the concerned  datasets are human-labeled with high quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' However, the  essential challenge lies with the real-world scenario as there  is no such high-quality annotated text-image aligned corpus  for text-only NLP applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Therefore, it is critical to  investigate a general method to take advantage of visual  information in a wide range of mono-modal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', text-only)  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In addition, it is still not clear the role of images  in language representation, as well as how to apply the  multimodality in the standard NLP scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Taking multimodal machine translation (MMT) as an  example, the starting point is to leverage visual information  to improve the quality of the translation from the source  to the target languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' However, the effectiveness heavily  relies on the availability of bilingual parallel sentence pairs  with manual image annotations, which hinders the image  applicability to neural machine translation (NMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' As a re-  sult, the visual information is only applied to the translation  task over specific multimodal datasets [25, 26, 27, 28, 29], in-  stead of general text-only NMT [30, 31, 32] and low-resource  text-only NMT [33, 34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In addition, because of the  high cost of annotation, the content of one bilingual parallel  Copyright © 20XX IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media,  including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists,  or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='03344v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='CL]  9 Jan 2023  2  sentence pair is paired with a single image, which is weak in  capturing the diversity of visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Therefore, the  current study of introducing visual information falls into a  bottleneck in the multimodal NMT and is not feasible for  text-only NMT and low-resource NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Our previous work [37] finds that using monolingual  corpora with image annotations can overcome the lack of  large-scale bilingual sentence-image pairs, thereby extend-  ing image applicability in NMT, with performance gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The method of using a lookup table is task-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This  work stimulates our further thinking, and we are interested  in answering the three major aspects of questions:  (i) Global Multimodality: Can we apply the multi-  modality to standard text-only NLP tasks to enhance the  language representations (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', natural lan-  guage generation (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1) and natural language un-  derstanding (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (ii) Interpretability: Why does the universal representa-  tion method work (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' How does multimodality  improve language representation, and what is the network  learned(Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (iii) Quality: How to control the quality of visual-text  alignment to reduce noise (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  In this paper, we present a universal visual represen-  tation (UVR) method relying only on a seed set of task-  independent annotations, instead of the existing approach  that depends on large-scale task-specific image-text anno-  tation, thus breaking the bottleneck of using visual infor-  mation in standard text-only NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For each sentence,  we retrieve diverse images from either a light topic-image  lookup table or pre-trained shared text-visual embedding  space that is pre-trained on a large-scale of text-image  pairs, to connect both the mono-modal paths of text and  image embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The text and images are encoded by  Transformer language model (LM) and a pre-trained convo-  lutional neural network (CNN), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' A simple and  effective attention layer is then designed to fuse the two  sequences of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Our approach can be easily applied to text-only tasks  without manually annotated multimodal parallel corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Therefore, our method is universal in terms of the task re-  quirements, in contrast to the recent vision-language models  that require large-scale and expensive annotation datasets  for each downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The proposed method is eval-  uated on 14 NLP benchmark datasets involving natural  language inference (NLI), semantic similarity, text classifica-  tion, and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The experiments and analysis  verify the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' To summa-  rize, our contributions are primarily three-fold:  (i) This work studies the universal visual representa-  tion for language representation in a broader view of the  natural language processing scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Besides neural ma-  chine translation, this work leverages visual information  as assistant signals for general NLP tasks, with the focus  on investigating the global multimodality for general NLP,  interpretability of effectiveness, and quality control of using  universal visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (ii) For the technical side, this work proposes new meth-  ods of semantic sentence-image matching from a shared  cross-modal space to give more accurately paired images  as topic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We also present a new multimodal  representation framework and systematically study the two  main instances, including the model with the original TF-  IDF topic-image lookup table and the newly proposed one  from the retrieval from cross-modal retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (iii) Experiments are extended to 14 representative NLP  tasks, which show the effectiveness of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  A series of in-depth analyses indicate that the visual signals  enrich textual representations of content words, provide  fine-grained grounding information about the relationship  between concepts and events, and potentially conduce to  disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  2  BACKGROUND  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Vision-Language Integration  This study is related to that of VL methods (VLMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Re-  cently, there has been a great deal of interest in integrating  image presentations in pre-trained Transformer architec-  tures [15, 16, 17, 18, 19, 20, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The common strategy is  to take a Transformer model [32], such as BERT, as the back-  bone and learn aligned representations of visual and lan-  guage in a pre-training manner inspired by the masked lan-  guage modeling mechanism in pre-trained language models  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' These studies require the annotation of task-dependent  sentence-image pairs, which are limited to VL tasks, such as  image captioning and visual question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Two studies [22, 37] are closely related to general image-  enhanced LM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Glyce [22] proposes incorporating glyph  vectors for Chinese character representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' However, it  can only be used for Chinese and only involves single  image enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Regarding the technical part, previous  methods only benefit from one image per sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We  propose taking advantage of a group of similar images using  a filtering mechanism to form a more fine-grained visual-  aware context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Our early version of this work [37] proposes  using multiple images for NMT, based on a text-image  lookup table trained over a sentence-image pair corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  However, the number of images is fixed because of the lack  of similarity measurement in the simple lookup method,  which possibly makes the resulting model suffer from the  noise of irrelevant images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This work is improved from the  perspectives of motivation and technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' It is motivated  by cross-modal semantic retrieval in the shared embedding  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' It adopts a neural matching method with a similarity  threshold to control the expected matching degree flexibly,  which is generally applicable to a broader range of NLP  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In addition, we conduct an in-depth analysis to inves-  tigate how the visual modality helps text representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Visual-Semantic Embeddings  Another research line is language grounding for images  whose major topic is multimodality and cross-modality be-  tween images and text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The major focus is to bridge the gap  between text and images through building visual-semantic  embeddings [39, 40, 41, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Prior studies have verified that representations of images  and text can be jointly leveraged to build visual-semantic  embeddings in a shared representation space [39, 40, 41, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  To this end, a popular approach is to connect both the mono-  modal text and image encoding paths using fully connected  3  layers [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The shared deep embedding can be used  for cross-modal retrieval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' thus, it can associate sentence  text with associated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Partly inspired by this line of  research, we are motivated to incorporate visual awareness  into sentence modeling by retrieving a group of images for  a given sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  One of the first techniques to align two views of hetero-  geneous data is the canonical correlation analysis method  [47], in which linear projections defined on both sides are  optimized to maximize the cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Recent studies  have followed the two-path architecture [45, 46], in which  the encoder consists of a joint embedding of textual and  image representations extracted from both the images and  corresponding caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Notably, Engilberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' [46] adopts  RNN to encode sentence embeddings in the same space  with extracted image representations from CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Portaz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' [48] enhances cross-modal retrieval using multilingual  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Inspired by the previous success of visual-semantic  embeddings, we apply neural image retrieval from the joint  space to fetch a group of associated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  3  UNIVERSAL REPRESENTATION FRAMEWORK  This section overviews our universal representation frame-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Given a sentence, we first fetch a group of matched  images from our retrieval methods (details of our retrieval  methods will be given in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The text and  images are encoded, respectively, by the text feature extrac-  tor and image feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then the two sequences of  representations are integrated using multi-head attention to  form a joint representation, which is passed to downstream  task-specific layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Figure 1 overviews the whole multi-  modal representation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Encoding Layer  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Text Encoder  We pair each sentence with the top matched m images  according to the retrieval method above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Following [5], the  sentence is fed into the multi-layer Transformer encoder [32]  to learn the text representation H ∈ Rn×d where n and d are  the input text length and dimension of hidden states for the  text representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Let X = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' , xn} be the input sentence in length  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We feed the sequence to a PRM encoder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', BERT  [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In the encoder, the input sequence is firstly mapped  to embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then, the embeddings are passed to multi-  head attention layers [32] to obtain the contextualized rep-  resentations, which is defined as  H = FFN(MultiHead(K, Q, V )),  (1)  where K, Q, V are packed from the input sequence repre-  sentation X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' As the common practice, we set K = Q = V  in the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' MultiHead is short for multi-head  attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Image Encoder  Similar to the standard way of retrieving word embeddings,  the image embeddings are fetched from a lookup table ∈  Rnm×dm that contains the image features encoded by a pre-  trained ResNet [49],1 where nm is the number of the total  number of unique images +1 and dm is the dimension of the  image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2 The first row of the lookup table is filled by  all-zero vectors, which will be used when no image is paired  for the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  After the feature lookup process, we obtain the image  embeddings E ∈ Rm×dm for the m input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then, the  embeddings are passed to a feedforward layer, to produce  the image representation M ∈ Rm×d with the same hidden  dimension as H:  M = FFN(ResNet(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (2)  There may exist cases when no word in the sentence can  be found in the topic-image lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' When there is no  paired image retrieved, we use the first-row all-zero vectors  of the image lookup table as the “blank features” in the  intuition to tell the model to ignore them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Multimodal Integration Layer  We connect the visual and text modalities by calculating the  attention between image and text features:  α = softmax(H(WgM + bg)⊤),  (3)  H′ = αM,  (4)  where Wg and bg are parameters to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' α ∈ Rn×m  denotes the weights assigned to the different hidden states  in the sentence and the image sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' H′ ∈ Rn×d is the  weighted sum of all the hidden states and it represents how  the sentence can be aligned to each hidden state in the image  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The retrieval process may possibly introduce noise of  irrelevant images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' To alleviate the influence, we use a neural  gating mechanism for information filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In detail, we  compute λ ∈ [0, 1]n×d to weight the expected importance  of image representation for each source word:  λ = sigmoid(WλH′ + UλH),  (5)  where Wλ and Uλ are model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We then fuse  H and H′ and pass the resulting representation to layer  normalization and learn an effective source representation:  ˆH = LayerNorm(H + λH′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (6)  The text and image representations are jointly encoded  as ˆH, which is fed to the task-specific layers for downstream  decoding or predictions depending on task settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  Task-specific Layer  In this section, we show how the joint representation ˆH is  used for downstream tasks by considering NMT and NLU  tasks as examples, generally following the standard proce-  dure of the concerned tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For NMT, ˆH is directly fed to  the decoder to learn a dependent-time context vector to pre-  dict the target translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For other tasks, ˆH is directly fed  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Note that this is the standard lookup table in embedding imple-  mentations, which is not our topic-image lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We used the maxpooling layer of ResNet, which is in the size of  Rnm×2400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  4  Add & Norm  Feed Forward  Add & Norm  Multi-head  Attention  Add & Norm  Multi-head  Attention  Feed Forward  ResNet  Text  Embedding  Sentence  Text Encoder  Image  Corpus  Image Retrieval  Pooling  Linear  Downstream Adaption  (NLU, NMT, etc)  Image Encoder  Integration  L×  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Overview of the universal representation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Given a sentence as input, a group of related images will be retrieved by our image  retrieval methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The text and images are encoded by the text feature extractor and image feature extractor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then the two sequences  of representations are integrated using multi-head attention to form a joint representation in the same shape as the original text sequence  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Finally, the joint representation is passed to downstream task-specific layers to give predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  to a feed-forward layer to make the prediction, which fol-  lows the same downstream procedure as the Transformer-  based LMs, like BERT [5] and RoBERTa [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Specifically,  for sentence-pair tasks, we maintain the pairwise input as  that in LMs and separate the encoded text representation  into two individual sentence representations {H1, H2}, ac-  cording to the positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The two text representations are  integrated with the corresponding image representations  respectively {M 1, M 2}, and then the resulting sequences  are concatenated for prediction, ˆH = ˆH1 ◦ ˆH2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  4  IMAGE RETRIEVAL METHODS  In this section, we describe our two visual retrieval models  used for image retrieval given sentence text:  (i) UVR-TILT: retrieval by topic-image lookup table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (ii) UVR-CMRM: retrieval from cross-modal embed-  ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Model-I: Retrieval by Topic-Image Lookup Table  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Topic-image Lookup Table Conversion  In this section, we will introduce the proposed universal  visual representation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Our basic intuition is to trans-  form the existing sentence-image pairs into a topic-image  lookup table,3 which assumes the topic words in a sentence  should be relevant to the paired image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The procedure can  be seen as the inverted index where a topic word is mapped  to a list of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Consequently, a sentence can possess a  group of images by retrieving the topic-image lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  To focus on the major part of the sentence and suppress  the noise such as stopwords and low-frequency words, we  design a filtering method to extract the “topic” words of  the sentence through the term frequency-inverse document  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We use the training set of the Multi30K dataset to build the topic-  image lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Algorithm 1 Topic-image Lookup Table Conversion  Require: Input sentences, S = {X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' XI} and paired  images E = {e1, e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' eI}  Ensure: Topic-image lookup table Q where each word is asso-  ciated with a group of images  1: Obtain the TF-IDF dictionary F = TF-IDF(S)  2: Transform sentence-image pair to topic-image lookup table  Q = LookUp(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' F)  3: procedure TF-IDF(S)  4:  for each sentence in S do  5:  Filter stop-words in the sentence  6:  Calculate the TF-IDF weight for each word  7:  end for  8:  return TF-IDF dictionary F  9: end procedure  10: procedure LOOKUP(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' F)  11:  for For each pair {Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' ei} ∈ zip{S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' E} do  12:  Rank and pick out the top-w “topic” words in the  sentence according to the TF-IDF score in the dictionary F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  and each sentence is reformed as T = {t1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' , tw}  13:  for For each word tjin T do  14:  if ei not in Q[tj] then  15:  Add ej to the corresponding image set Q[tj]  for word tj  16:  end if  17:  end for  18:  end for  19:  return Topic-image lookup table Q  20: end procedure  frequency (TF-IDF),4 inspired by [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Specifically, given an  original input sentence X = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' , xI} of length I  and its paired image e, X is first filtered by a stopword list,5  and then the sentence is treated as a document g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We then  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We describe our methods by regarding the processing unit as word  though this method can also be applied to a subword-based sentence  for which the subword is considered to be the processing unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='com/stopwords-iso/stopwords-en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  UBS  SOONERS  MEGA  OMEGA  0  SOON5  dog is playing in the snow  dog (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='512)  playing (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='531)  snow (439)  (a) a black dog and a spotted dog are fighting  (b) a dog is running in the snow  (c) a dog is playing with a hose  (d) a family playing on a tractor on a beautiful day  (e) two people working on removing snow from a roof  (f) a black dog and a white dog are standing on snow  corpus (29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='000)  (a)  (b)  (c)  (d)  (e)  (f)  sentence-image pairs  topic-image lookup table  associated images for input sentence  tokenize,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' filtering  word-image transform  sampling  ranking  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Illustration of the TILT method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We first transform the existing sentence-image pairs from seed small-scale sentence-image datasets into a  topic-image lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For a given sentence, we extract its topic words and the associated images will be retrieved from the lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  compute TF-IDF TIi,j for each word xi in g,  TIi,j =  oi,j  �  k ok,j  × log  |G|  1 + |j : xi ∈ g|,  (7)  where oi,j represents the number of occurrences of the word  xi in the input sentence g, |G| the total number of source  language sentences in the training data, and |j : xi ∈ g|  the number of source sentences including word xi in the  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We then select the top-w high TF-IDF words as  the new image description T = {t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' , tw} for the input  sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' After the preprocessing, each filtered sentence T  is paired with an image e, and each word ti ∈ T is regarded  as the topic word for image e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' After processing the whole  corpus (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', Multi30K), we form a topic-image lookup table  Q as described in Algorithm 1, in which each topic word ti  would be paired with dozens of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Image Retrieval  For the input sentence, we first obtain its topic words  according to the text preprocessing method described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Then we retrieve the associated images for each topic word  from the lookup table Q and group all the retrieved images  together to form an image list G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We observe that an image  might be associated with multiple topic words so that it  would occur multiple times in the list G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Thus, we sort the  images according to the frequency of occurrences in G to  maintain the same total number of images for each sentence  at m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Figure 2 illustrates the retrieval process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In the left block,  we show six examples of sentence-image pairs in which the  topic words are in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then we process the corpus  using the topic-image transformation method demonstrated  above and obtain the topic-image lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For example,  the word dog is associated with 1,512 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For an input  source sentence, we obtain the topic words (in boldface)  using the same preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then we retrieve the corre-  sponding images from the lookup table for each topic word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Now we have a list of images, and some images appear  multiple times as they have various topics (like the boxed  image in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' So we sort the retrieved image list by the  count of occurrence to pick out the top-m images that cover  the most topics of the sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  At test time, the process of getting images is done using  the image lookup table built by the training set, so we do  not need to use the images from the validation and test sets  in Multi30K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6 Intuitively, we do not strictly require  the manual alignment of the word (or concept) and image  but rely on the co-occurrence of the topic word and image,  which is simpler and more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In this way, we call our  method universal visual retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Model-II: Retrieval from Cross-modal Embedding  Following Engilberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' [46], we train a semantic-visual  embedding on a text-image corpus, which is then used for  image retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The semantic-visual embedding architec-  ture comprises two paths to encode the text and images into  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Based on our preliminary experiments, we maintain  the same settings in Engilberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' [46] by using the simple  recurrent unit as text encoder, and the fully convolutional  residual ResNet-152 [52] with Weldon pooling [53] as image  encoder for our cross-modal retrieval model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  During training, each text X is paired with (i) a posi-  tive image Y that is paired with the text and (ii) a hard  negative Z, which is selected as the image that has the  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The lookup table can be easily adapted to a wide range of other  NLP tasks even without any paired image, and therefore opens our  proposed model to generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6  highest similarity to the text while not being associated  with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Triplet loss [54, 55, 56] is used to enable the images  to converge correctly to improve the performance of the  proposed method:  loss(X, Y, Z) = max(0, γ − E(X) · E(Y ) + E(X) · E(Z)),  (8)  where E(X), E(Y ), and E(Z) are the embeddings of X, Y ,  and Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' γ is the minimum margin between the  similarity of the correct caption and the unrelated caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The loss function enables the sentence X to be closer to the  corresponding image Y than the unrelated image Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' During  the prediction time, the relationship between the text and  images is calculated using the cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  For general use, it is reasonable that some sentences,  such as social constructs or metaphorical usage, are not  paired with images after retrieval and have a low similarity  score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In these cases, visual information might not be help-  ful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' To measure how similar the retrieved images should be,  we set a threshold δ to choose the top-ranked images for  each sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5  EXPERIMENTS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Task Settings  Our evaluation is performed on the widely-used natural  language generation and understanding tasks involving 14  NLP benchmark datasets that involve machine translation,  natural language inference (NLI), semantic similarity, and  text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Part of the NLU tasks is available from  the GLUE benchmark [57], which is a collection of nine NLU  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Neural Machine Translation  Five widely-used translation tasks are used for model eval-  uation, including WMT’16 English-to-Romanian (En-Ro),  WMT’14 English-to-German (En-De), WMT’14 English-to-  French (En-Fr), and Multi30K dataset for WMT’16 and  WMT’17, which are standard corpora for NMT and MMT  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (i) For the En-Ro task, we experiment with the officially  provided parallel corpus: Europarl v7 and SETIMES2 from  WMT’16 with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6M sentence pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We use newsdev2016 as  the validation set and newstest2016 as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (ii) The En-De task has 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='43M bilingual sentence pairs  of the WMT14 dataset used as training data, including  Common Crawl, News Commentary, and Europarl v7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The  newstest2013 and newstest2014 datasets are used as the vali-  dation set and test set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (iii) The En-Fr task has 36M bilingual sentence pairs  from the WMT14 dataset used as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Newstest12  and newstest13 are combined for validation and newstest14  is used as the test set, following the setting of [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (iv) Multi30K dataset contains 29K English→{German,  French} parallel sentence pairs with visual annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The 1,014 English→{German, French} sentence pairs with  visual annotations serve as the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For WMT’16  and WMT’17 tasks, we have two test sets, test2016 and  test2017, with 1,000 pairs for each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Natural Language Understanding  The NLU task involves natural language inference, semantic  similarity, and classification subtasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Natural Language Inference involves reading a pair of sen-  tences and assessing the relationship between their mean-  ings, such as entailment, neutral, and contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We  evaluate the proposed method on four diverse datasets:  SNLI [58], MNLI [59], QNLI [60], and RTE [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Semantic Similarity aims to predict whether two sentences  are semantically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Three datasets are used: Mi-  crosoft Paraphrase Corpus (MRPC) [62], Quora Question  Pairs (QQP) dataset [63], and Semantic Textual Similarity  benchmark (STS-B) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Classification CoLA [65] is used to predict whether an  English sentence is linguistically acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' SST-2 [66] pro-  vides a dataset for sentiment classification that needs to  determine whether the sentiment of a sentence extracted  from movie reviews is positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Retrieval Setup  This part describes the implementation of the image re-  trieval by the topic-image lookup table (TILT) and cross-  modal retrieval model (CMRM):  TILT: We segment the sentences using the same BPE  vocabulary as that for each source language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We select top-  8 (w = 8) high TF-IDF words, and the default number of  images m is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7 The detailed case study is shown in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Image features are extracted from the averaged  pooled features of a pre-trained ResNet50 CNN [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The  dimension of the feature maps is V ∈ R2048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  CMRM: The cross-modal retrieval model is trained on  the MS-COCO dataset [67], which contains 123,287 images  with five English captions per image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' It is split into 82,783  training images, 5,000 validation images, and 5,000 test  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We use the Karpathy split [40] that forms 113,287  training, 5,000 validation and 5,000 test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The model  is implemented following the same settings as Engilberge  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' [46], and produces state-of-the-art results (94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0% R@10)  for cross-modal retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' To ensure that each task can enjoy  enough images, we set the similarity threshold δ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4 and  rank the paired images according to the similarity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The  maximum number of retrieved images m for each sentence  is set to eight according to our preliminary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Multi30K and COCO datasets are used as the candidate  seed image retrieval corpus for our downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  Model Implementation  Since our task involves text generation and understanding,  we have two kinds of baselines, the NLG model for transla-  tion and the NLU model for the other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  NLG Model  Our baseline for NLG is encoder-decoder Transformer [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  We use six layers for the encoder and the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The  number of dimensions of all input and output layers is set to  512 and 1024 for base and big models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For MMT experiments  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In some cases when there is no paired image retrieved, we use  the first-row all-zero vectors of the image lookup table as the “blank  features”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  7  TABLE 1  Results for the NMT tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' “++/+” after the BLEU score indicates that the proposed method (base: 5-8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' large:9-12) was significantly better than the  corresponding baseline Transformer (base or big) at significance level p <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  #  Model  En→Ro  En→De  En→Fr  BLEU  #Param  BLEU  #Param  BLEU  #Param  Text-only Transformer  1  Transformer-Base  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='66  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='54M  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='83M  2  Transformer-Big  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='85  207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='02M  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='78++  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='43M  on the Multi30K dataset, we also use the tiny setting where  the dimension of the input and output layer is 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='  The other settings were the same as our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The byte pair encoding algorithm is adopted to segment  sentences into subword sequences, with the vocabulary size  set to 40,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In each training batch, a set of sentence pairs  contains approximately 4096×4 source tokens and 4096×4  target tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' During training, the value of label smoothing  is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1, and the attention dropout and residual dropout  rates are p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We used Adam optimizer [74] to tune the  parameters of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The learning rate is varied under  a warm-up strategy with 8,000 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For evaluation, we  validate the model with an interval of 1,000 batches on the  validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For the Multi30K dataset, we train the model  up to 10,000 steps, and the training will be early-stopped  if the validation set BLEU score does not improve for ten  epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For the En-De, En-Ro, and En-Fr tasks, following  the training of 200,000 batches, the model with the highest  BLEU score of the validation set is selected to evaluate  the test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' During the decoding, the beam size is set to  five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Multi-bleu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='perl is used to compute case-sensitive 4-  gram BLEU scores for all test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8 We follow the model  configurations of [32] to train big models for WMT En-Ro,  En-De, and En-Fr translation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The experiments of NLG  are conducted with fairseq [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  For the statistical tests, we perform the paired bootstrap  resampling test [76] to measure the reliability of the con-  clusion that our system is better than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Our im-  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='com/moses-smt/mosesdecoder/tree/  RELEASE-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0/scripts/generic/multi-bleu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='perl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='com/pytorch/fairseq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  8  TABLE 3  Comparison with public methods on the Multi30K MMT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The results of existing methods are from [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' “++/+” after the BLEU score  indicates that the proposed method was significantly better than the corresponding baseline Transformer (tiny) at significance level p <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  #  Model  En-De  En-Fr  Test2016  Test2017  #Param  Test2016  Test2017  #Param  Text-only Transformer  1  Transformer-Tiny  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='38  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='86  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6M  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='00  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='42  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6M  Existing MMT systems  2  GMNMT [69]  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0M  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9 3  DCCN [70]  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1M  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9M  Our MMT systems  4  UVR-TILTTiny  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='27++  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='62++  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9M  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='60+  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='83++  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9M  5  UVR-CMRMTiny  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='94+  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='11+  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9M  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='50+  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='64++  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9M  TABLE 4  Test results on the GLUE benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The best results are marked in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  #  Model  Classification  Semantic Similarity  Language Inference  Average  CoLA  SST-2  MRPC  STS-B  QQP  MNLI  QNLI  RTE  SNLI  Public Systems  1  BERT [5]  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  2  MT-DNN [71]  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  3  BERT + Voken-cls [72] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6 4  UniT [73] 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0 Our Systems  5  Baseline (BERTWWM)  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='9  6  UVR-TILTMulti30K  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='4  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='1  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='8  7  UVR-TILTCOCO  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='7  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  8  UVR-CMRMMulti30K  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  9  UVR-CMRMCOCO  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  plementation is based on the public toolkit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='10 Two thousand  bootstrap samples are used for each significance test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  NLU Model  For the NLU tasks, the baseline is encoder-only BERT [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='11  We use the whole-word-mask (WWM) version of the pre-  trained weights due to its more stable, reproducible, and  slightly better performance than the original large version  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The initial learning rate is set in the range {2e-5, 3e-5}  with a warm-up rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1 and L2 weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The batch size is selected from {16, 24, 32}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The maximum  number of epochs is set in the range [2, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Texts are  tokenized using SentencePiece,12 with a maximum length  of 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  Main Results  Tables 1-4 show the results for the 14 NMT, MMT, and NLU  tasks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' According to the results, we have the  following observations:  (i) According to the machine translation results in Tables  1-2, the proposed UVR methods significantly outperform  the baselines according to the statistical test, demonstrating  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='com/neubig/util-scripts/blob/master/  paired-bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='py  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='com/huggingface/transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='com/google/sentencepiece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  the effectiveness of modeling visual information for text-  only NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In particular, the superiority is observed in  the translation tasks of three language pairs with different  training data scales, verifying that the proposed approach is  a universal method for improving translation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (ii) Our method introduces only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5M and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0M param-  eters for the base and big Transformers, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The  number is less than 3% of the baseline parameters as we use  the fixed image embeddings from the pre-trained ResNet  feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Besides, the training time is basically the  same as the baseline model (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (iii) Results in Tables 2-3 show that our model can  generally outperform the Transformer baseline in multi-  modal settings that could benefit from the gold sentence-  image annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Compared with the results in text-only  NMT, we find that the image enhancement sometimes gives  marginal contribution, which is consistent with the findings  in previous work [77, 78, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The most plausible reason  might be that the sentences in Multi30K are quite simple,  short, and repetitive, so that the source text itself is sufficient  to perform the translation [10, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We also see that the big  models sometimes show inferior results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The possible reason  is that the dataset is too small to effectively train such big  models, which easily suffer from over-fitting issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The  hypothesis is also supported by our superior results with  the tiny model setting in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The observation verifies  our assumption of the current bottleneck of MMT due to  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  Percentage  D-value  Coverage  CoLA  SST-2  MRPC  STS-B  QQP  MNLI  QNLI  RTE  SNLI  0  5  10  Accuracy  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Accuracy difference between our method and baseline compared  with the coverage percentage of tokens that can be paired with images  in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  TABLE 5  Validation results on GLUE datasets in different sizes: small datasets  with less than 10k examples (RTE and STS-B), and a large dataset  with more than 10k examples (QNLI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The MT-DNN results are  reproduced using the released weights [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Model  RTE  STS-B  QNLI  MT-DNNbase  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='83  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='14±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='40  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='12  w/ UVR-TILT  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='59±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='63  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='07  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='19  MT-DNNlarge  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='70±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='65  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='17  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='27  w/ UVR-TILT  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='19±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='37  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='32±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='12  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='12  the limitation of Multi30K and shows the necessity of our  new methodology of transferring multimodality into more  standard and mature text-only NMT tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (iv) Table 4 shows our method is generally helpful for a  wide range of NLU tasks in the GLUE benchmark, which  verifies the effectiveness of modeling visual information  for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We are interested in whether  public methods, such as MT-DNN, can be further enhanced  by our method, we apply our UVR-TILT method to the MT-  DNN model based on the same implementation in BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  According to the results in Table 5, both the base and  large models are enhanced, and we observe consistent gains  in different datasets, especially the small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For the  results in Tables 4-5, we notice a few inferior or marginally  better performances in the CoLA, MNLI, and QNLI tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  We calculate the accuracy difference (D-value) between our  method and baseline compared with the coverage percent-  age of tokens that can be paired with images in each dataset  using UVR-TILTMulti30K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' From Figure 3, we see that those  datasets are commonly paired with a relatively small num-  ber of images so that the visual signals can enhance only a  small proportion of token representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In addition, some  datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', ColA, mainly require linguistic knowledge for  solving the tasks, so introducing visual modality might not  benefit such tasks, which corresponds to the common short-  coming of vision injection for language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For MNLI and  QNLI, the possible reason for the marginal improvements  would be that both of the datasets are quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Still, the  task is relatively simple, so the model might well solve the  tasks directly via the text representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In this scenario,  the visual features might only provide the regularization  effect to improve the model robustness [80, 81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  0  5  10  15  20  epoch  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  value  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Illustration of the gate values λ with the UVR-TILT method on  Multi30K En-De Test2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (v) For the two retrieval methods, we observe that UVR-  CMRM is slightly better than UVR-TILT in general, and  the results of the two methods are pretty close for the  MMT task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We find a performance tradeoff between them:  there might be more accurate similarity calculation after  cross-modal pre-training than the direct topic extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  However, controlling the proper similarity threshold would  be a heuristic for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In contrast, the advantage of  UVR-TILT is the simple preprocessing, which only requires  TF-IDF-based topic extraction and matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (vi) We also compare the results using different seed cor-  pora, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', Multi30K and COCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For the MMT task in Table  2, using Multi30k is basically similar to or slightly better  than COCO in general, as the task could enjoy the images  in the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For the out-of-domain evaluations in  Table 1 and Table 4, we obverse that using COCO generally  achieves better results because the size of COCO images  is three times that of Multi30K, which could provide more  diverse image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6  ANALYSIS  This section presents our exploration on the role of visual  contexts, which involves two aspects, when the visual con-  text helps and how the visual context helps language repre-  sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In the following analysis part, we use Multi30K  for UVR-TILT and COCO for UVR-CMRM by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  Dynamics of the visual information  To explore the role of visual context in the training process,  we illustrate the gate values λ (defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 5) in Figure  4 where a larger value indicates more dependence on the  visual context in the fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We observe that the  model relies on the visual context in the early stages, and  the role of visual information changes dynamically across  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' When the training starts, the model accommodates  the visual information to a large extent (λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6), indicating  that the model tends to trust the visual context, which could  provide useful information in the early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' With more  knowledge captured as the training continues, the contribu-  tions of the visual contexts appeal to decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In the following  parts, we will further discuss the specific effectiveness of  visual representations in language modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  10  A girl in a purple tutu dances in the yard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  A little girl is walking over a path of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  A girl jumping rope on a sidewalk near a parking garage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  A young girl washes an automobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Examples of sentences that share the same retrieved images, in which the common topic is about “girl”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Sentences with similar topics tend to  be paired with similar or even the same images, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This means that images may provide topic information, which benefits the modeling  of similar sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  Pairwise relationship across modalities  The benefits of the universal representation method could  be two folds: (i) the content connection of the sentences and  images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' (ii) the topic-aware co-occurrence of similar images  and sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' According to Distributional Hypothesis [83],  which states that words that occur in similar contexts tend to  have similar meanings, we are inspired to extend the concept  in the multimodal world, the sentences with similar meanings  would be likely to pair with similar even the same images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' There-  fore, the consistent images (with a related topic) could play  the role of topic or type hints for similar sentence modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  After using our image retrieval method, sentences with  similar topics tend to be paired with similar or even the  same images, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Figure 5 shows examples in  which the common topic is about “girl”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This means that  images may provide topic information, which benefits the  modeling of similar sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Thus, aside from the image  embeddings’ inner meaning (vectors), there is a mapping  relationship between the sentence and images after the  retrieval process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  For the image embeddings, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2,  we adopt the embedding lookup to fetch the embedding  features for each image, which is very similar to the way of  using word embedding by treating each image as a “word”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The weights of the embedding features are derived from  the average pooled output of ResNet, where each image is  represented as a 2400-d vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For all the 29,000 images  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', using Multi30K), we have an embedding layer with  size (29000, 2400).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The “content” of the image can be seen  as embedding initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The pre-initialized embedding  weights might yield slight improvement gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' However,  the neural network can also be effectively trained with  random initialization [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In contrast, whether to use  the embedded feature is more critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In other words, the  mapping relationship of the sentences and images in image  embedding would be essential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', similar sentences (with  the same topic words) tend to map the same or similar  image after the word-image lookup process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  To verify the hypotheses, we conduct the following  ablations: we replace the ResNet50 feature extractor in our  UVR-TILT model with (1) ResNet101 and (2) ResNet152;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  additionally, we compare the results with the following  operations: (3) Shuffle: shuffle the image features but retain  TABLE 6  Ablation for the image embedding operation on En-Ro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The scores are  reported by means and standard deviations for three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Method  BLEU Score  Baseline  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='10  UVR-TILT  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='08  w/ (1) Res101  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='06  w/ (2) Res152  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='07  w/ (3) Shuffle  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='40±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='14  w/ (4) Random Init  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='08±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='17  w/ (5) Random Mapping  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='14  the lookup table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' (4) Random Init: randomly initialize the  image embedding but keep the lookup table;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' (5) Random  Mapping: randomly retrieve unrelated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Table 6 shows the ablation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The BLEU scores  of models 1-4 are close to the proposed UVR method,  and those ablated methods still outperform the baseline,  indicating that using image features generally yields better  performance than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In addition, either replacing  the trained image features (model 4) or disturbing the  mapping information (model 5) leads to a performance  drop (↓0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='64/↓1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='67, respectively), which indicates that both  the image features and mapping information are contribut-  ing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Compared with image features, the mapping  information has a larger impact, which verifies our prior  hypothesis that the consistent images with a related topic  could play the role of topic or type hints for similar sentence  modeling in the whole training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' With the mapping  information, the same images will be assigned to the same  context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' During training, the image features will be learned  just like word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Therefore, it does not mean that  the image features are not very helpful, but the mapping  information in the lookup table reduces the dependence on  the trained image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  From the view of an individual image, image content  (embedding) has an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' If we maintain the pairwise  relationship between the sentence and image, the result is  still higher than the baseline, even with shuffled or random  image embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This indicates that the pairwise rela-  tionship is a vital contributor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' From the macro perspective  of sentence-image co-occurrence, image information plays  the role of topic information, where similar sentences tend  11  a) baseline method without visual features  b) our method with visual features  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Visualization of (a) attention weights of the input tokens with regards to the probed token “lock” across different layers (Y-axis) using the  BERT baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' (b) image-to-word attention from our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The illustration shows that the images provide fine-grained grounding information about  the relationship between concepts and events, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', “lock”, “door”, “fancy”, “hotel”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  TABLE 7  Results of MMT with incomplete source texts by removing visually  grounded tokens in the Multi30K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The scores are reported by  means and standard deviations for three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Model  En-De  En-Fr  Test2016  Test2017  Test2016  Test2017  Baseline  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='21  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='24  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='61±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='16  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='01±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='15  UVR-TILT  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='80±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='18  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='19  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='60±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='17  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='20  to pair with similar images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The observation corresponds  to the distributional hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This explains the potential  effects of the pairwise relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  This finding may potentially facilitate future research  because most existing studies focus on the content of the  individual image itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We highlight the pairwise relation-  ship across modalities as a different research line to bridge  the gap between language and image modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  Handling incomplete source texts  Content words are naturally related to specific content, such  as car, room, play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We collect a list of tokens that have  more than ten occurrences in the Multi30K training set  after removing all stop words following [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We remove  those tokens in the source sentence, which occupy 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='87%  of tokens in the token dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Table 7 shows the results  of the baseline model and our model with the incomplete  source texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We observe that our model achieves noticeable  gains over the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The results verify that the visual  representation can reduce the gap of the missing informa-  tion from the content words in the source texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  Knowledge grounding with the visual context  To gain an insight into the process of multimodal integra-  tion by our model, we analyze the attention distributions  (α in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3) at the multimodal integration layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Figure 6  shows the attention distributions of (a) the baseline and  (b) our model 13 for an example randomly selected from  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Our model is the UVR-TILT trained on the CoLA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  TABLE 8  Results (BLEU score) of the multimodal disambiguation experiments  on WAT’19 English to Hindi dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The scores are reported by means  and standard deviations for three random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Model  Validation  Test  Challenge  Baseline  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='04±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='27  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='24  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='14  UVR-CMRM  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='49±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='06  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='81±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='12  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='08  our GLUE validation sets, “You should always lock your door,  no matter how fancy the hotel might be.” For comparison,  the baseline is implemented following Abnar and Zuidema  [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Concretely, we collect the attention weights of all the  input tokens with regards to a targeted token “lock” across  different layers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', {2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' , 24}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' As the baseline uses the  representation of the last layer for prediction, we focus on  the attention distributions of the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Compared with the baseline that only captures partial  relations in the last layer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', with a lack of relationship  among {“lock”, “door”, “hotel”}, our model provides more  fine-grained connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In detail, two generic patterns are  observed in these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (i) the images appear to match the concepts and actions  with the texts, in other words, the images tend to provide  fine-grained grounding information about the relationship  between concepts and events, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=', {”lock”, ”door”, ”fancy”,  ”hotel”}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  (ii) our model can resist irrelevant information from  noisy images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' For example, the second and third images  yield low attention scores for the texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  Disambiguation  A natural intuition of using visual clues for text represen-  tation is the advantage of alleviating the ambiguation of  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' To evaluate the model performance for disam-  biguation, we use a dataset from the HVG [88], which serves  as a part of the WAT’19 Multimodal Translation Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='14 The  dataset consists of a total of 31525 randomly selected images  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' http://lotus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='kuee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='jp/WAT/WAT2019/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='html  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  always  lock  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  Joop  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='4  how  fancy  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  might  be  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0Toilet  f1忆  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='175  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='150  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='125  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='100  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='075  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='050  9  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='025  2  2  matt12  a) baseline method without visual features  b) our method with visual features  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Visualization of (a) attention weights of the input tokens with regards to the ambiguous token “apple” across different layers (Y-axis) using the  BERT baseline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' b) image-to-word attention from our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The illustration shows that the images bridge the connection between “apple”, “watch”,  “time”, helping disambiguate the meaning of “apple”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  TABLE 9  Selected eight probing tasks [87] to study what syntactic and semantic properties are captured by the encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Probing Tasks  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Content  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Syntactic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='TrDep  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Checking whether an encoder infers the hierarchical structure of sentence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='ToCo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Sentences should be classified in terms of the sequence of top constituents immediately below the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='sentence node  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='BShif  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Testing whether two consecutive tokens within the sentence have been inverted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Semantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Tense  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Asking for the tense of the main clause verb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='SubN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Focusing on the number of the main clause’s subject  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='ObjN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Testing for the number of the direct object of the main clause  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='SoMo  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Some sentences are modified by replacing a random noun or verb with another one and the classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='should tell whether a sentence has been modified  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='CoIn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Containing sentences made of two coordinate clauses  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='TABLE 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='Classification accuracy on eight probing tasks of evaluating linguistics embedded in the encoder outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Model  Syntactic  Semantic  TrDep  ToCo  BShif  Tense  SubN  ObjN  SoMo  CoIn  Baseline  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='34  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='33  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='34  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='66  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='02  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='57  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='42  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='51  UVR-CMRM  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='53  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='64  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='72  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='97  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='79  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='66  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='44  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='23  from Visual Genome [89] and a parallel image caption  corpus in English-Hindi for selected image segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The  training part consists of 29K English and Hindi short cap-  tions of rectangular areas in photos of various scenes, and  it is complemented by three evaluation subsets: validation,  test, and challenge test set (Challenge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The challenge test set  is created by searching for (particularly) ambiguous English  words based on the embedding similarity and manually  selecting those where the image helps to resolve the am-  biguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We do not use the images but follow the same  settings as the experiments on Multi30K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' As the results  shown in Table 8, we observe that our UVR model works  effectively on the challenge disambiguation set, indicating  that the visual information induced by retrieved images  allows disambiguation of translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Figure 7 shows a heatmap of the attention visualization  on an ambiguous sentence, “apple watch keeps telling me  the time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In Figure 7(a), the baseline model fails to capture  the relationship between “apple” with “time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In Figure  7(b), the retrieved images bridge the connection among  “apple”, “watch” and “time”, which helps disambiguate the  meaning of “apple”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  Linguistic Analysis  We are interested in what knowledge is learned in the  universal representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In this section, we select eight  widely-used language probing tasks [87] (see Table 9) to  study what kind of syntactic and semantic properties are  captured by the encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Specifically, we use the encoders  of the baseline BERT-based SNLI model,15 and our UVR-  CMRM visual representation model to generate the sentence  representations of input, which are used to carry out the  above eight probing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The results are as shown in Table  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We select the SNLI model because NLI models show good  generalization capacity for language representation [90, 91], which is  supposed to be a strong test-bed for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  apple  watch  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  telling  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0AppleWatchEdition  WATCHEDITIONWATCHWATCH7AppStore&iTunes忆  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='25  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='20  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='15  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='10  8  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='05  4  2  me  the  telling13  0  1  3  5  7  9  15 20 30  33  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  Number of images per sentence  BLEU  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Influence of the number of images on the BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  TABLE 11  Experiments on different source languages using the Multi30K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The baseline is Transformer-Tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Model  De-En  Fr-En  Test2016  Test2017  Test2016  Test2017  Baseline  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='88  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='57  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='60  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='45  UVR-TILT  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='10  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='07  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='92  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='10  Concerning semantic properties, our model gains the  most significant improvement on the SubN and ObjN tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  The result indicates that visual information helps NMT to  identify and represent the subject and object information,  which is consistent with our hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  Effectiveness across languages  Table 11 shows the experiment results on different source  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We observe that our method is applicable when  the source texts are in other languages such as German  and French.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Our proposed methods are supposed to be  independent of languages because the calculation for image  retrieval only relies on the light lookup table, which can be  extracted from an off-the-shelf seed corpus that is available  for many languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='8  Joint Training and Fine-tuning  Since the multimodal and text-only machine translation  tasks can benefit from the visual modality after retrieving  images from the seed corpus, it is possible to bridge both  tasks to train an even more powerful model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The connec-  tions between the texts and images inside the Multi30K  datasets could be strong indications to bridge the gap be-  tween text and image modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Therefore, we train a unified model based on UVR-  CMRM by using the joint En-De datasets of Multi30K and  WMT’14 (Joint Model), and respectively train the Multi30K  (Fine-tuned Multi30K) and WMT’14 (Fine-tuned WMT) mod-  els by initializing the trainable model parameters using the  joint model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' According to the results shown in Table 12, we  summarize the following observations:  (i) Two-stage training (joint training and fine-tuning) can  boost the performance on the two concerned datasets, which  indicates that the highly relevant Multi30K dataset can play  the role of the seed data for training a text-only NMT model  using our topic-image lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  0  50  100  Percentage  BLEU  Data Size  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  34  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  35  Percentage  BLEU  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' BLEU score for different similarity thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  TABLE 12  Results of joint training and fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Model  Multi30K Task  WMT Task  Joint training baseline  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='28  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='68  + Fine-tuned Multi30K 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='96  + Fine-tuned WMT  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='13 (ii) The major gain is achieved by fine-tuning the smaller  dataset, showing that the large-scale text-only dataset with  our lookup table can also provide valuable complementary  information for training a multimodal model on a much  smaller dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The result further verifies the effectiveness  of our method in low-resource settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  Parameter Sensitivity Analysis  In this section, we analyze our model sensitivity against  parameter settings, including similarity threshold, number  of images, and gating weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Influence of the similarity threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We investigate the  influence of the similarity threshold δ that is set to filter the  top-ranked images for each sentence in UVR-CMRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Figure  9 shows the performance for thresholds in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='6] on the En-Ro test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We observe that setting  the threshold around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='4 can yield a good balance of data  size and BLEU score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' It is reasonable that the best thresholds  vary for different datasets because of the domain divergence  of the image corpus for pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Influence of the number of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' To evaluate the  influence of the number of paired images m for UVR-TILT,  we constrain m in {0, 1, 3, 5, 7, 9, 15, 20, 30} for experiments  on the En-Ro test set, as shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' When m = 0,  the model is the baseline NMT model, whose BLEU score  is lower than all the models with images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' As the number  of images increases, the BLEU score also increases at the  beginning (from 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='66 to 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='78) and then slightly decreases  when m exceeds 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The reason might be that too many  images for a sentence would have a higher chance of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Therefore, we set m = 5 in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Influence of gating weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In our model, the weight  λ of the gated aggregation method is learned automatically  to measure the importance of the visual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We  compare by manually setting the weight λ to scalar values  in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9} for experiments of UVR-TILT on the  En-Ro test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Figure 10 shows that all models with manual  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='9  33  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='5  Weight λ  BLEU  Manual weight  Transformer-Base  Ours  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Quantitative study of the gating weight λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  λ outperform the baseline Transformer-base, indicating the  effectiveness of image information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' In contrast, they are in-  ferior to the performance of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' This means that the  degree of dependency for image information varies for each  source sentence, indicating the necessity of automatically  learning the gating weights of image representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='10  Computation Efficiency  There are mainly two extra computation costs using our  method, including (i) obtaining image data for sentences  and (ii) learning image representations, which are negligible  compared with training an NMT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The time of retriev-  ing image data for MT sentences for the En-Ro dataset is  less than 1 minute using GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The lookup table is formed  as the mapping of the token (only topic words) index to  the image id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Then, the retrieval method is applied as the  tensor indexing from the sentence token indices (only topic  words) to image ids, which is the same as the procedure of  word embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The retrieved image ids are then sorted  by frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Learning image representations takes about  2 minutes for all the 29,000 images in Multi30K using 6G  GPU memory for feature extraction and eight CPU threads  for transforming images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The extracted features are formed  as the “image embedding layer” in the size of (29000, 2400)  for quick access in the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  7  CONCLUSIONS  This work investigates a flexible framework to incorporate  visual information into sentence modeling by image re-  trieval from a light lookup table and learned cross-modal  embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Extensive empirical experiments on 14  benchmark datasets verify the effectiveness of the proposed  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' A series of case studies are conducted to evaluate  visual benefits and influence factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Our method is general  and can be easily implemented in existing deep-learning  NLP systems for different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Through the proposed  retrieval methods, we can provide a group of images that  disclose a diversity of implicit topics that might be entailed  in sentences, yielding better context grounding with fine-  grained information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' We show that our method enriches  the representation of content words, provide fine-grained  grounding information about the relationship between con-  cepts and events, and potentially enhances the accuracy  of disambiguation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Besides incorporating images to build  the pairwise relationship across modalities, it is potential  to incorporate various extra knowledge as alignment topic  information in the future, such as audio, not only images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  Part of this study has been published as “Neural Machine  Translation with Universal Visual Representation” [37] in  the Eighth International Conference on Learning Represen-  tations (ICLR 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' The extension includes three sides: (i)  general tasks: this work studies the universal visual rep-  resentation for language representation in a broader view  of the natural language processing scenario, with experi-  ments on 14 representative NLP tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' (ii) new method: this  work investigates new methods of semantic sentence-image  matching from a shared cross-modal space, to give more  accurately paired images as topic information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' (iii) in-depth  analysis to interpret the benefits from the visual modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='  Zhuosheng Zhang received his Bachelor’s de-  gree in internet of things from Wuhan University  in 2016, his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' degree in computer science  from Shanghai Jiao Tong University in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' degree in computer  science with the Center for Brain-like Computing  and Machine Intelligence of Shanghai Jiao Tong  University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' He was an internship research fellow  at NICT from 2019-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' His research interests  include natural language processing, machine  reading comprehension, dialogue systems, and  machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Kehai Chen is an Assistant Professor at Harbin  Institute of Technology (Shenzhen) since 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Before that, he was a researcher at Japan Na-  tional Institute of Information and Communica-  tions Technology (NICT) from 2018 to 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' degree in computer science  from Harbin Institute of Technology in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='  Rui Wang is an associate professor at Shanghai  Jiao Tong University since 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Before that, he  was a researcher (tenured in 2020) at Japan  National Institute of Information and Communi-  cations Technology (NICT) from 2016 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' degree from Harbin Institute  of Technology in 2009, his M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' degree from the  Chinese Academy of Sciences in 2012, and his  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+page_content='  He was a joint Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' at Centre Nationnal de  la Recherche Scientifique, France in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' His research interests are  machine translation and natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Masao Utiyama is a research manager of the  National Institute of Information and Communi-  cations Technology, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' He completed his  doctoral dissertation at the University of Tsukuba  in 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' His main research field is machine  translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Eiichiro Sumita received the Bachelor and Mas-  ter degree in computer science from The Univer-  sity of Electro-Communications, Japan in 1980  and 1982 and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='D degree in Engineering  from Kyoto University, Japan in 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' He is cur-  rently Director of Multilingual Translation Labora-  tory of National Institute of Information and Com-  munication Technology from 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' He worked  at Advanced Telecomunications Research Insti-  tute International from 1992 to 2009 and IBM  Research-Tokyo from 1980 to 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' His re-  search interests include machine translation and e-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Zuchao Li received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' degree from Wuhan  University, Wuhan, China, in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' Since 2017,  he has been a Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' student with the Center for  Brain-like Computing and Machine Intelligence  of Shanghai Jiao Tong University, Shanghai,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' His research focuses on natural language  processing, especially syntactic and semantic  parsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  Hai Zhao received the BEng degree in sensor  and instrument engineering, and the MPhil de-  gree in control theory and engineering from Yan-  shan University in 1999 and 2000, respectively,  and the PhD degree in computer science from  Shanghai Jiao Tong University, China in 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content='  He is currently a full professor at department  of computer science and engineering, Shanghai  Jiao Tong University after he joined the university  in 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' He was a research fellow at the City  University of Hong Kong from 2006 to 2009, a  visiting scholar in Microsoft Research Asia in 2011, a visiting expert in  NICT, Japan in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' He is an ACM professional member, and served as  area co-chair in ACL 2017 on Tagging, Chunking, Syntax and Parsing,  (senior) area chairs in ACL 2018, 2019 on Phonology, Morphology and  Word Segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
+page_content=' His research interests include natural language  processing and related machine learning, data mining and artificial  intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/E9E1T4oBgHgl3EQfqgWQ/content/2301.03344v1.pdf'}
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+arXiv:2301.01282v1  [cs.CR]  31 Dec 2022
+RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA
+KRISHNAN SHANKAR
+Introduction
+The RSA algorithm has been around for nearly five decades ([RSA78]) and remains one
+of the most studied public key cryptosystems. Many attempts have been made to break
+it or improve it and questions remain about the equivalence of the strength of its security
+to well known hard problems in computational number theory. A basic question which has
+received much attention (cf. [AM09], [BV98]1) is: Is breaking RSA equivalent to factoring?
+In this note we propose a modified version which we call RSA+ which is at least as secure as
+RSA and show that breaking RSA+ is probably computationally equivalent to factoring n,
+the public modulus. The motivation came from wanting to obscure the encryption exponent
+in RSA.
+1. The RSA+ Algorithm
+RSA: Bob wishes to send a message m to Alice whose RSA public key is (n, e) and private
+key is (p, q, d) where n = pq and de ≡ 1 (mod ϕ(n)).
+In the usual implementation of
+RSA Bob computes c ≡ me (mod n) and sends it to Alice. Decryption is straightforward:
+cd ≡ (md)e ≡ mde ≡ m (mod n) since de ≡ 1 (mod ϕ(n)).
+RSA+: We start with the same setup as above namely that Bob has a message m to
+transmit to Alice whose RSA public key is (n, e).
+Encryption:
+1. Bob finds a random number x (mod n) and computes y ≡ x2 (mod n).
+2. Bob computes c ≡ mx (mod n) and r ≡ ye (mod n).
+3. Bob transmits the pair (c, r) to Alice.
+Decryption:
+1. Alice computes y ≡ (ye)d (mod n). This step is similar to decrypting in RSA.
+2. Alice then writes down the equation y ≡ x2 (mod n). She uses her knowledge of
+the factorization of n = pq to compute all four square roots of y (see Section 4).
+3. For each square root, say {x1, x2, x3, x4}, Alice sequentially computes the inverse
+ui ≡ x−1
+i
+(mod ϕ(n)) and evaluates cui (mod n) until she sees an intelligible mes-
+sage m (but see 1.2 below).
+Theorem A. Breaking RSA+ is probably computationally equivalent to factoring n = pq.
+1In [BV98] there seems to be an issue in the proof of Lemma 3.2, where it is assumed that a cyclotomic
+polynomial Φd(x) is irreducible over Fp, which is not always true.
+
+2
+KRISHNAN SHANKAR
+1.1.
+Similar to RSA care must be taken in choosing the random integer x in Step 1 of
+encryption. If Bob chooses x < √n, then y = x2 as integers and it is easy to find the square
+root if one knows y. In this case the system is as secure as RSA since y is RSA-encrypted.
+It is also important that gcd(x, ϕ(n)) = 1 since otherwise Alice cannot decrypt the message
+even if she can find x (Euler’s theorem). So, Bob could choose x to be a prime of at least
+150 digits assuming n is around 300 digits.
+1.2.
+In Step 3 of decryption as stated Alice must compute all four square roots and then
+try to uncover an intelligible message by sequentially decrypting each mxi (mod n) which is
+onerous in practice. We describe a way around this problem as long as both primes dividing
+n are congruent to 3 mod 4 (this is the workaround suggested by Blum and Williams).
+Consider the setup with y ≡ x2 (mod n), where Bob chooses x, y. Then y has four square
+roots mod n which come from combining the two square roots each mod p and mod q using
+the Chinese Remainder Theorem (CRT). Suppose both primes p, q in the factorization of n
+are chosen (by Alice) to be congruent to 3 mod 4. Now Bob chooses x, y such that x itself
+is a square mod n. Then solving the equation X2 ≡ y (mod n) has solutions (say) X ≡ ±a
+(mod p) and X ≡ ±b (mod q). These are combined using CRT to yield X ≡ xi (mod n),
+for i = 1, 2, 3, 4 and let us suppose x = x1. Note that p, q ≡ 3 (mod 4) so
+�
+−1
+p
+�
+= −1 and
+�
+−1
+q
+�
+= −1 i.e., −1 is not a quadratic residue mod p nor mod q. This implies that exactly
+one of the roots ± (mod p) and one of the roots ±b (mod q) is a quadratic residue (mod the
+respective primes) and the other is not. Since Bob picked x to be a square mod n, it follows
+that the congruences that yielded x1 must also be squares i.e., suppose X ≡ a (mod p)
+and X ≡ b (mod q) yields X ≡ x1 (mod n), then it follows that
+�
+a
+p
+�
+= +1,
+�
+b
+q
+�
+= +1 and
+�
+−a
+p
+�
+= −1,
+�
+−b
+q
+�
+= −1. From this it follows that none of x2, x3, x4 is a square mod n. We
+summarize this via the following
+Proposition 1.1. Suppose n = pq, where p and q are congruent to 3 mod 4. Given x, y
+such that y ≡ x2 (mod n), gcd(x, n) = 1. Then y has four square roots mod n exactly one
+of which is a square mod n.
+This suggests a way to pinpoint x without having to search all square roots, namely, Bob
+picks x to be a square mod n and then computes y ≡ x2 (mod n). By the above Proposition
+this will be the only square root of y mod n. It also follows from the discussion above that:
+(i) If one of the primes is congruent to 1 mod 4, then the number of square roots that are
+themselves perfect squares is either 0 or 2; (ii) If both primes are congruent to 1 mod 4,
+then the number of square roots that are themselves perfect squares is 0 or 4.
+1.3.
+Decryption is more involved if at least one of the primes dividing n is congruent to 1
+(mod 8). Nevertheless, the overall algorithm’s security is tied to the difficulty of factoring
+and as such lies in class NP, although it is unknown whether it is in P.
+
+RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA
+3
+2. RSA+ is at least as secure as RSA
+In this section we show that RSA+ is at least as secure as RSA. We will show this via
+the use of a so-called black box i.e., an unknown machine or method or algorithm that
+takes in a specified input and returns an output that is otherwise currently computationally
+intractable.
+Theorem B. RSA+ is at least as secure as RSA.
+Proof. Suppose we have a black box that is able to decrypt RSA+ messages i.e., this black
+box takes as input (n, e, mx (mod n), ye (mod n)) and returns m. Here y ≡ x2 (mod n).
+Given an RSA ciphertext c ≡ me (mod n) one would then input (n, e, c, (e2)e (mod n))
+which should generate the output of m.
+This shows that any system that can decrypt
+RSA+ ciphertexts can also decrypt RSA ciphertexts.
+Conversely, suppose there is a black box that is able to decrypt RSA messages i.e., this
+black box takes as input (n, e, c ≡ me (mod n)) and returns m. Given an RSA+ ciphertext
+(mx (mod n), ye (mod n)) one can only decrypt y. In order to get to m one would need to
+know the exponent x for input into this black box, i.e., one would now need x in order to
+input (n, x, mx (mod n)) into the black box. This means computing a square root of y. It
+is not known whether this black box which can decrypt RSA ciphertexts can also compute
+square roots (but see Theorem C).
+□
+3. Computational Black Boxes
+In this section we explore further computational black boxes which may circumvent known
+methods like factoring. Suppose one is able to decrypt y by ascertaining d. One needs to
+compute the particular square root x used to encrypt m. Is it possible to compute just one
+(pair of) square root(s) without knowledge of the factors? If there exists a computational
+black box that can produce one square root when the input is (y, n), then there are two
+possibilities.
+3.1.
+Suppose the black box spits out a random square root mod n for the equation y ≡ x2
+mod n. In this case one simply inputs the same pair (y, n) repeatedly until we get two
+distinct square roots that do not add up to zero mod n. By Lemma 4.1 this will yield a
+factorization of n.
+3.2.
+Suppose the black box spits out a random square root mod n but always the same
+square root for the equation y ≡ x2 mod n i.e., for a given input (y, n), the output is
+always the same x rather than one of the four distinct square roots at random. In this case
+we start with some known x and square it to obtain y ≡ x2 (mod n). Now input (y, n)
+into the black box; if the output is ±x, then discard and try again with a different x. If
+the output, say x′ is different from ±x (mod n), then by Lemma 4.1 we can factor n by
+computing gcd(x − x′, n).
+
+4
+KRISHNAN SHANKAR
+3.3.
+If the adversary has instead a black box that computes d given the input (n, e),
+then ϕ(n) | (de − 1). Therefore, for any x we have xde−1 ≡ 1 (mod ϕ(n)). Since ϕ(n) =
+(p−1)(q−1) is divisible by 4, we may compute y ≡ x(de−1)/2 (mod n). Then, by construction
+y2 ≡ 1 (mod n) which means n | (y − 1)(y + 1). Then either y ≡ ±1 (mod n) or by Lemma
+2.1 gcd(y ± 1, n) yields a factor of n. Again, we can repeat this for several different x as
+needed to factor n with high probability.
+Whether RSA+ is computationally equivalent to factoring depends on the following
+thought experiment. Suppose we have a black box like the one in the previous Section
+which takes as input (n, e, mx (mod n), ye (mod n)), where y ≡ x2 (mod n) and produces
+as output m. Does this allow us to factor n?
+Theorem C. A black box with input (n, e, mx (mod n), ye (mod n)) and output m can
+probably factor n.
+Proof. Start with a known pair (x, y), where y ≡ x2 (mod n). Then note that:
+yx ≡ (x2)x ≡ x2x ≡ (xx)2
+(mod n)
+xy ≡ xx2 ≡ xx·x ≡ (xx)x
+(mod n)
+If we were to input (n, e, xy (mod n), ye (mod n)) into the black box, then this is the same
+as the input (n, e, (xx)x (mod n), ye (mod n)). The output should therefore be xx (mod n)
+which is a square root of yx (mod n). From 3.1 and 3.2 above, repeating this procedure for
+several different x should yield a factorization of n.
+□
+3.4. Remark. The above proof carries over for RSA as well i.e., if we had a black box for
+RSA, then the same argument above yields a procedure to compute square roots mod n.
+3.5. Remark. The only caveat in Theorem C above is the nature of the black box. Suppose
+the black box were to always return the distinguished square root xx (mod n) for the square
+yx ≡ (xx)2 (mod n), then this will not allow us to factor n.
+4. Computing square roots and factoring
+In the absence of a black box an adversary Eve would need y and then a square root
+of y to attempt to decrypt m. Computing y ≡ (ye)d mod n without (p, q, d) is as hard
+as breaking RSA. Even if Eve can deduce d without factoring n, she would next have to
+solve the equation y ≡ x2 (mod n). See Section 3 for a discussion on black boxes that may
+compute square roots.
+Lemma 4.1. Given n = pq if we can solve the (generic) equation y ≡ x2 (mod n) and find
+all four roots ±a, ±b mod n, then we can factor n.
+Conversely, if one can solve the equation y ≡ x2 (mod p) and y ≡ x2 (mod q), then one
+can combine the roots using the Chinese Remainder Theorem to produce (in general) four
+square roots mod pq.
+
+RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA
+5
+Proof. Since a ̸≡ ±b (mod n) and a2 ≡ b2 mod n implies that n | (a − b)(a + b). Thus,
+computing gcd(a ± b, n) yields a non-trivial factor of n.
+□
+Lemma 4.2. If p ≡ 3 (mod 4) is prime, then one can solve y ≡ x2 (mod p) by computing
+x ≡ y(p+1)/4 (mod p). Then (±x)2 ≡ y (mod p).
+Lemma 4.3. If p = 8k + 5 is prime, i.e., p ≡ 5 (mod 8), then one can solve y ≡ x2
+(mod p).
+Proof. Since y is a square mod p, we have that y(p−1)/2 ≡ 1 (mod p). Since (p − 1)/4 is an
+integer we can take a square root and we obtain that y(p−1)/4 ≡ ±1 (mod p).
+Case 1: If y(p−1)/4 ≡ 1 (mod p) and p = 8k + 5, then x ≡ yk+1 ≡ y(p+3)/8 (mod p) is a
+desired square root. This is because
+x2 ≡ y(p+3)/4 ≡ y(p−1)/4 · y ≡ y
+(mod p)
+Case 2: If y(p−1)/4 ≡ −1 (mod p), then x ≡ 22k+1yk+1 ≡ 2(p−1)/4y(p+3)/8 (mod p) is a
+desired square root. This is because
+x2 ≡ 2(p−1)/2y(p+3)/4 ≡ (−1) · y(p−1)/4 · y ≡ (−1)(−1)y ≡ y
+(mod p)
+Note that 2(p−1)/2 ≡ −1 (mod p) since p ≡ 5 (mod 8).
+□
+Lemma 4.4. If p ≡ 1 (mod 8) is prime, then there exists an algorithm terminating in at
+most r steps to compute y ≡ x2 (mod p), where p − 1 = 2rs.
+Proof. This is the most involved case of the Tonelli–Shanks algorithm, which covers all the
+above Lemmas; see [Sh73], [Ton1891]. See [Tur94] for a nice exposition.
+□
+Now we can prove Theorem A.
+Proof of Theorem A. If an adversary Eve can factor n, then she can certainly decrypt the
+message m: compute ϕ(n) = (p−1)(q −1), find d using the Extended Euclidean Algorithm
+and use this to find y ≡ (ye)d mod n. Now using Lemmas 4.2, 4.3, 4.4, depending on
+whether p, q are congruent to 1 (mod 4) or 3 (mod 4), Eve can find the square roots of y
+mod p and mod q. Then she can combine them to obtain all square roots mod n using the
+Chinese Remainder Theorem. Finally Eve solves for x−1 (mod ϕ(n)) to decrypt m.
+If Eve possesses a black box as described in Theorem C, then Eve can probably factor n
+(with the caveat introduced in Remark 3.5). If Eve has instead a black box or method to
+find square roots mod n, then by Lemma 4.1 she can factor n.
+□
+5. Similar cryptosystems
+A search of the literature by the author yielded similar cryptosystems and these are
+described briefly here. Any other omissions are entirely accidental. To the author’s knowl-
+edge this particular protocol has not been described before. For each protocol below the
+description is brief and only intended to serve as a reference or comparison.
+
+6
+KRISHNAN SHANKAR
+5.1. The Rabin Cryptosystem. This cryptosystem ([Rab79]) is the closest in spirit to
+the method proposed here. Its security is tied to square roots and factoring. Alice finds
+primes p, q both congruent to 3 mod 4 and computes n = pq and exponents e, d with de ≡ 1
+(mod ϕ(n)) same as in RSA.
+Bob encrypts his message m as c ≡ m2 (mod n) and transmits c.
+To decrypt Alice
+computes the square root mod p and mod q by the method outlined in Lemma 4.2. Then
+she combines them using the Chinese Remainder Theorem to obtain all four square roots
+and then examines which of these is intelligible. This latter disambiguation problem was
+addressed by Blum and Williams. The Rabin cryptosystem was shown to be vulnerable
+to a chosen ciphertext attack. This attack can be thwarted by adding redundancies to the
+message.
+5.2. D–RSA (Dependent RSA). This cryptosystem, described in [Po99], was in re-
+sponse to finding a so-called semantically secure cryptosystem as efficient as RSA. Similar
+to RSA Alice has public key (n, e). Then Bob picks a random k ∈ (Zn)∗ and computes
+A ≡ ke (mod n) and B ≡ m · (k + 1)e (mod n). Decryption is via computing k ≡ (ke)d
+(mod n) and then B · (k + 1)−e (mod n). This is also similar in spirit to what we have
+described in that Bob picks a random number during the encryption process. Among the
+theorems in the paper it is shown that a slight modification of this protocol is semantically
+secure against adaptive chosen ciphertext attacks relative to the Decision D–RSA Problem
+in the random oracle model.
+5.3. RSA–CRT. This is a variant of RSA ([Vig08]) in which the public key is the same,
+namely (n, e), but the private key is split up as dp ≡ d (mod p − 1), dq ≡ d (mod q − 1),
+qinv ≡ q−1 (mod p). The encrypted message c ≡ me (mod n) is then decrypted as m1 ≡
+mdp (mod p), m2 ≡ mdq (mod q), h ≡ qinv(m1 − m2) (mod p) and finally, m = m2 + hq.
+This variant runs faster than RSA in practice but it was found in 1996 to be vulnerable to
+a differential fault attack by the Bellcore Institute.
+5.4. Multi-prime RSA. In this variant more than two primes are used in the public
+modulus. The system yields some advantages in efficiency but may be more vulnerable to
+attacks to factor n.
+5.5. Multi-power RSA. In this variant one considers public moduli of the form n = prqs,
+where gcd(r, s) = 1 (when s = 1 this is also called Takagi’s variant; [Ta98]). The method
+offers advantages in speedier decryption using Hensel’s lemma and the Chinese Remainder
+theorem mod pr and mod qs.
+5.6. More RSA variants. Several other variants exist: Dual RSA (using two distinct
+moduli that share the same d, e), Batch RSA (encrypting using two different small expo-
+nents; [F89]) etc. Most RSA variants are vulnerable to small private exponent attacks first
+described by Boneh and Durfee; [BD00]. A fairly exhaustive survey of RSA variants and
+their cryptanalysis can be found in the book by Hinek; [Hin09].
+
+RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA
+7
+5.7. Hofheinz–Kiltz–Shoup Cryptosystem. This more recent system is described as a
+so-called Key Exchange Mechanism (KEM); [HKS13]. The precise details would take up a
+few pages so an abbreviated version is outlined here. To set up Alice picks two safe primes
+P, Q i.e., Sophie Germain primes, P = 2p + 1, Q = 2q + 1. Moreover, P, Q ≡ 3 (mod 4).
+Then N = PQ is a so-called Blum integer.
+Let QRN ⊆ (ZN)∗ denote the subset of quadratic residues mod N; this has pq elements.
+A signed quadratic residue is an element of QRN which is normalized to lie in the set,
+[− (N−1)
+2
+, (N−1)
+2
+]; this normalized set is denoted QR+
+N. We also a target collision resistant
+function (also known as universal one way hash function), say T. The desired key to be
+transmitted is K and the notation ℓK denotes the number of bits in K. Finally, for u ∈ ZN
+(thought of as an element in [− (N−1)
+2
+, (N−1)
+2
+]) define the function LBSN(u) = u (mod 2) and
+then use this to define bits of the key using the Blum–Blum–Shub pseudorandom number
+generator ([BBS86]):
+BBSN(u) := (LBSN(u), LBSN(u2), . . . , LBSN(u2ℓK −1)) ∈ {0, 1}ℓK
+Public and Private keys: Given N, Alice picks a signed quadratic residue g ∈ QR+
+N
+and an element α ∈ {1, 2, . . . , (N−1)
+4
+}. Let X ≡ gα2ℓK +ℓT (mod N). Then the public key is
+(N, g, X) and the private key is (N, g, α).
+Encapsulation: Bob chooses r ∈ {1, 2, . . . , (N−1)
+4
+} and computes
+R ≡ gr2ℓK +ℓT
+(mod N),
+t = T(R),
+S ≡ (gtX)r
+(mod N)
+The ciphertext is C = (R, S) and the key transmitted is K = BBSN(gr·2ℓT ) ∈ {0, 1}ℓK .
+Decapsulation: The receiver computes t = T(R) and verifies whether
+S2ℓK +ℓT
+?≡ Rt+α2ℓK +ℓT
+(mod N)
+Assuming this equality is satisfied the receiver (Alice) computes integers a, b, c such that
+2c = gcd(t, 2ℓK+ℓT ) = at + b2ℓK+ℓT . Alice then derives
+U ≡ (SaRb−aα)2ℓT −c (mod N),
+=⇒
+K = BBSN(U)
+We don’t add any more details; the reason for describing this algorithm is to point out
+that the ciphertext has the same flavor of the algorithm presented in this paper. Note that
+decapsulation here does not require knowing the factorization of N.
+6. Remarks
+6.1.
+The method proposed is probably computationally equivalent to factoring.
+To be
+certain one would have to solve the following thought exercise: Suppose there is a black box
+which takes in input (c, r, n, e) in the proposed algorithm and returns m. Can one factor n
+given this information? A similar formulation can be made for RSA itself: Suppose there is
+a black box which takes input (c, n, e) and outputs m; can one factor m? This is similar in
+spirit to a chosen plaintext attack on either system. While Theorem C indicates this may
+be true the caveat of Remark 3.4 suggests uncertainty.
+
+8
+KRISHNAN SHANKAR
+Acknowledgments
+The author is grateful to Kimball Martin, Steven Miller and Larry Washington for their
+careful reading of early drafts and for providing valuable comments that helped greatly
+improve this paper.
+References
+[AM09] D. Aggarwal and U. Maurer, Breaking RSA generically is equivalent to factoring, In A. Joux, editor,
+EUROCRYPT 2009, volume 5479 of LNCS, pages 36–53. Springer, Heidelberg, April 2009.
+[BBS86] L. Blum, M. Blum and M. Shub, A simple unpredictable pseudo-random number generator, SIAM
+J. Comput. 15(2), 364–383 (1986).
+[BD00] D. Boneh and G. Durfee, Cryptanalysis of RSA with Private Key d Less than n0.292, IEEE Trans.
+Information Theory, 46(4):1339–1349, July 2000.
+[BDH-G99] D. Boneh, G. Durfee, and N. Howgrave-Graham, Factoring N = prq for Large r, In M. Weiner,
+ed., Proceedings of Crypto ’99, vol. 1666 of LNCS, pp. 326–337. Springer–Verlag, Aug. 1999.
+[BV98] D. Boneh and R. Venkatesan, Breaking RSA may not be equivalent to factoring, In K. Nyberg, editor,
+EUROCRYPT’98, volume 1403 of LNCS, pages 59–71. Springer, Heidelberg, May / June 1998.
+[F89] A. Fiat., Batch RSA, In G. Brassard, ed., Proceedings of Crypto 1989, vol. 435 of LNCS, pp. 175–185.
+Springer–Verlag, Aug. 1989.
+[Hin09] M. J. Hinek, Cryptanalysis of RSA and its variants, Cryptography & Network Security series,
+Chapman and Hall publishers, 2009.
+[HKS13] D. Hofheinz, E. Kiltz and V. Shoup, Practical Chosen Ciphertext Secure Encryption from Factoring,
+J. of Cryptology (2013) vol. 26, 102–118.
+[Po99] D. Pointcheval, New Public Key Cryptosystems based on the Dependent RSA-Problems, EURO-
+CRYPT ’99, Lecture Notes in Computer Science 1592, 239–254.
+[RSA78] R. Rivest, A. Shamir, L. Adleman, A method for obtaining digital signatures and public-key cryp-
+tosystems, Communications of the Association for Computing Machinery, 21(2):120–126.
+[Rab79] M. Rabin, Digital signatures and public key functions as intractable as factorization., Technical
+Report MIT/LCS/TR-212, Massachusetts Institute of Technology, January 1979.
+[Sh73] D. Shanks, Five Number Theoretic Algorithms, Proceedings of the Second Manitoba Conference on
+Numerical Mathematics. Pp. 51–70, 1973.
+[Ta98] T. Takagi, Fast RSA-type Cryptosystem Modulo pkq, In H. Krawczyk, ed., Proceedings of Crypto
+1998, vol. 1462 of LNCS, pp. 318–326. Springer-Verlag, Aug. 1998
+[Ton1891] A. Tonelli, Bemerkung ¨uber die Aufl¨osung quadratischer Congruenzen, Nachrichten von der
+K¨oniglichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universit¨at zu G¨ottingen, 344–
+346, 1891.
+[Tur94] S. Turner, Square roots mod p, The American Mathematical Monthly, May 1994, vol. 101, no. 5,
+443–449.
+[Vig08] D. Vigilant, RSA with CRT: A New Cost-Effective Solution to Thwart Fault Attacks, In: Cryp-
+tographic Hardware and Embedded Systems – CHES 2008. CHES 2008. Lecture Notes in Computer
+Science, vol 5154.
+[Wie90] M. Wiener, Cryptanalysis of Short RSA Secret Exponents, IEEE Trans. Information Theory
+36(3):553–558. May 1990.
+Department of Mathematics & Statistics, James Madison University, 60 Bluestone Drive,
+Harrisonburg VA 22807.
+Email address: shankakx@jmu.edu
+
diff --git a/JdAzT4oBgHgl3EQfVPzH/content/tmp_files/load_file.txt b/JdAzT4oBgHgl3EQfVPzH/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,381 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf,len=380
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='01282v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='CR]  31 Dec 2022  RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA  KRISHNAN SHANKAR  Introduction  The RSA algorithm has been around for nearly five decades ([RSA78]) and remains one  of the most studied public key cryptosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Many attempts have been made to break  it or improve it and questions remain about the equivalence of the strength of its security  to well known hard problems in computational number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' A basic question which has  received much attention (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' [AM09], [BV98]1) is: Is breaking RSA equivalent to factoring?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  In this note we propose a modified version which we call RSA+ which is at least as secure as  RSA and show that breaking RSA+ is probably computationally equivalent to factoring n,  the public modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The motivation came from wanting to obscure the encryption exponent  in RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The RSA+ Algorithm  RSA: Bob wishes to send a message m to Alice whose RSA public key is (n, e) and private  key is (p, q, d) where n = pq and de ≡ 1 (mod ϕ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  In the usual implementation of  RSA Bob computes c ≡ me (mod n) and sends it to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Decryption is straightforward:  cd ≡ (md)e ≡ mde ≡ m (mod n) since de ≡ 1 (mod ϕ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  RSA+: We start with the same setup as above namely that Bob has a message m to  transmit to Alice whose RSA public key is (n, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Encryption:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Bob finds a random number x (mod n) and computes y ≡ x2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Bob computes c ≡ mx (mod n) and r ≡ ye (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Bob transmits the pair (c, r) to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Decryption:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Alice computes y ≡ (ye)d (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This step is similar to decrypting in RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Alice then writes down the equation y ≡ x2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' She uses her knowledge of  the factorization of n = pq to compute all four square roots of y (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' For each square root, say {x1, x2, x3, x4}, Alice sequentially computes the inverse  ui ≡ x−1  i  (mod ϕ(n)) and evaluates cui (mod n) until she sees an intelligible mes-  sage m (but see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Breaking RSA+ is probably computationally equivalent to factoring n = pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  1In [BV98] there seems to be an issue in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2, where it is assumed that a cyclotomic  polynomial Φd(x) is irreducible over Fp, which is not always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  2  KRISHNAN SHANKAR  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Similar to RSA care must be taken in choosing the random integer x in Step 1 of  encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If Bob chooses x < √n, then y = x2 as integers and it is easy to find the square  root if one knows y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' In this case the system is as secure as RSA since y is RSA-encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  It is also important that gcd(x, ϕ(n)) = 1 since otherwise Alice cannot decrypt the message  even if she can find x (Euler’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' So, Bob could choose x to be a prime of at least  150 digits assuming n is around 300 digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  In Step 3 of decryption as stated Alice must compute all four square roots and then  try to uncover an intelligible message by sequentially decrypting each mxi (mod n) which is  onerous in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' We describe a way around this problem as long as both primes dividing  n are congruent to 3 mod 4 (this is the workaround suggested by Blum and Williams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Consider the setup with y ≡ x2 (mod n), where Bob chooses x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then y has four square  roots mod n which come from combining the two square roots each mod p and mod q using  the Chinese Remainder Theorem (CRT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Suppose both primes p, q in the factorization of n  are chosen (by Alice) to be congruent to 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Now Bob chooses x, y such that x itself  is a square mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then solving the equation X2 ≡ y (mod n) has solutions (say) X ≡ ±a  (mod p) and X ≡ ±b (mod q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' These are combined using CRT to yield X ≡ xi (mod n),  for i = 1, 2, 3, 4 and let us suppose x = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Note that p, q ≡ 3 (mod 4) so  �  −1  p  �  = −1 and  �  −1  q  �  = −1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', −1 is not a quadratic residue mod p nor mod q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This implies that exactly  one of the roots ± (mod p) and one of the roots ±b (mod q) is a quadratic residue (mod the  respective primes) and the other is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Since Bob picked x to be a square mod n, it follows  that the congruences that yielded x1 must also be squares i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', suppose X ≡ a (mod p)  and X ≡ b (mod q) yields X ≡ x1 (mod n), then it follows that  �  a  p  �  = +1,  �  b  q  �  = +1 and  �  −a  p  �  = −1,  �  −b  q  �  = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' From this it follows that none of x2, x3, x4 is a square mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' We  summarize this via the following  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Suppose n = pq, where p and q are congruent to 3 mod 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Given x, y  such that y ≡ x2 (mod n), gcd(x, n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then y has four square roots mod n exactly one  of which is a square mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  This suggests a way to pinpoint x without having to search all square roots, namely, Bob  picks x to be a square mod n and then computes y ≡ x2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' By the above Proposition  this will be the only square root of y mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' It also follows from the discussion above that:  (i) If one of the primes is congruent to 1 mod 4, then the number of square roots that are  themselves perfect squares is either 0 or 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' (ii) If both primes are congruent to 1 mod 4,  then the number of square roots that are themselves perfect squares is 0 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Decryption is more involved if at least one of the primes dividing n is congruent to 1  (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Nevertheless, the overall algorithm’s security is tied to the difficulty of factoring  and as such lies in class NP, although it is unknown whether it is in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' RSA+ is at least as secure as RSA  In this section we show that RSA+ is at least as secure as RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' We will show this via  the use of a so-called black box i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', an unknown machine or method or algorithm that  takes in a specified input and returns an output that is otherwise currently computationally  intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' RSA+ is at least as secure as RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Suppose we have a black box that is able to decrypt RSA+ messages i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', this black  box takes as input (n, e, mx (mod n), ye (mod n)) and returns m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Here y ≡ x2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Given an RSA ciphertext c ≡ me (mod n) one would then input (n, e, c, (e2)e (mod n))  which should generate the output of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  This shows that any system that can decrypt  RSA+ ciphertexts can also decrypt RSA ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Conversely, suppose there is a black box that is able to decrypt RSA messages i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', this  black box takes as input (n, e, c ≡ me (mod n)) and returns m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Given an RSA+ ciphertext  (mx (mod n), ye (mod n)) one can only decrypt y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' In order to get to m one would need to  know the exponent x for input into this black box, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', one would now need x in order to  input (n, x, mx (mod n)) into the black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This means computing a square root of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' It  is not known whether this black box which can decrypt RSA ciphertexts can also compute  square roots (but see Theorem C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Computational Black Boxes  In this section we explore further computational black boxes which may circumvent known  methods like factoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Suppose one is able to decrypt y by ascertaining d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' One needs to  compute the particular square root x used to encrypt m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Is it possible to compute just one  (pair of) square root(s) without knowledge of the factors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If there exists a computational  black box that can produce one square root when the input is (y, n), then there are two  possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Suppose the black box spits out a random square root mod n for the equation y ≡ x2  mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' In this case one simply inputs the same pair (y, n) repeatedly until we get two  distinct square roots that do not add up to zero mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1 this will yield a  factorization of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Suppose the black box spits out a random square root mod n but always the same  square root for the equation y ≡ x2 mod n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', for a given input (y, n), the output is  always the same x rather than one of the four distinct square roots at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' In this case  we start with some known x and square it to obtain y ≡ x2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Now input (y, n)  into the black box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' if the output is ±x, then discard and try again with a different x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If  the output, say x′ is different from ±x (mod n), then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1 we can factor n by  computing gcd(x − x′, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  4  KRISHNAN SHANKAR  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  If the adversary has instead a black box that computes d given the input (n, e),  then ϕ(n) | (de − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Therefore, for any x we have xde−1 ≡ 1 (mod ϕ(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Since ϕ(n) =  (p−1)(q−1) is divisible by 4, we may compute y ≡ x(de−1)/2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then, by construction  y2 ≡ 1 (mod n) which means n | (y − 1)(y + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then either y ≡ ±1 (mod n) or by Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1 gcd(y ± 1, n) yields a factor of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Again, we can repeat this for several different x as  needed to factor n with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Whether RSA+ is computationally equivalent to factoring depends on the following  thought experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Suppose we have a black box like the one in the previous Section  which takes as input (n, e, mx (mod n), ye (mod n)), where y ≡ x2 (mod n) and produces  as output m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Does this allow us to factor n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' A black box with input (n, e, mx (mod n), ye (mod n)) and output m can  probably factor n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Start with a known pair (x, y), where y ≡ x2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then note that:  yx ≡ (x2)x ≡ x2x ≡ (xx)2  (mod n)  xy ≡ xx2 ≡ xx·x ≡ (xx)x  (mod n)  If we were to input (n, e, xy (mod n), ye (mod n)) into the black box, then this is the same  as the input (n, e, (xx)x (mod n), ye (mod n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The output should therefore be xx (mod n)  which is a square root of yx (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' From 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2 above, repeating this procedure for  several different x should yield a factorization of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The above proof carries over for RSA as well i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', if we had a black box for  RSA, then the same argument above yields a procedure to compute square roots mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The only caveat in Theorem C above is the nature of the black box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Suppose  the black box were to always return the distinguished square root xx (mod n) for the square  yx ≡ (xx)2 (mod n), then this will not allow us to factor n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Computing square roots and factoring  In the absence of a black box an adversary Eve would need y and then a square root  of y to attempt to decrypt m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Computing y ≡ (ye)d mod n without (p, q, d) is as hard  as breaking RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Even if Eve can deduce d without factoring n, she would next have to  solve the equation y ≡ x2 (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' See Section 3 for a discussion on black boxes that may  compute square roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Given n = pq if we can solve the (generic) equation y ≡ x2 (mod n) and find  all four roots ±a, ±b mod n, then we can factor n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Conversely, if one can solve the equation y ≡ x2 (mod p) and y ≡ x2 (mod q), then one  can combine the roots using the Chinese Remainder Theorem to produce (in general) four  square roots mod pq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Since a ̸≡ ±b (mod n) and a2 ≡ b2 mod n implies that n | (a − b)(a + b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Thus,  computing gcd(a ± b, n) yields a non-trivial factor of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If p ≡ 3 (mod 4) is prime, then one can solve y ≡ x2 (mod p) by computing  x ≡ y(p+1)/4 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then (±x)2 ≡ y (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If p = 8k + 5 is prime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', p ≡ 5 (mod 8), then one can solve y ≡ x2  (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Since y is a square mod p, we have that y(p−1)/2 ≡ 1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Since (p − 1)/4 is an  integer we can take a square root and we obtain that y(p−1)/4 ≡ ±1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Case 1: If y(p−1)/4 ≡ 1 (mod p) and p = 8k + 5, then x ≡ yk+1 ≡ y(p+3)/8 (mod p) is a  desired square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This is because  x2 ≡ y(p+3)/4 ≡ y(p−1)/4 · y ≡ y  (mod p)  Case 2: If y(p−1)/4 ≡ −1 (mod p), then x ≡ 22k+1yk+1 ≡ 2(p−1)/4y(p+3)/8 (mod p) is a  desired square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This is because  x2 ≡ 2(p−1)/2y(p+3)/4 ≡ (−1) · y(p−1)/4 · y ≡ (−1)(−1)y ≡ y  (mod p)  Note that 2(p−1)/2 ≡ −1 (mod p) since p ≡ 5 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If p ≡ 1 (mod 8) is prime, then there exists an algorithm terminating in at  most r steps to compute y ≡ x2 (mod p), where p − 1 = 2rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This is the most involved case of the Tonelli–Shanks algorithm, which covers all the  above Lemmas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' see [Sh73], [Ton1891].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' See [Tur94] for a nice exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  □  Now we can prove Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If an adversary Eve can factor n, then she can certainly decrypt the  message m: compute ϕ(n) = (p−1)(q −1), find d using the Extended Euclidean Algorithm  and use this to find y ≡ (ye)d mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Now using Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='4, depending on  whether p, q are congruent to 1 (mod 4) or 3 (mod 4), Eve can find the square roots of y  mod p and mod q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then she can combine them to obtain all square roots mod n using the  Chinese Remainder Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Finally Eve solves for x−1 (mod ϕ(n)) to decrypt m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  If Eve possesses a black box as described in Theorem C, then Eve can probably factor n  (with the caveat introduced in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' If Eve has instead a black box or method to  find square roots mod n, then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1 she can factor n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Similar cryptosystems  A search of the literature by the author yielded similar cryptosystems and these are  described briefly here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Any other omissions are entirely accidental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' To the author’s knowl-  edge this particular protocol has not been described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' For each protocol below the  description is brief and only intended to serve as a reference or comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  6  KRISHNAN SHANKAR  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The Rabin Cryptosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This cryptosystem ([Rab79]) is the closest in spirit to  the method proposed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Its security is tied to square roots and factoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Alice finds  primes p, q both congruent to 3 mod 4 and computes n = pq and exponents e, d with de ≡ 1  (mod ϕ(n)) same as in RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Bob encrypts his message m as c ≡ m2 (mod n) and transmits c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  To decrypt Alice  computes the square root mod p and mod q by the method outlined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then  she combines them using the Chinese Remainder Theorem to obtain all four square roots  and then examines which of these is intelligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This latter disambiguation problem was  addressed by Blum and Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The Rabin cryptosystem was shown to be vulnerable  to a chosen ciphertext attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This attack can be thwarted by adding redundancies to the  message.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' D–RSA (Dependent RSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This cryptosystem, described in [Po99], was in re-  sponse to finding a so-called semantically secure cryptosystem as efficient as RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Similar  to RSA Alice has public key (n, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then Bob picks a random k ∈ (Zn)∗ and computes  A ≡ ke (mod n) and B ≡ m · (k + 1)e (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Decryption is via computing k ≡ (ke)d  (mod n) and then B · (k + 1)−e (mod n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This is also similar in spirit to what we have  described in that Bob picks a random number during the encryption process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Among the  theorems in the paper it is shown that a slight modification of this protocol is semantically  secure against adaptive chosen ciphertext attacks relative to the Decision D–RSA Problem  in the random oracle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' RSA–CRT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This is a variant of RSA ([Vig08]) in which the public key is the same,  namely (n, e), but the private key is split up as dp ≡ d (mod p − 1), dq ≡ d (mod q − 1),  qinv ≡ q−1 (mod p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The encrypted message c ≡ me (mod n) is then decrypted as m1 ≡  mdp (mod p), m2 ≡ mdq (mod q), h ≡ qinv(m1 − m2) (mod p) and finally, m = m2 + hq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  This variant runs faster than RSA in practice but it was found in 1996 to be vulnerable to  a differential fault attack by the Bellcore Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Multi-prime RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' In this variant more than two primes are used in the public  modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The system yields some advantages in efficiency but may be more vulnerable to  attacks to factor n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Multi-power RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' In this variant one considers public moduli of the form n = prqs,  where gcd(r, s) = 1 (when s = 1 this is also called Takagi’s variant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' [Ta98]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The method  offers advantages in speedier decryption using Hensel’s lemma and the Chinese Remainder  theorem mod pr and mod qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' More RSA variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Several other variants exist: Dual RSA (using two distinct  moduli that share the same d, e), Batch RSA (encrypting using two different small expo-  nents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' [F89]) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Most RSA variants are vulnerable to small private exponent attacks first  described by Boneh and Durfee;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' [BD00].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' A fairly exhaustive survey of RSA variants and  their cryptanalysis can be found in the book by Hinek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' [Hin09].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  RSA+: AN ALGORITHM AT LEAST AS SECURE AS RSA  7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Hofheinz–Kiltz–Shoup Cryptosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This more recent system is described as a  so-called Key Exchange Mechanism (KEM);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' [HKS13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The precise details would take up a  few pages so an abbreviated version is outlined here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' To set up Alice picks two safe primes  P, Q i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=', Sophie Germain primes, P = 2p + 1, Q = 2q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Moreover, P, Q ≡ 3 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Then N = PQ is a so-called Blum integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Let QRN ⊆ (ZN)∗ denote the subset of quadratic residues mod N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' this has pq elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  A signed quadratic residue is an element of QRN which is normalized to lie in the set,  [− (N−1)  2  , (N−1)  2  ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' this normalized set is denoted QR+  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' We also a target collision resistant  function (also known as universal one way hash function), say T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' The desired key to be  transmitted is K and the notation ℓK denotes the number of bits in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Finally, for u ∈ ZN  (thought of as an element in [− (N−1)  2  , (N−1)  2  ]) define the function LBSN(u) = u (mod 2) and  then use this to define bits of the key using the Blum–Blum–Shub pseudorandom number  generator ([BBS86]):  BBSN(u) := (LBSN(u), LBSN(u2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' , LBSN(u2ℓK −1)) ∈ {0, 1}ℓK  Public and Private keys: Given N, Alice picks a signed quadratic residue g ∈ QR+  N  and an element α ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' , (N−1)  4  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Let X ≡ gα2ℓK +ℓT (mod N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Then the public key is  (N, g, X) and the private key is (N, g, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Encapsulation: Bob chooses r ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' , (N−1)  4  } and computes  R ≡ gr2ℓK +ℓT  (mod N),  t = T(R),  S ≡ (gtX)r  (mod N)  The ciphertext is C = (R, S) and the key transmitted is K = BBSN(gr·2ℓT ) ∈ {0, 1}ℓK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Decapsulation: The receiver computes t = T(R) and verifies whether  S2ℓK +ℓT  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='≡ Rt+α2ℓK +ℓT  (mod N)  Assuming this equality is satisfied the receiver (Alice) computes integers a, b, c such that  2c = gcd(t, 2ℓK+ℓT ) = at + b2ℓK+ℓT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Alice then derives  U ≡ (SaRb−aα)2ℓT −c (mod N),  =⇒  K = BBSN(U)  We don’t add any more details;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' the reason for describing this algorithm is to point out  that the ciphertext has the same flavor of the algorithm presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Note that  decapsulation here does not require knowing the factorization of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Remarks  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  The method proposed is probably computationally equivalent to factoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  To be  certain one would have to solve the following thought exercise: Suppose there is a black box  which takes in input (c, r, n, e) in the proposed algorithm and returns m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Can one factor n  given this information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' A similar formulation can be made for RSA itself: Suppose there is  a black box which takes input (c, n, e) and outputs m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' can one factor m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' This is similar in  spirit to a chosen plaintext attack on either system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' While Theorem C indicates this may  be true the caveat of Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='4 suggests uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  8  KRISHNAN SHANKAR  Acknowledgments  The author is grateful to Kimball Martin, Steven Miller and Larry Washington for their  careful reading of early drafts and for providing valuable comments that helped greatly  improve this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
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+page_content=' CHES 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content=' Lecture Notes in Computer  Science, vol 5154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
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+page_content=' May 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Department of Mathematics & Statistics, James Madison University, 60 Bluestone Drive,  Harrisonburg VA 22807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='  Email address: shankakx@jmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdAzT4oBgHgl3EQfVPzH/content/2301.01282v1.pdf'}
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+MNRAS 000, 1–24 (2022)
+Preprint 23 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Measuring the variability of directly imaged exoplanets using vector
+Apodizing Phase Plates combined with ground-based differential
+spectrophotometry
+Ben J. Sutlieff,1,2★ Jayne L. Birkby,3,1 Jordan M. Stone,4 David S. Doelman,2
+Matthew A. Kenworthy,2 Vatsal Panwar,1,5 Alexander J. Bohn,2 Steve Ertel,6,7
+Frans Snik,2 Charles E. Woodward,8 Andrew J. Skemer,9 Jarron M. Leisenring,7
+Klaus G. Strassmeier,10 and David Charbonneau11
+1Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
+2Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
+3Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, United Kingdom
+4Naval Research Laboratory, Remote Sensing Division, 4555 Overlook Ave. SW, Washington, DC 20375, USA
+5Department of Physics, University of Warwick, Coventry West Midlands, CV4 7AL, United Kingdom
+6Large Binocular Telescope Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA
+7Steward Observatory and Department of Astronomy, University of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA
+8Minnesota Institute for Astrophysics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA
+9Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High St, Santa Cruz, CA 95064, USA
+10Leibniz-Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
+11Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+Clouds and other features in exoplanet and brown dwarf atmospheres cause variations in bright-
+ness as they rotate in and out of view. Ground-based instruments reach the high contrasts and
+small inner working angles needed to monitor these faint companions, but their small fields-
+of-view lack simultaneous photometric references to correct for non-astrophysical variations.
+We present a novel approach for making ground-based light curves of directly imaged compan-
+ions using high-cadence differential spectrophotometric monitoring, where the simultaneous
+reference is provided by a double-grating 360° vector Apodizing Phase Plate (dgvAPP360)
+coronagraph. The dgvAPP360 enables high-contrast companion detections without blocking
+the host star, allowing it to be used as a simultaneous reference. To further reduce systematic
+noise, we emulate exoplanet transmission spectroscopy, where the light is spectrally-dispersed
+and then recombined into white-light flux. We do this by combining the dgvAPP360 with the
+infrared ALES integral field spectrograph on the Large Binocular Telescope Interferometer. To
+demonstrate, we observed the red companion HD 1160 B (separation ∼780 mas) for one night,
+and detect 8.8% semi-amplitude sinusoidal variability with a ∼3.24 h period in its detrended
+white-light curve. We achieve the greatest precision in ground-based high-contrast imaging
+light curves of sub-arcsecond companions to date, reaching 3.7% precision per 18-minute
+bin. Individual wavelength channels spanning 3.59-3.99 µm further show tentative evidence
+of increasing variability with wavelength. We find no evidence yet of a systematic noise floor,
+hence additional observations can further improve the precision. This is therefore a promis-
+ing avenue for future work aiming to map storms or find transiting exomoons around giant
+exoplanets.
+Key words: infrared: planetary systems – instrumentation: high angular resolution – planets
+and satellites: detection – brown dwarfs – planets and satellites: atmospheres – techniques:
+imaging spectroscopy
+★ E-mail: b.j.sutlieff@uva.nl
+1
+INTRODUCTION
+Planets do not have a homogeneous appearance. When we look at
+the planets in our own solar system, we see distinct cloud structures
+© 2022 The Authors
+arXiv:2301.08689v1  [astro-ph.EP]  20 Jan 2023
+
+2
+B. J. Sutlieff et al.
+Figure 1. An on-sky example image of a 𝐾 ∼ 7 mag target observed with the dgvAPP360 coronagraph on LBT, copied and annotated on the right. The
+dgvAPP360 produces a dark hole of deep flux suppression, seen here surrounding the target star, allowing high-contrast companions to be detected in this
+region. The star itself remains unsaturated in the middle, allowing it to be used as a simultaneous reference PSF when making a differential light curve for a
+companion. The outer edge of the dark hole lies beyond the field of view in the LBT observations used in this work.
+and giant storms that show great diversity in size, shape, lifetime,
+and brightness (e.g. Simon et al. 2016, 2021; Stauffer et al. 2016;
+Ge et al. 2019; Coulter et al. 2022). These features rotate in and
+out of view throughout the planet’s rotation period, modulating its
+overall brightness and thus allowing us to map out its atmosphere
+(e.g. Kostov & Apai 2013; Karalidi et al. 2015, 2016; Fletcher et al.
+2016; Apai et al. 2017; Plummer & Wang 2022). Beyond the solar
+system, variations in the light curves of stars deliver information on
+the distribution of features such as star spots (Barnes et al. 2002;
+Jeffers et al. 2007; Frasca et al. 2009; Strassmeier 2009; Morales
+et al. 2010; Goulding et al. 2012; Herbst 2012; Nielsen et al. 2013;
+Thiemann et al. 2021; Park et al. 2021). By measuring the pho-
+tometric variability of exoplanets and brown dwarfs in the same
+way, we can gain not only an insight into their visual appearance,
+but key information on the distribution of condensate clouds that
+strongly affect the infrared spectra of directly imaged companions,
+allowing degeneracies between atmospheric models to be broken
+(e.g. Yang et al. 2016; Rajan et al. 2017; Charnay et al. 2018; Zhou
+et al. 2019; Zhang 2020; Ward-Duong et al. 2021; Tan & Showman
+2021). Space-based photometric monitoring with the Hubble Space
+Telescope (HST) has already shown that giant planetary-mass and
+brown dwarf companions do exhibit such variability, at a range
+of amplitudes and periods (Apai et al. 2013; Buenzli et al. 2014,
+2015a,b; Zhou et al. 2016, 2020a; Manjavacas et al. 2018, 2019b;
+Miles-Páez et al. 2019; Lew et al. 2020a; Bowler et al. 2020b). These
+results are in good agreement with observations of isolated brown
+dwarfs and giant exoplanet analogues (Biller et al. 2018; Vos et al.
+2018, 2020; Lew et al. 2020b; Ashraf et al. 2022), including a large
+Spitzer survey by Metchev et al. (2015) who found that photospheric
+spots causing ≥0.2% variability at 3-5 µm are ubiquitous. Several
+studies have identified objects with much stronger variability, at the
+>10% level, with some even varying with peak-to-peak amplitudes
+as high as 26% (e.g. Radigan et al. 2012; Wilson et al. 2014; Biller
+et al. 2015; Eriksson et al. 2019; Bowler et al. 2020b). Vos et al.
+(2022) further found that young, low-mass brown dwarfs with simi-
+lar colours and spectra to directly imaged exoplanetary companions
+are highly likely to display variability in the L2-T4 spectral type
+range, with an enhancement in maximum amplitudes compared to
+field dwarfs.
+The rotation periods of brown dwarf and planetary-mass com-
+panions are consistent with those of the isolated low-mass brown
+dwarf population, suggesting that they may share similar angular
+momentum histories (Bryan et al. 2018; Vos et al. 2022). These pe-
+riods are generally short, ranging from ∼1 hour to ≳20 hours (e.g.
+Radigan et al. 2014; Metchev et al. 2015; Apai et al. 2021; Tannock
+et al. 2021), within the range expected when evolutionary models
+and the age- and mass-dependent breakup velocities are considered
+(Leggett et al. 2016; Vos et al. 2020). These periods, derived from
+photometric measurements, are complementary to measurements
+of companion spin obtained from their spectra (Snellen et al. 2014;
+Schwarz et al. 2016; Bryan et al. 2018, 2020b; Xuan et al. 2020;
+Wang et al. 2021b). When combined, rotation period and spin mea-
+surements can be used to constrain companion obliquities (Bryan
+et al. 2020a, 2021). However, the population of directly imaged
+companions accessible to space-based facilities such as HST re-
+mains small as most companions lie at close angular separations
+within the inner working angles of these facilities.
+Equipped with coronagraphs and extreme adaptive optics (AO)
+systems operating in the infrared, large ground-based observatories
+have the resolution and photon collecting power needed to overcome
+the glare of the host star and reach the high contrasts and close an-
+gular separations of substellar companions currently inaccessible to
+space telescopes (Bowler 2016; Hinkley et al. 2021; Currie et al.
+2022). Although this provides the opportunity for variability studies
+of such companions, precise photometric monitoring is difficult as
+the companion light curve is contaminated by variability arising
+from Earth’s atmosphere and other systematics. Therefore, a simul-
+taneous, unsaturated photometric reference is required to remove
+this contaminant variability from the companion light curve. For
+non-coronagraphic, ground-based observations of isolated brown
+dwarfs and planetary-mass objects, nonvariable stars present in the
+field of view have often been used as photometric references to en-
+able many successful measurements of variability (e.g. Gelino et al.
+2002; Girardin et al. 2013; Biller et al. 2013, 2015; Radigan et al.
+2014; Wilson et al. 2014; Naud et al. 2017; Eriksson et al. 2019;
+MNRAS 000, 1–24 (2022)
+
+Outer working
+Unsaturated
+angle
+reference PSF
+Dark holeExoplanet variability with vAPP coronagraphs
+3
+Vos et al. 2019; Manjavacas et al. 2021, 2022). However, the typi-
+cally narrow field of view of ground-based coronagraphic imagers
+generally precludes the use of field stars as photometric references
+for observations of companions, and widely used focal-plane coron-
+agraphs block the host star to enable the detection of the companion
+(Soummer 2005; Mawet et al. 2012; Ruane et al. 2018).
+One solution to this problem is to use off-axis satellite Point
+Spread Functions (PSFs), or satellite spots, which can act as simul-
+taneous photometric references even when a host star is blocked by
+a coronagraph (Sivaramakrishnan & Oppenheimer 2006; Marois
+et al. 2006b). Satellite spots can be created by adding a periodic
+modulation to the deformable mirror of an AO-equipped telescope
+or by placing a square grid in the pupil plane to produce spots
+through diffraction of starlight (Langlois et al. 2013; Wang et al.
+2014; Jovanovic et al. 2015b). The former approach has been used
+by Apai et al. (2016), Biller et al. (2021), and Wang et al. (2022)
+for observations of the multi-planet HR 8799 system (Marois et al.
+2008, 2010). The first two of these studies observed HR 8799 with
+the Spectro-Polarimetric High-contrast imager for Exoplanets RE-
+search (SPHERE, Beuzit et al. 2019) instrument at the Very Large
+Telescope (VLT), and the latter used the CHARIS integral field spec-
+trograph in combination with the Subaru Coronagraphic Extreme
+Adaptive Optics instrument at the Subaru Telescope (Jovanovic et al.
+2015a; Groff et al. 2017). Biller et al. (2021) used the satellite spots
+with a broadband-H filter to successfully constrain their sensitivity
+to variability to amplitudes >5% for HR 8799b for periods <10
+hours, and amplitudes >25% for HR 8799c for similar periods, not-
+ing that the observed amplitude of any variability would be muted
+by the likely pole-on viewing angle of these planets (Vos et al. 2017;
+Wang et al. 2018; Ruffio et al. 2019). They also rule out non-shared
+variability between HR 8799b and HR 8799c at the <10-20% level
+over a 4-5 hour timescale by using one planet as a photometric refer-
+ence for the other. Using a spectrophotometric approach, Wang et al.
+(2022) further constrained the variability amplitudes of HR 8799c
+to the 10% level, and HR 8799d to the 30% level, and found that
+there was no significant variability in the planet’s colours. However,
+all three studies found that satellite spots are anti-correlated with
+each other and can demonstrate individual variations of their own,
+potentially setting a limit to the precision that can be achieved with
+this technique (although Biller et al. (2021) and Wang et al. (2022)
+note that the satellite spot light curves can be flat in their most stable
+epochs).
+1.1
+Ground-based differential spectrophotometry
+In this paper we present a novel, alternative ground-based approach
+for constructing light curves of high-contrast companions directly
+through the technique of differential spectrophotometric monitor-
+ing, akin to that used highly successfully to study exoplanet trans-
+mission spectra (e.g. Kreidberg et al. 2014; Stevenson et al. 2014;
+Wilson et al. 2020; Arcangeli et al. 2021; Panwar et al. 2022a,b).
+We use a double-grating 360° vector Apodizing Phase Plate (dg-
+vAPP360) coronagraph (Doelman et al. 2017, 2020, 2021; Wagner
+et al. 2020), which enables high-contrast companions to be detected
+without blocking the host star, hence leaving an unsaturated image
+of the host star available for use as a simultaneous reference. The
+more widely used grating vector Apodizing Phase Plate (gvAPP)
+coronagraph adjusts the phase of the incoming wavefront to modify
+the PSFs of all objects in the field of view, creating two images of
+the target star each with a 180° D-shaped ‘dark hole’, a region of
+deep flux suppression in which high-contrast companions can be
+observed, on opposing sides (Snik et al. 2012; Otten et al. 2014a,
+2017; Bos et al. 2020; Doelman et al. 2021; Sutlieff et al. 2021). The
+dgvAPP360 instead creates a 360° dark hole surrounding each of
+the two images of the target star, and then uses an additional grating
+to overlap the images to produce a single image of the star (Doelman
+et al. 2022). An example image obtained with the dgvAPP360 is
+shown in Figure 1, with the target star in the centre and the dark
+hole surrounding it.
+In addition, we combine the dgvAPP360 with an integral field
+spectrograph (IFS), enabling us to use differential spectrophotom-
+etry for high-contrast directly imaged companions. The incoming
+light is first dispersed into individual spectra, and then recombined
+into a single ‘white-light’ data point. This has the advantage of
+smoothing out wavelength-dependent flat-fielding errors and allows
+wavelength regions with instrumental absorption or highly variable
+telluric bands to be excluded, meaning systematic effects can be
+significantly reduced, thus yielding greater stability and precision
+in the final white-light curve.
+1.2
+The HD 1160 system
+We test this approach using observations of the HD 1160 system,
+which is located at a distance of 120.4±0.6 pc (Gaia Collaboration
+et al. 2016, 2021; Bailer-Jones et al. 2021) and consists of host
+star HD 1160 A (spectral type A0V, Houk & Swift 1999) and two
+high-contrast companions, HD 1160 B and C (Nielsen et al. 2012).
+HD 1160 B and C lie at separations of ∼80 au (∼0.78′′) and ∼530 au
+(∼5.1′′), respectively. Several key properties of HD 1160 A are listed
+in Table 1. HD 1160 A is bright (K = 7.040±0.029 mag, Cutri et al.
+2003), and the contrast ratio between HD 1160 A and HD 1160 B is
+Δ𝐿′ = 6.35 ± 0.12 mag (Nielsen et al. 2012). This makes it an ideal
+target for demonstrating our technique, as a high signal-to-noise
+detection of the companion allows high cadence monitoring and a
+deep investigation into any residual systematic effects. A-type stars
+such as HD 1160 A generally vary below the millimagnitude level,
+which corresponds to variability amplitudes comfortably below the
+∼1% level (Ciardi et al. 2011). We assess the variability of the host
+star in Section 4.2 using observations from the Transiting Exoplanet
+Survey Satellite (TESS) mission. The age of the system is poorly
+constrained, with estimates ranging from 50+50
+−40 Myr (Nielsen et al.
+2012) to 100+200
+−70 Myr (Maire et al. 2016). The HD 1160 system may
+be a member of the Pisces-Eridanus stellar stream, which would
+place its age at ∼120 Myr, but this has yet to be confirmed (Curtis
+et al. 2019).
+HD 1160 B has a spectral type close to the brown dwarf/stellar
+boundary; Nielsen et al. (2012) found a spectral type of ∼L0, but
+more recent papers suggest that it lies between M5-M7 (Maire et al.
+2016; Garcia et al. 2017; Mesa et al. 2020). However, Mesa et al.
+(2020) found its spectrum to be highly peculiar and that no spectral
+model or template in current libraries can produce a satisfactory fit.
+Although the cause of this peculiarity has not yet been explained,
+Mesa et al. (2020) hypothesise that possible causes could include
+a young system age, dust in the photosphere of HD 1160 B, or
+ongoing evolutionary processes. The mass of HD 1160 B also re-
+mains unclear, primarily due to the poorly constrained age of the
+system, with estimates ranging from that of a low mass brown dwarf
+(∼20 MJup, Mesa et al. 2020) to decisively in the stellar mass regime
+(0.12±0.01M⊙ ≈ 123MJup, Curtis et al. 2019) if the system is in-
+deed a member of the Pisces-Eridanus stellar stream. HD 1160 C is
+a low-mass star (spectral type M3.5), and its separation of ∼530 au
+(∼5.1′′) places it beyond the field of view of most vAPPs currently
+in use (Maire et al. 2016; Doelman et al. 2021).
+The observations carried out on the HD 1160 system are
+MNRAS 000, 1–24 (2022)
+
+4
+B. J. Sutlieff et al.
+Table 1. Properties of host star HD 1160 A.
+Parameter
+Value
+Reference(s)
+Spectral Type
+A0V
+(1)
+Right Acension (J2000)
+00:15:57.32
+(2)
+Declination (J2000)
++04:15:03.77
+(2)
+Age (Myr)
+10-300
+(3, 4)
+Parallax (mas)
+8.2721±0.0354
+(2)
+Distance (pc)
+120.4±0.6
+(2, 5)
+Proper motion (RA, mas yr−1)
+20.150±0.040
+(2)
+Proper motion (Dec, mas yr−1)
+-14.903±0.034
+(2)
+Mass (M⊙)
+∼2.2
+(3)
+Teff (K)
+9011±85
+(6)
+log(L/L⊙)
+1.12±0.07
+(6)
+log(g) (dex)
+∼4.5
+(7)
+[Fe/H]
+∼solar
+(7)
+V (mag)
+7.119±0.010
+(8)
+G (mag)
+7.1248±0.0004
+(2)
+J (mag)
+6.983±0.020
+(9)
+H (mag)
+7.013±0.023
+(9)
+K (mag)
+7.040±0.029
+(9)
+References: (1) Houk & Swift (1999); (2) Gaia Collaboration et al. (2016,
+2021); (3) Nielsen et al. (2012); (4) Maire et al. (2016); (5) Bailer-Jones
+et al. (2021); (6) Garcia et al. (2017); (7) Mesa et al. (2020); (8) Tycho-2
+(Høg et al. 2000); (9) 2MASS (Cutri et al. 2003; Skrutskie et al. 2006)
+described in Section 2, and in Section 3 we describe the spec-
+tral extraction and data reduction processes. In Section 4 we pro-
+duce and present our differential spectrophotometric light curves of
+HD 1160 B. We then examine various factors that may be correlated
+with the light curves and detrend them in Section 5. These results
+and their implications are then discussed in Sections 6 and 7. Lastly,
+the conclusions of the work are summarised in Section 8.
+2
+OBSERVATIONS
+We observed the HD 1160 system on the night of 2020 September
+25 (03:27:31 - 11:16:14 UT) with the double-grating 360° vector
+Apodizing Phase Plate (dgvAPP360) coronagraph (see Section 1.1)
+and the Arizona Lenslets for Exoplanet Spectroscopy (ALES) inte-
+gral field spectrograph (IFS) (Skemer et al. 2015; Hinz et al. 2018;
+Stone et al. 2018). ALES is integrated inside the Large Binocu-
+lar Telescope Interferometer (LBTI) (Hinz et al. 2016; Ertel et al.
+2020), which works in conjunction with the LBT mid-infrared cam-
+era (LMIRcam), on the 2 x 8.4-m LBT in Arizona (Skrutskie et al.
+2010; Leisenring et al. 2012). For these observations, ALES was in
+single-sided mode and was therefore fed only by the left-side aper-
+ture of LBT. Atmospheric turbulence was corrected for by the LBTI
+adaptive optics (AO) system (Bailey et al. 2014; Pinna et al. 2016,
+2021). We used the ALES L-band prism, which covers a simulta-
+neous wavelength range of 2.8-4.2 µm with a spectral resolution of
+R∼40 (Skemer et al. 2018). The plate scale is ∼35 mas spaxel−1.
+The other LBT aperture was used to feed the Potsdam Echelle Po-
+larimetric and Spectroscopic Instrument (PEPSI), which obtained
+R = 50,000 combined optical spectra of HD 1160 A and B in the
+383-907nm wavelength range (Strassmeier et al. 2015, 2018), which
+is subject to analysis in other forthcoming works.
+Conditions were exceptionally clear and stable throughout the
+night, with no time lost to weather, and seeing ranged from 0.7-
+1.4′′. We acquired 2210 on-target ALES frames, with an integration
+time of 5.4 s per frame, giving a total on-target integration time of
+11934.0 s (∼3.32 h) spread over ∼7.81 h once readout time, nod-
+ding, and wavelength calibrations are included. The integration time
+was chosen such that the stellar PSF remained unsaturated in the
+core so that it can be used as the photometric reference for the com-
+panion. We used an on/off nodding pattern to enable background
+subtraction, nodding to a position 5′′ away for the off-source nod
+position. As HD 1160 C is located at a similar separation (∼5.1′′),
+we nodded in a direction away from this companion to prevent
+it from contaminating the frames obtained in the off-source nod
+position. Beam-switching in this way is possible because of the
+intrinsic stability provided by the dgvAPP360 coronagraph’s place-
+ment in the pupil plane. We obtained dark frames with the same
+exposure time at the end of the night, and 6 wavelength calibrations
+were acquired at irregular intervals during the night. LBTI operates
+in pupil-stabilized mode, such that the field of view was rotating
+throughout the observations. The total field rotation across the ob-
+serving sequence was 109.7°. The HD 1160 system was observed
+from an elevation of 29.4° at the start of the night to a maximum
+elevation of 61.7°, and then back down to an elevation of 27.5°. The
+dgvAPP360 creates an annular dark hole around the target PSF,
+with an inner radius close to the PSF core and an outer radius at the
+edge of the 2.2′′ field of view of the detector (2.7 - 15 𝜆/D in ALES
+mode) (Doelman et al. 2020, 2021). For these observations, this
+meant that HD 1160 B was located in the dark hole of HD 1160 A
+across the entire wavelength range covered by the ALES L-band
+prism. HD 1160 C (separation ∼5.1′′) remained beyond the ALES
+field of view.
+3
+DATA REDUCTION
+The raw ALES images contain the spectra that have been projected
+onto the detector. Ultimately, we are aiming to produce a light curve
+for the companion that is made from the ‘white-light’, i.e. combined
+in the wavelength dimension. To do this, we must first extract the
+spectra from the raw data along with bad pixel correction and flat-
+fielding. We can then extract the photometry of the star and the
+companion at each wavelength before collapsing the data in the
+wavelength dimension to obtain white-light fluxes. A light curve
+for the companion is then obtained by dividing the companion flux
+by the stellar flux to remove systematic trends shared by both. In
+the following subsections we describe the methods used to carry
+out each of these steps and obtain the white-light curve of the
+companion.
+3.1
+Spectral data cube extraction
+Raw ALES data consists of a two-dimensional grid of 63×67 micro-
+spectra over a 2.2′′x 2.2′′ field of view, which must be extracted
+into three-dimensional data cubes of 𝑥-position, 𝑦-position, and
+wavelength 𝜆 (Stone et al. 2022). To do this, we first performed a
+background subtraction using the sky frames obtained in the off-
+source nod position. For each ALES image, we subtracted the me-
+dian combination of the 100 sky frames closest in time. We then
+extracted the micro-spectra into cubes using optimal extraction,
+which is an inverse variance and spatial profile weighted extraction
+approach (Horne 1986; Briesemeister et al. 2018; Stone et al. 2020).
+The extraction weights were obtained by measuring the spatial pro-
+file of each micro-spectrum in the dark-subtracted sky frames. As
+there is no significant change in the spatial profile as a function of
+wavelength, we average the spatial profile of each micro-spectrum
+MNRAS 000, 1–24 (2022)
+
+Exoplanet variability with vAPP coronagraphs
+5
+over wavelength to obtain higher signal-to-noise (S/N). The wave-
+length calibration of the raw ALES data was then obtained using
+four narrow-band photometric filters at 2.9, 3.3, 3.5, and 3.9 µm,
+positioned upstream of the ALES optics. These filters are each of
+a higher spectral resolution
+𝜆
+Δ𝜆∼100 than ALES, so are therefore
+unresolved and provide four single-wavelength fiducial spots with
+which each micro-spectra can be calibrated (Stone et al. 2018, 2022).
+For each micro-spectrum, we performed this calibration by fitting a
+second-order polynomial to the calculated pixel positions of these
+four spots, therefore mapping pixel position to wavelength. Each of
+the 63×67 micro-spectra was thereby converted into a correspond-
+ing spaxel in the three-dimensional data cube (Briesemeister et al.
+2019; Doelman et al. 2022). The resulting data cube contained 100
+channels in the wavelength dimension ranging from 2.8-4.2 µm.
+The primary wavelength calibration used to process this data
+set was obtained at 08:25:00 UT on the night of observations, i.e.
+4 hours 57 minutes into the observing sequence. To test whether
+a wavelength calibration obtained at a different point in the night
+has a significant effect on the photometry of the target star and
+the companion, we also separately processed the data using an
+alternative wavelength calibration obtained at 04:54:00 UT, 1 hour
+26 minutes into the observing sequence. We compare and discuss
+the results of the two wavelength calibrations in Section 4.3.
+3.2
+Data processing
+Once the spectra were extracted into background-subtracted three-
+dimensional cubes of images for each exposure, we applied several
+data reduction steps to remove systematics and improve the S/N at
+the location of the companion. We first removed 8 time frames from
+the data in which the AO loop opened while the data was being col-
+lected. Next, we identified bad pixels using a 6𝜎 filter and replaced
+them with the mean of the neighbouring pixels. We then applied
+a flat-field correction to calibrate the data against the response of
+the detector. For each wavelength channel, the corresponding flat
+was a time-average of the frames obtained in the ‘off’ nod position,
+which had then been corrected for bad pixels in the same way as
+the science frames and smoothed over using a Gaussian filter. These
+flats were then divided by the maximum value in the frame such that
+the value of every pixel was between zero and one. We then divided
+the science frames by these smoothed, normalised sky flats. This
+flat-fielding process was also repeated separately using a median
+filter instead of a Gaussian filter as a means to test the robustness
+of this step in our method. We proceed with the Gaussian filter and
+discuss the impact of the choice of flat frame on the photometry of
+the star and companion in Section 4.3.
+Doelman et al. (2022) previously identified that background-
+subtracted, flat-fielded ALES images contain residual structure that
+cannot be described by purely Gaussian noise, in the form of time-
+varying row and column discontinuities (faintly visible in the top
+panel of Figure 2). Such discontinuities are expected and arise from
+the way in which the micro-spectra lie across multiple channels
+of the LMIRcam detector (Doelman et al. 2022). We followed the
+method of Doelman et al. (2022) to characterise and remove these
+discontinuities by fitting a third-order polynomial to each row and
+column in each frame (Figure 2). Removing these systematics is
+important as they could impact the precision of our differential light
+curve, or even generate a false variability signal if the target moves
+over them throughout the observing sequence. Prior to fitting, we
+applied circular masks at the locations of the star and the compan-
+ion in each frame such that their flux did not contaminate the fit.
+To find the position of the star in each time frame, we selected a
+wavelength channel with a high stellar flux per frame (channel 52,
+𝜆 ≈ 3.69 µm) and fit the PSF core with a 2D Gaussian. The position
+of the companion in each time frame was then identified using its
+separation and position angle relative to the star and accounting for
+the effect of the field rotation over time. We then masked the star
+and the companion across all wavelength channels using circular
+masks with diameters of 18 pixels and 5 pixels, respectively, before
+fitting the third-order polynomials to each column. The resulting
+values were then subtracted from the data to remove the column
+discontinuities. This process was then repeated for each row in the
+resulting image to remove the row discontinuities.
+In addition to removing these systematic discontinuities, this
+process has the effect of removing residual background flux not
+eliminated by earlier processing steps. This is indicated by the his-
+tograms in Figure 2, which show that the noise distribution of the
+data was offset from zero prior to the removal of the discontinuities
+(in blue) but is approximately consistent with zero after this process
+has been applied (in orange). We then used the position of the star
+in each frame, found when applying the masks in the previous step,
+to spatially align the data such that the star was in the centre of each
+frame. Finally, we rotationally aligned the images by applying an
+anticlockwise rotation corresponding to their parallactic angles.
+We did not use any further post-processing methods that reduce
+quasistatic speckle noise through the subtraction of reference PSFs,
+such as Angular Differential Imaging (ADI, Marois et al. 2006a).
+Although the field rotation over the night of observations was suffi-
+cient enough to use these to remove noise with minimal companion
+self-subtraction, doing so would also remove the unsaturated stel-
+lar reference PSF, which is required to eliminate systematics in the
+companion photometry. It would not be possible to use the host
+star PSF prior to ADI subtraction as a photometric reference for the
+companion PSF after ADI subtraction, as the two would no longer
+share the same systematic trends. Furthermore, HD 1160 B is suffi-
+ciently bright that it can be detected at ample S/N for our purposes
+without further noise reduction.
+3.3
+Wavelength channel selection
+Although the final processed cubes contain data from 100 wave-
+length channels across the observed wavelength range of 2.8-4.2 µm,
+not all of these channels are suitable for further analysis. The first
+3 and final 10 channels contain flux from the adjacent spaxel in the
+dispersion direction as an oversized spectral length is required to
+extract the spectrum at each position, so the extracted data overlaps
+slightly in the wavelength dimension (Stone et al. 2022). Further-
+more, the dgvAPP360 contains a glue layer that causes up to 100%
+absorption between 3.15 and 3.55 µm (Otten et al. 2017; Doelman
+et al. 2021). As described in Section 1.1, one of the key advantages
+of a spectrophotometric approach is the option to exclude channels
+that are known to cause systematic variability in the ‘white-light’
+curve, hence improving the companion S/N and stability in the com-
+bined image when compared to combining all wavelength channels
+without any selection. This is key to reducing large systematic ef-
+fects that may otherwise dominate the variability signals that we are
+aiming to measure. For the purposes of this technique demonstra-
+tion, we proceed using 30 sequential wavelength channels (45-74,
+spanning 3.59-3.99 µm) which all have a high throughput and do
+not lie in the regions affected by the dgvAPP360 glue absorption,
+significant telluric absorption, or the overlapping spectral traces. In
+the left panel of Figure 3 we show the median combination of these
+channels in both wavelength and time, while the centre and right
+panels respectively show example images from the median com-
+MNRAS 000, 1–24 (2022)
+
+6
+B. J. Sutlieff et al.
+0
+20
+40
+60
+Columns only
+Rows only
+Combined
+0
+20
+40
+60
+Pixels
+0
+25
+50
+0
+20
+40
+60
+0
+25
+50
+Pixels
+0
+25
+50
+100
+0
+100
+0.000
+0.005
+0.010
+0.015
+100
+0
+100
+Counts
+100
+0
+100
+125
+100
+75
+50
+25
+0
+25
+50
+75
+100
+125
+150
+175
+Counts
+Figure 2. Demonstration of the process for removing systematic row and column discontinuities present in background-subtracted ALES images, as applied
+to a single frame of data in the 52nd ALES wavelength channel (𝜆 ≈ 3.69 µm). Top panels: input frame prior to the removal of the discontinuities, which are
+faintly visible as a chequered pattern. All three panels are the same. Both HD 1160 A and B are masked. Second row: results of the third-order polynomial
+fits individually to the columns (left panel) and rows (centre panel), and the combination of both (right panel). The combination of both was produced by first
+fitting and removing the column discontinuities, and then repeating the process on the resulting image to fit and remove the rows. We show row and column fits
+to the input frame separately here to highlight their individual contribution to the original systematics. The star and the companion were masked during this
+process, and the values to be removed at their locations were found through interpolation of the fits. Third row: data frame with the discontinuities removed by
+subtracting the fits. The histograms in the bottom row show the distributions of the counts in the unmasked regions of the third row images, with the original
+noise distribution in blue and the noise distribution after the discontinuities were removed in orange. The bottom right panel shows the version used in the
+analysis.
+bined cubes in the time and wavelength dimensions only. The images
+shown are those processed using the flat frame that was smoothed
+using a Gaussian filter; the equivalent images as processed using
+the median-smoothed flat frame are visually indistinguishable from
+these.
+4
+GENERATING DIFFERENTIAL
+SPECTROPHOTOMETRIC LIGHT CURVES
+Variability arising from instrumental systematics and the effects
+of Earth’s atmosphere, such as airmass, seeing, and tellurics, con-
+taminate the raw flux of the companion. Simultaneous flux mea-
+surements of a photometric reference are required to eliminate this
+contaminant variability and produce a differential light curve of the
+companion, relative to the photometric reference. Although suitable
+photometric references are generally absent when using corona-
+graphs (Section 1), the dgvAPP360 coronagraph uniquely provides
+an image of the host star simultaneously to the companion, allowing
+the star to be used as the photometric reference when it is not satu-
+rated. Its placement in the pupil plane also makes it inherently stable
+and insensitive to tip/tilt instabilities (Otten et al. 2017; Doelman
+et al. 2022).
+MNRAS 000, 1–24 (2022)
+
+Exoplanet variability with vAPP coronagraphs
+7
+-0.8
+0.0
+0.8
+RA [arcsec]
+-0.8
+0.0
+0.8
+Dec [arcsec]
+-0.8
+0.0
+0.8
+RA [arcsec]
+-0.8
+0.0
+0.8
+RA [arcsec]
+Figure 3. Reduced images of the HD 1160 system obtained with LBT/ALES and the dgvAPP360 coronagraph after all data processing steps and wavelength
+channel selection. Left: the median combination, in both wavelength and time, of all wavelength channels in the 3.59-3.99 µm range, covering a total integration
+time of 11891 s (∼3.31 h). Centre: example image from the same data cube but median combined only in the time dimension, along the 3.64 µm wavelength
+channel. Right: example time frame (integration time = 5.4 seconds) resulting from a median combination in wavelength over the 3.59-3.99 µm wavelength
+range. All three images use the same arbitrary logarithmic colour scale, and are aligned to north. North is up, and east is to the left.
+4.1
+Aperture photometry
+We used version 1.4.0 of the Photutils Python package (Bradley et al.
+2022) to simultaneously extract aperture photometry of HD 1160 A
+and B. We carried out this process for every individual frame in
+each of the 30 wavelength channels in the 3.59-3.99 µm range
+chosen in Section 3.3, with the aim of then combining these in
+the wavelength dimension to produce the white-light flux for each
+object. Circular apertures with radii of 9 pixels (3.1 𝜆/D) and 2.5
+pixels (0.9 𝜆/D) were used for the host star and the companion,
+respectively. To estimate the background flux at the position of the
+star, we also extracted photometry in a circular annulus centred on
+the stellar location with inner and outer radii of 11 and 16 pixels,
+respectively. It was not possible to use this method to estimate the
+background flux at the position of the companion as the companion
+lies close to the edge of the field of view in some frames, limiting
+the space available to place an annulus that would be statistically
+wide enough. We therefore instead followed an approach used by
+Biller et al. (2021), estimating the background at the companion
+location by masking the companion and extracting flux in a circular
+annulus centred on the host star, with a width of 6 pixels at the radial
+separation of the companion. As most of the residual background in
+each frame was eliminated by the data processing steps in Section
+3.2, these background values are close to zero. These apertures and
+annuli are shown in Figure 4, overlaid on a single processed time
+frame of data in the 52nd ALES wavelength channel (𝜆 ≈ 3.69 µm).
+We then removed the residual background from our stellar and
+companion flux measurements by multiplying the mean flux per
+pixel in the background annuli by the area of the corresponding
+apertures and subtracting the resulting values from the aperture
+photometry. We then produced single white-light measurements for
+both the companion and the star at each time frame by taking the
+median combination of the photometric measurements across the 30
+wavelength channels. These raw time series, uncorrected for shared
+variations introduced by Earth’s atmosphere (i.e. before division),
+are shown in grey in the top two panels of Figure 5. The discrete
+gaps in integration reflect time spent off-target due to the two-point
+on/off nodding pattern used to enable background subtraction. We
+also plot the data binned to 18 minutes of integration time. We
+binned the data by taking the median value in each time bin. The
+error on the binned fluxes are the Gaussian approximation of the root
+mean square (RMS) i.e. median absolute deviation (MAD) × 1.48
+of the flux measurements inside each bin divided by
+√
+𝑁 − 1, where
+𝑁 is the number of frames per bin. Next, we removed variability due
+to Earth’s atmosphere and other systematics from the unbinned raw
+flux of the companion using the unbinned raw flux of the host star,
+which acts as a simultaneous photometric reference. By dividing the
+unbinned companion flux by the unbinned stellar flux, we eliminate
+trends common to both and produce a differential light curve that
+only contains non-shared variations. Assuming that the host star is
+not itself varying (see Section 4.2), the resulting differential light
+curve reflects the intrinsic variability of the companion plus any
+contamination arising from non-shared systematics. We show this
+raw differential light curve in the third panel of Figure 5. The bottom
+panel shows a zoomed-in view of the binned version, with tighter
+limits on the y-axis.
+In the following sections we examine a number of physical,
+instrumental, and processing factors that may be correlated with
+non-astrophysical features in the differential white-light curve, and
+in Section 5 we model and remove non-shared variations arising
+from some of these factors.
+4.2
+TESS light curves of host star HD 1160 A
+Although the vast majority of A-type stars generally vary well below
+the ∼1% level, a small fraction vary at a much higher level, showing
+up to ∼15% variations (Ciardi et al. 2011). To test our assumption
+that the host star HD 1160 A is not varying at a level that impacts our
+differential light curve, we used data from the TESS mission, which
+is publicly available on the Mikulski Archive for Space Telescopes
+(MAST) data archive1. The TESS mission observed HD 1160 A
+for 25 days in Sector 42 (from 2021 August 21 to 2021 September
+14) and for 26 days Sector 43 (from 2021 September 16 to 2021
+1 MAST data archive portal: https://mast.stsci.edu/portal/
+Mashup/Clients/Mast/Portal.html
+MNRAS 000, 1–24 (2022)
+
+8
+B. J. Sutlieff et al.
+-1.0
+0.0
+1.0
+RA [arcsec]
+-1.0
+0.0
+1.0
+Dec [arcsec]
+Figure 4. The apertures (continuous lines) and annuli (dashed lines) used to
+extract photometry and background measurements for host star HD 1160 A
+(in yellow) and companion HD 1160 B (in orange). The image is a single time
+frame from the 52nd ALES wavelength channel (𝜆 ≈ 3.69 µm). North is up,
+and east is left. The orange aperture is placed at the location of HD 1160 B,
+which is too faint to be visible in a single frame. The companion was masked
+when extracting photometry in the annulus for the companion background.
+October 11), 51 days in total, with 2 minute cadence. The TESS de-
+tector bandpass covers a broad-band wavelength range of 0.6-1.0 µm
+(Ricker et al. 2015), which does not overlap with our LBT/ALES
+observations in the 2.8-4.2 µm range. However, as stars are generally
+less variable in the infrared than in the optical regime, any variations
+in the TESS light curve of HD 1160 A should represent an upper
+limit for its variability at the wavelengths covered by ALES (e.g.
+Solanki & Unruh 1998; Unruh et al. 1999; Fröhlich & Lean 2004;
+Davenport et al. 2012; Goulding et al. 2012; Ermolli et al. 2013;
+Rackham et al. 2022). Each TESS pixel covers 21′′ on sky. This
+means that HD 1160 A, HD 1160 B (at a separation of ∼0.78′′), and
+HD 1160 C (at a separation of ∼5.1′′) are not resolved separately
+in the TESS images and appear as a single object. However, as both
+HD 1160 B (Δ𝐽 = 8.85 ± 0.10 mag, Δ𝐿′ = 6.35 ± 0.12 mag) and C
+(Δ𝐽 = 6.33 ± 0.04 mag, Δ𝐿′ = 4.803 ± 0.005 mag) are far fainter
+than HD 1160 A, especially at shorter wavelengths, it is reasonable
+to assume that flux of HD 1160 A will dominate in the TESS data
+(Nielsen et al. 2012).
+We first masked out any bad quality exposures using the one-
+hot encoded quality mask in the ‘QUALITY’ keyword in the header
+of the light curve files provided by the TESS Science Processing
+Operations Center (SPOC, Jenkins et al. 2016) on MAST. We then
+used the ‘CROWDSAP’ keyword in the header to get an estimate
+of the ratio of target flux to total flux in the optimal aperture used
+for the PDC SAP (Pre-search Data Conditioning Simple Aperture
+Photometry) flux (e.g. Panwar et al. 2022b). The ‘CROWDSAP’
+value for sector 42 and 43 indicates that 0.17% and 0.2% flux is
+from dilution by nearby sources. We subtract the estimated diluted
+flux from each exposure in both the sectors. The resultant light
+curves for both the sectors are shown in Figure 6. Although the
+TESS observations are not contemporaneous with our LBT/ALES
+observations, we do not see variations above 0.03% in the light
+curve of HD 1160 A over the timescale covered by the two TESS
+sectors (51 days). As this is far smaller than the precision of our
+differential light curve, we proceed with the assumption the host star
+HD 1160 A is non-varying within the flux precision of our analysis
+of the variations in the light curve.
+4.3
+Impact of wavelength calibration and flat-field smoothing
+In Section 3.1, we described the process used to perform the wave-
+length calibration of the raw data and to extract the micro-spectra
+into a three-dimensional image cube. This step was repeated sepa-
+rately using the wavelength calibration that was the most divergent
+of the 6 obtained throughout the observing sequence, i.e. the 3.9 µm
+fiducial spots for this wavelength calibration were the most signifi-
+cantly offset compared to the one that was originally used. Over the
+course of the night, the projection of the micro-spectra onto the de-
+tector drifts slightly. If this drift is significant then a particular wave-
+length calibration may not remain accurate for the entire observing
+sequence, potentially producing a false variability signal when the
+wrong part of the spectrum is assigned to a given channel. Repeat-
+ing our spectral extraction using a wavelength calibration obtained
+at a different point during the observations allows us to test whether
+this effect has a significant impact on the photometry of the target
+star and the companion. After extracting the micro-spectra using
+the alternative wavelength calibration, we then processed the data
+again in full to produce an alternative differential light curve. The
+original and alternative wavelength calibrations were obtained at 4
+hours 57 minutes (08:25:00 UT) and 1 hour 26 minutes (04:54:00
+UT) after the beginning of the observing sequence, respectively. We
+plot the resulting alternative stellar and companion fluxes, and the
+differential white-light curve (binned to 18 minutes) in Figure 7,
+alongside the originals from Figure 5 for comparison. The differ-
+ential light curves in each case are consistent within 1𝜎, indicating
+that the extracted photometry is sufficiently robust to changes in the
+wavelength calibration and sub-pixel mis-registration of the spatial
+profiles for each microspectrum.
+In Section 3.2, we described our method for applying a flat-
+field correction to the data to calibrate for the non-uniform response
+of the detector. Incorrect flat-fielding can lead to a false variability
+signal if the companion moves over regions of the detector with a
+non-uniform response that has not been properly calibrated. To test
+the robustness of our differential light curve to differences in the
+flat used we processed the data in two separate streams, using a flat
+that had been smoothed over using a Gaussian filter and a median
+filter, respectively. We plot the resulting differential light curves in
+Figure 8 for comparison. The two differential light curves are in
+close agreement and every binned data point lies well within their
+1𝜎 error bars, indicating that the method for producing the flat is
+robust and does not significantly affect the final images or extracted
+photometry of the star or companion.
+5
+DETRENDING THROUGH LINEAR REGRESSION
+Trends shared by the star and companion fluxes are removed in
+the differential light curve (see bottom panel of Figure 5), but any
+non-shared trends will still be present, including the intrinsic com-
+panion variability signal that we aim to measure. To improve our
+sensitivity to the companion’s variability, we needed to remove
+the non-astrophysical residual trends in the differential light curve,
+which can arise from both telluric and instrumental sources. In
+MNRAS 000, 1–24 (2022)
+
+Exoplanet variability with vAPP coronagraphs
+9
+0.8
+0.9
+1.0
+1.1
+Raw white-light Star-only
+Raw white-light Star-only (18 min. bins)
+1
+0
+1
+2
+3
+Raw white-light Companion-only
+Raw white-light Companion-only (18 min. bins)
+1
+0
+1
+2
+3
+Raw white-light Companion/Star
+Raw white-light Companion/Star (18 min. bins)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (hours)
+0.8
+0.9
+1.0
+1.1
+1.2
+Raw white-light Companion/Star (18 min. bins)
+Normalised Flux
+Figure 5. The top two panels show the normalised raw white-light flux (𝜆 =3.59-3.99 µm wavelength range) of host star HD 1160 A (in grey, top panel) and
+companion HD 1160 B (in grey, second panel). The blue and orange lines are the same fluxes of the star and companion, respectively, binned to 18 minutes
+of integration time per bin. Each has been normalised by dividing by the mean value across the full sequence. The third panel shows, in grey, the resulting
+differential white-light curve when the white-light of the companion is divided by the white-light of the star to remove trends shared by both. The same data
+binned to 18 minutes of integration time per bin is shown in purple. The light curve in the bottom panel is the same as the third panel but zoomed in on the
+y-axis with a dashed line at normalised flux = 1 for clarity. Provided that there is no contamination from stellar variability, variations in this light curve are a
+combination of any intrinsic companion variability and trends arising from non-shared systematics. The gaps in the data are due to the two-point on/off nodding
+pattern used to collect sky frames for background subtraction.
+MNRAS 000, 1–24 (2022)
+
+10
+B. J. Sutlieff et al.
+Figure 6. TESS 2 minute cadence PDC SAP light curve of HD 1160 A
+obtained from TESS sectors 42 (black points) and 43 (red points). Light
+curves from both sectors have been normalized to the median count level for
+the respective sector to remove the systematic change in the base flux level
+from one sector to the next. Overplotted is the binned TESS light curve for
+visual aid. The standard deviation of the stellar flux is 0.027% and 0.03%
+for sectors 42 and 43, respectively.
+the field of exoplanet transmission spectroscopy, such systematic
+trends are generally modelled and removed from light curves using
+either a polynomial model created by simultaneously fitting several
+decorrelation parameters (e.g. de Mooij & Snellen 2009; de Mooij
+et al. 2011; Brogi et al. 2012; Stevenson et al. 2014; Diamond-
+Lowe et al. 2018, 2020a; Todorov et al. 2019), or a non-parametric
+model produced using Gaussian processes (e.g. Gibson et al. 2012,
+2013; Evans et al. 2013, 2015; Montet et al. 2016; Carter et al.
+2020; Diamond-Lowe et al. 2020b; Panwar et al. 2022a,b). Un-
+like traditional transmission spectroscopy observations, our target
+is significantly fainter than the simultaneous reference that we use
+for detrending. Furthermore, the target was not pixel-stabilised for
+these observations and moved across the detector throughout the
+night, so we might predict that the measured light curves could be
+significantly correlated with the change in position of the compan-
+ion and the star on the detector over time. Knowing how to remove
+these systematics is key to obtaining high-precision light curves in
+future observations of directly imaged exoplanets. For space-based
+observations the instrumentation is generally sufficiently stable such
+that the systematics are repeatable over time, allowing them to be
+well characterised. This is more challenging for ground-based ob-
+servations, like those here, which are inherently less stable as Earth’s
+atmosphere introduces systematics that can vary night by night.
+As a basic demonstration of how to remove such residual trends
+from ground-based differential light curves of directly imaged plan-
+ets, we here used a multiple linear regression approach to simul-
+taneously fit several possible sources of systematics. This is not
+intended as a strictly rigorous statistical analysis of the trends in
+the light curve, but is done to perform an initial investigation into
+which parameters have the greatest impact and to illustrate an exam-
+ple approach of how to do this for future observations. As studies
+move towards increasing precision to measure smaller amplitude
+variability, one might instead consider approaches using Gaussian
+processes or similar.
+An investigation of the LBT telemetry and white light im-
+ages revealed eight physical and instrumental factors, shown plotted
+against time in Figure 9, that varied notably during the observing
+sequence and may be correlated with the residual trends in the dif-
+ferential light curve. We therefore included these parameters in the
+linear regression. The first three of these were air temperature, wind
+speed, and wind direction, shown in the three panels on the left-hand
+side of Figure 9. We also considered airmass, which is shown in
+the top-right panel. While the light from the companion and its host
+star pass through almost identical airmass (maximum difference
+∼10−5), their significantly different colours mean that atmospheric
+extinction due to absorption and e.g. Rayleigh scattering can result
+in a differing airmass dependence, even when such scattering effects
+are reduced at our longer 2.8-4.2 µm wavelength range (Allen 1955;
+Broeg et al. 2005; Croll et al. 2015; Panwar et al. 2022a).
+The remaining four parameters included in the linear regres-
+sion were the x- and y- pixel positions of the star and the companion
+in the images prior to spatial and rotational alignment, shown in the
+centre-right and bottom-right panels of Figure 9. The dgvAPP360
+is located in the pupil plane, meaning that drifts in the locations
+of the target PSFs on the detector do not affect its response and
+performance as it applies the phase modification to every source in
+the field of view (Otten et al. 2017; Doelman et al. 2022). How-
+ever, systematics could still be introduced by such drifts if there are
+variations in the instrumentation or detector response. A number of
+sharp discontinuities in the y-position, and a singular discontinuity
+in the x-direction, can be seen in Figure 9. These discontinuities
+are the result of manual positional offsets applied during the ob-
+servations to keep the star close to the centre of the small (∼2.2′′x
+2.2′′) ALES field of view. These offsets were always applied while
+in the off-source nod position, and always along one axis at a time.
+The largest discontinuities in the x- and y-positions correspond to
+shifts of 0.1′′ along the given axis. Neglecting the discontinuities,
+the drift of the stellar PSF in both the x- and y-directions follow
+arcs with turn-overs approximately 4.5 hours into the observing se-
+quence. This slow drift is correlated with the pointing altitude of
+the telescope and arises from flexure of the ALES lenslet array as
+the telescope rotates. As the observations were obtained in pupil-
+stabilized mode such that the field of view was rotating over time,
+the change in position of the companion throughout the data has an
+additional rotational component compared to the star. The drift aris-
+ing from the flexure of the lenslet array therefore instead produces
+an inflection point ∼4.5 hours in the case of the companion.
+We used the linear regression tools in version 1.0.2 of the
+scikit-learn Python package (Pedregosa et al. 2011) to simultane-
+ously fit these input parameters and produce a model fit to our
+differential light curve. The coefficients of the linear model pro-
+duced by the linear regression are shown in Table 2, and the model
+itself is shown in green in the top panel of Figure 10, relative to the
+raw differential white-light curve in grey. We then divided the raw
+differential light curve by this model to produce a detrended version,
+shown in red in the bottom panel of Figure 10. We also overplot the
+raw differential light curve from the bottom panel of Figure 5, prior
+to detrending, in purple to allow the two to be compared. We also
+repeated this detrending process to produce detrended differential
+light curves for each of the 30 individual wavelength channels over
+the 3.59-3.99 µm wavelength range that comprise the white-light
+curve. These are shown in Figure 11, again binned to 18 minutes.
+We note that while ALES has a resolution of R∼40, the raw data
+were spectrally extracted into 100 wavelength channels and so there
+is some correlation between wavelength channels.
+MNRAS 000, 1–24 (2022)
+
+300
+S
+electrons/s
+200
+100
+NMMMM
+0
+xn
+-100
+-200
+-300
+HD 1160 A, Sector 42
+-400
+HD 1160 A. Sector 43
+450
+460
+480
+470
+490
+500
+Time[BiD]
++2.459e6Exoplanet variability with vAPP coronagraphs
+11
+0.94
+0.96
+0.98
+1.00
+1.02
+1.04
+Raw white-light Star-only (original wavecal.)
+Raw white-light Star-only (alternate wavecal.)
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+Raw white-light Companion-only (original wavecal.)
+Raw white-light Companion-only (alternate wavecal.)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (hours)
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+Raw white-light Companion/Star (original wavecal.)
+Raw white-light Companion/Star (alternate wavecal.)
+Normalised Flux
+Figure 7. Impact of different wavelength calibrations on the star-only and companion-only fluxes, and the differential white-light curve, shown in 18 minute
+binning. The alternate wavelength calibration was chosen as the one that diverged most from the one originally used. The times at which the wavelength
+calibrations were obtained are indicated by vertical dashed lines in the same colours as the corresponding light curve.
+6
+RESULTS
+The detrended differential white-light curve (Figure 10) shows
+sinusoidal-like variability over a timescale of a few hours. In the
+binned light curve, the normalised flux ranges from a minimum of
+0.91 at 0.874 hours to a maximum of 1.13 at 2.961 hours. To better
+allow us to estimate the differential precision that we achieve in our
+light curve, we fitted the variability and removed it from our light
+curve. As the variability signal appears periodic, we used the NASA
+Exoplanet Archive periodogram service2 to apply the Lomb-Scargle
+algorithm to the unbinned detrended differential white-light curve
+2 https://exoplanetarchive.ipac.caltech.edu/cgi-bin/
+Pgram/nph-pgram
+and thereby search for sinusoidal periodic signals (Lomb 1976;
+Scargle 1982).
+The strongest peak in the resulting periodogram (top panel,
+Figure 12) is at a period of 3.242 hours. We then fit a sinusoid to the
+light curve using this period as an initial guess, which returned a
+function with the same period (3.239), a semi-amplitude of 0.088, a
+phase shift of 0.228, and a 𝑦-offset of 0.993. This sinusoid is shown
+overplotted on the detrended light curve in the centre panel of Fig-
+ure 12. The differential light curve was then divided by the fitted
+sinusoid to remove the variability signal to the first order (centre
+panel residuals, Figure 12). The bottom panel of this figure is the
+same as the panel above but with the data phase-folded to the period
+of the fitted sinusoid. Next, we followed the method of Kipping &
+Bakos (2011) to assess the degree of “red” (correlated) noise in our
+light curve. We binned our detrended differential white-light curve,
+MNRAS 000, 1–24 (2022)
+
+12
+B. J. Sutlieff et al.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (hours)
+0.8
+0.9
+1.0
+1.1
+1.2
+Raw white-light Companion/Star (Gaussian-smoothed flat)
+Raw white-light Companion/Star (Median-smoothed flat)
+Normalised Flux
+Figure 8. The raw differential companion/star white-light curves that are
+produced when the flat-field correction uses a flat that was smoothed using
+a median filter and a Gaussian filter, in turquoise and purple, respectively.
+The latter light curve is the same as that in the bottom panel of Figure 5,
+reproduced here for comparison.
+Table 2. The decorrelation parameters 𝑥𝑖 used for the linear regression and
+the corresponding coefficients 𝑐𝑖 and intercept 𝑐0 of the resulting linear
+model fit to the raw differential light curve. The linear model fit is then
+given by 𝑦 = (�𝑛
+𝑖=1 𝑐𝑖 𝑥𝑖) + 𝑐0.
+Parameter (𝑥𝑖)
+Value (𝑐𝑖)
+Airmass
+0.28782524
+Air temperature
+0.10690617
+Star x-position
+0.04819219
+Star y-position
+-0.04368692
+Companion x-position
+-0.02608288
+Companion y-position
+0.02129148
+Wind speed
+0.00059976
+Wind direction
+0.00050927
+Intercept (𝑐0)
+-1.318553799
+with the sinusoid removed, to a range of bin sizes, before normal-
+ising and subtracting one to centre around zero. We then measured
+the RMS of each resulting binned light curve. These RMS values
+are plotted against bin size in Figure 13, alongside the expectation
+of independent random numbers as a function of bin size i.e. the
+white noise. For our chosen binning of 18 minutes of integration
+time per bin, which has a bin size of 200 frames per bin, we find
+an RMS of 0.037. For comparison, the RMS of the detrended dif-
+ferential white-light curve prior to the removal of the sinusoid, but
+with the same binning, is 0.073. We therefore conclude that the
+light curve of HD 1160 B shows variations with a semi-amplitude
+of ∼8.8% or peak-to-peak amplitude of ∼17.6%, and that the dif-
+ferential precision achieved in the binned light curve is at the 3.7%
+level. The amplitude of the variations is therefore above the mea-
+sured precision. Furthermore, this estimate of the precision is likely
+a conservative one; the variability signal is unlikely to have been
+perfectly removed by the sine fit and so the measured RMS values
+may be higher than the true limiting precision. A caveat of this result
+is that the baseline of our observations is only ∼7.81 hours, so we
+only cover ∼2.4 periods. Additional data is therefore needed to con-
+firm the periodicity and amplitude of the variability of HD 1160 B.
+We discuss these results further and compare the precision achieved
+to similar studies in the literature in Section 7.2.
+7
+DISCUSSION
+In this section we discuss the relative impact of the decorrelation
+parameters on our results, and the physical explanations of the sys-
+tematics they introduce. We also compare the precision that we
+achieve to other variability studies in the literature that use differ-
+ent techniques, and discuss the potential application of differential
+spectrophotometry in future work.
+7.1
+Impact of decorrelation parameters
+Using the linear regression coefficients of each parameter from Ta-
+ble 2, we can assess which parameters have the greatest impact on
+the light curve of HD 1160 B. The small angular separation of the
+companion and star might suggest that the effect of airmass would
+be small. Airmass can be an important systematic for differential
+spectrophotometric observations where other stars are used as the
+simultaneous photometric reference as the angular separation be-
+tween the target and the reference can be large, causing the light
+from the two objects to pass through different atmospheric columns
+(e.g. Broeg et al. 2005; Panwar et al. 2022a). However, in our case
+we use the companion’s host star as the photometric reference, so
+the angular separation between the two is much smaller than is gen-
+erally the case for observations using reference stars in the field.
+However, there is a significant colour difference between the star
+and the companion, which leads to different degrees of extinction at
+a given airmass. Therefore, we expect an airmass dependence and
+indeed it has the largest coefficient in the detrending model. Similar
+extinction effects are often seen in studies of transiting exoplan-
+ets, and can also exhibit a non-linear wavelength dependence (e.g.
+Panwar et al. 2022a).
+We also find the air temperature at LBT to be one of the
+parameters that is most correlated with our differential light curve.
+As the air temperature changes, this can potentially cause slight
+changes in the optical path of the telescope or instrument that lead
+to this correlation.
+We further predicted that the positions of the companion and
+the star could introduce significant non-shared systematics to our
+measurements; as HD 1160 A was not pixel-stabilised during our
+observations, we are sensitive to both intra- and interpixel varia-
+tions as the star and the companion move across the detector. Both
+the star and the companion changed position on the detector due
+to flexure of the ALES lenslet array as the instrument moved and
+positional offsets applied intentionally to keep the target close to
+the centre of the field of view. We also nodded on and off of the
+target to enable background subtraction. While the process of nod-
+ding is itself relatively accurate, it is not repeatable at a sub-pixel
+pointing precision, introducing a slight offset error between nods.
+Furthermore, LBTI data is always pupil-stabilized, so the field of
+view was rotating throughout the night. Although it may have been
+optimal to fix the star and companion positions to the same detector
+pixels for the duration of the observations, this was not possible due
+to the effect of the lenslet array flexure and the lack of instrument
+derotator in the LBTI architecture (Doelman et al. 2022). However,
+we find that these positional changes are not the most correlated
+with our differential light curve. It is possible that this is because
+the light from the star and companion is spread out across multiple
+detector pixels when spectrally dispersed, reducing the impact of
+MNRAS 000, 1–24 (2022)
+
+Exoplanet variability with vAPP coronagraphs
+13
+11.00
+11.25
+11.50
+11.75
+12.00
+12.25
+12.50
+12.75
+13.00
+Air Temperature (°C)
+1.2
+1.4
+1.6
+1.8
+2.0
+2.2
+Airmass
+0
+1
+2
+3
+4
+5
+Wind speed (m s
+1)
+30
+32
+34
+36
+38
+Pixel position
+Star y-position
+Star x-position
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (hours)
+200
+250
+300
+350
+400
+Wind direction (degs E of N)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (hours)
+10
+20
+30
+40
+50
+60
+Pixel position
+Companion y-position
+Companion x-position
+Figure 9. The decorrelation parameters used when detrending the differential light curve via linear regression. From top to bottom, the left panels show the air
+temperature (in ℃), wind speed (in m s−1), and wind direction (in degrees east of north) at the observatory as a function of time, as extracted from the FITS
+headers of the raw data. The panels on the right show airmass, the x- and y-positions of the star in pixels, and the x- and y-positions of the companion in pixels,
+as a function of time. The gaps in time reflect the on/off nodding pattern of the observations. The companion and star positions are those in the images prior
+to spatial and rotational alignment, and the sharp discontinuities in pixel position are due to manual offsets applied to keep the star close to the centre of the
+small field of view. The stellar positions follow arc-shaped trends aside from these discontinuities, which correlate with the pointing altitude of the telescope
+and arise from flexure of the ALES lenslet array as the telescope rotates throughout the night. The change in the companion position has an additional trend
+due to the 109.7° rotation of the field of view over the observing sequence.
+MNRAS 000, 1–24 (2022)
+
+14
+B. J. Sutlieff et al.
+1
+0
+1
+2
+3
+Raw white-light Companion/Star
+Linear regression model
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Time (hours)
+0.80
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+Raw white-light Companion/Star (18 min. bins)
+Companion/Star, model removed (18 min. bins)
+Normalised Flux
+Figure 10. The model produced by the linear regression using the decorrelation parameter coefficients (in Table 2) is shown in the top panel in green alongside
+the raw differential light curve in grey. The bottom panel then shows in red the light curve produced when the raw differential light curve is divided by the
+linear regression model to remove the modelled trends that are not shared by the stellar and companion fluxes, binned to 18 minutes of integration time per bin.
+The raw differential light curve (i.e. prior to detrending) is also shown faintly in purple, reproduced for comparison from the bottom panel of Figure 5.
+any systematic issues arising from any single pixel. The light of our
+target is dispersed across wavelength, similar to a technique often
+used for high-precision differential photometry whereby a telescope
+is intentionally defocused to disperse starlight over the detector (e.g.
+de Mooij et al. 2011, 2013; Crossfield et al. 2012; Croll et al. 2015).
+This has the effect of reducing systematics due to intrapixel varia-
+tions and minimises the impact of any residual flat-field errors. The
+step of recombining our data into a white-light curve may therefore
+have helped to reduce systematic trends that would otherwise have
+been introduced by the positional movement of the targets through-
+out the night. The use of a spectrograph also has the additional
+benefit of allowing us to remove individual wavelength channels
+that are found to contain defects. For this dataset, this allowed us
+to leave out channels affected by overlapping spectral traces, sig-
+nificant telluric absorption, and absorption by the glue layer of the
+dgvAPP360, as described in Section 3.3.
+Lastly, we find that the wind speed and direction are the least
+correlated with our differential light curve, but note that the condi-
+tions on the night were exceptionally stable and so cannot rule out
+that these factors could have an impact in less optimal conditions.
+For future observations, residual systematics in differential
+light curves of directly imaged companions could be removed us-
+ing more advanced methods from the exoplanet transmission spec-
+troscopy and high-precision secondary eclipse literature such as
+fitting the data using a Gaussian process regression (e.g. Gibson
+et al. 2012; Evans et al. 2017; Nikolov et al. 2018a,b; Diamond-
+Lowe et al. 2020a,b; Wilson et al. 2021). In general, methods used
+to identify trends in transmission spectroscopy data also include a
+model of the exoplanet transit itself, allowing the transit to be de-
+tected even when the strength of the signal is very low. However,
+this is possible because the expected shape of the signal is well un-
+derstood. This is more difficult for searches for variability in directly
+imaged companions, where the expected shape of the variability sig-
+nal is not necessarily well-known in advance. Furthermore, many
+of these methods assume a linear relationship between systematics
+and trends in the light curve, while the telluric and instrumental sys-
+tematics present in time-series data can be complex and non-linear.
+In the future, an optimal approach should account for the functional
+form of the correlated parameters (Panwar et al. 2022a,b).
+MNRAS 000, 1–24 (2022)
+
+Exoplanet variability with vAPP coronagraphs
+15
+0
+2
+4
+6
+8
+Time (hours)
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+ = 3.59 m, RMS = 0.221
+ = 3.605 m, RMS = 0.13
+ = 3.62 m, RMS = 0.15
+ = 3.635 m, RMS = 0.15
+ = 3.65 m, RMS = 0.167
+ = 3.664 m, RMS = 0.158
+ = 3.679 m, RMS = 0.103
+ = 3.693 m, RMS = 0.149
+ = 3.708 m, RMS = 0.23
+ = 3.722 m, RMS = 0.122
+Companion/Star, model removed (18 min. bins)
+Normalised Flux
+0
+2
+4
+6
+8
+Time (hours)
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+ = 3.736 m, RMS = 0.163
+ = 3.75 m, RMS = 0.163
+ = 3.764 m, RMS = 0.247
+ = 3.778 m, RMS = 0.259
+ = 3.791 m, RMS = 0.209
+ = 3.805 m, RMS = 0.312
+ = 3.819 m, RMS = 0.357
+ = 3.832 m, RMS = 0.12
+ = 3.846 m, RMS = 0.123
+ = 3.859 m, RMS = 0.176
+Normalised Flux
+0
+2
+4
+6
+8
+Time (hours)
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+ = 3.872 m, RMS = 0.241
+ = 3.885 m, RMS = 0.314
+ = 3.898 m, RMS = 0.398
+ = 3.911 m, RMS = 0.564
+ = 3.924 m, RMS = 0.36
+ = 3.937 m, RMS = 0.298
+ = 3.95 m, RMS = 0.34
+ = 3.963 m, RMS = 0.361
+ = 3.976 m, RMS = 0.201
+ = 3.988 m, RMS = 0.335
+Normalised Flux
+Figure 11. In red, we show the detrended versions of the individual differential light curves in each of the 30 wavelength channels that were combined to
+produce the white-light curve. These wavelength channels cover 𝜆 =3.59-3.99 µm, and are binned to 18 minutes of integration time per bin. An offset factor of
+2 has been applied between each light curve to separate them from each other. Overall variability appears to increase with longer wavelength.
+7.2
+Differential light curves
+7.2.1
+Variability interpretation
+In Section 6 we found that HD 1160 B shows variations with a
+semi-amplitude of ∼8.8% (or peak-to-peak amplitude of ∼17.6%).
+To compare this result to literature observations of similar objects,
+we must consider both spectral type and the wavelengths at which
+variability has been observed. Vos et al. (2022) found that virtually
+all L dwarfs are likely to be variable at the 0.05-3% range, and
+several studies have measured higher variability, up to 26% (e.g.
+Radigan et al. 2012; Radigan 2014; Lew et al. 2016; Biller et al.
+2018; Bowler et al. 2020b). However, there is evidence that brown
+dwarf variability amplitude may have a a strong wavelength depen-
+dence. For example, HST observations of highly variable L dwarf
+companion VHS 1256-1257 b identified a large variability ampli-
+tude of 24.7% at 1.27 µm, while Spitzer observations at 4.5 µm
+found a far lower amplitude of 5.76±0.04% (Bowler et al. 2020b;
+Zhou et al. 2020b; Miles et al. 2022). Zhou et al. (2020b) do note,
+however, that the HST and Spitzer observations were not obtained
+contemporaneously and so the atmosphere of VHS 1256-1257 b and
+hence its variability properties are likely to have changed substan-
+tially over the intervening timescale. Comparisons of large surveys
+also suggest that variability amplitudes are lower in the mid-infrared
+than in the near-infrared, although there is evidence for a weaker
+wavelength dependence and enhanced mid-infrared variability am-
+plitudes for the young isolated brown dwarfs most similar to sub-
+stellar companions (e.g. Radigan et al. 2014; Metchev et al. 2015;
+Biller et al. 2018; Vos et al. 2022).
+The amplitude of the L-band variability we measure for
+HD 1160 B is quite extreme compared to literature results, but
+it is important to note that there have been very few variability stud-
+ies for substellar companions in the mid-infrared, making direct
+comparison difficult. High-amplitude variability in brown dwarfs is
+generally attributed to heterogeneous surface features, such as spots
+or clouds of varying thickness, rotating in and out of view as the
+object rotates (e.g. Apai et al. 2013; Biller 2017; Artigau 2018).
+Some light curves show more complex features that cannot be mod-
+elled with a single atmospheric feature, or features that evolve over
+short or long timescales (e.g. Artigau et al. 2009; Metchev et al.
+2015). These features and time evolution may arise from changing
+weather systems, or bands of clouds which rotate within the target’s
+atmosphere and generate waves on a global scale (e.g. Apai et al.
+2017; Tan & Showman 2021). For HD 1160 B, observations over
+MNRAS 000, 1–24 (2022)
+
+16
+B. J. Sutlieff et al.
+Time (hours)
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+1.30
+Flux
+Detrended Companion/Star
+Detrended Companion/Star (18 min. bins)
+Sinusoidal fit
+0
+1
+2
+3
+4
+5
+6
+7
+Time (hours)
+0.9
+1.0
+1.1
+1.2
+Flux
+Phase
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+1.20
+1.25
+Flux
+Detrended Companion/Star
+Detrended Companion/Star (18 min. bins)
+Sinusoidal fit
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Phase
+0.9
+1.0
+1.1
+1.2
+Flux
+Figure 12. Top panel: the Lomb-Scargle periodogram for the unbinned
+detrended differential white-light curve. The strongest peak is at a period
+of 3.242 hours. Centre panel: the same detrended differential white-light
+curve from the bottom panel of Figure 10, unbinned in grey and binned
+to 18 minutes of integration time per bin in red. The blue line is the fitted
+sinusoid with a semi-amplitude of 0.088 and a phase shift of 0.228. The
+residuals when the fitted sinusoid is divided out from the light curves are
+shown underneath. Bottom panel: the same as the panel above, but phase-
+folded to a period of 3.24 hours.
+a longer baseline are required to be able to characterise any time
+evolution in the variability signal.
+However, the spectral type of HD 1160 B is unclear as it has
+a highly peculiar spectrum that cannot be satisfactorily fit with
+spectral models or templates in current libraries, and some studies
+suggest that it could instead be a late-M dwarf (Maire et al. 2016;
+Garcia et al. 2017; Mesa et al. 2020). If HD 1160 B is an M dwarf,
+its variability would most likely arise from cool star spots caused
+by magnetic activity in its photosphere, which are common in M-
+100
+101
+102
+103
+Frames per bin (bin size)
+10
+2
+10
+1
+100
+RMS
+Light curve RMS after linear regression
+and sinusoidal variability removed
+White noise model
+Figure 13. The RMS of the binned detrended differential white-light curve,
+after removing sinusoidal variability, is shown in orange as a function of
+bin size. The black line shows the theoretical white noise model, or the
+expectation of independent random numbers for a given bin size.
+dwarfs and would rotate in and out of view in much the same way
+as the cloud features of lower mass objects (e.g. Barnes et al. 2002;
+Frasca et al. 2009; Scholz et al. 2009; Goulding et al. 2012; Ghosh
+et al. 2021; Johnson et al. 2021). The properties of such variability
+can be highly dependent on the spot distribution and fractional spot
+coverage of a given object; some M-dwarfs have a very high cov-
+erage with multiple starspots covering as much as 20-50% of their
+fractional surface area inhomogeneously (e.g. O’Neal et al. 2004;
+Morales et al. 2010; Irwin et al. 2011; Goulding et al. 2012; Jackson
+& Jeffries 2013). Spot-induced variability amplitudes for M-dwarfs
+generally range from the subpercent level up to around ∼5% (e.g.
+Rockenfeller et al. 2006; Charbonneau et al. 2009; Birkby et al.
+2012; Nefs et al. 2013; de Mooij et al. 2012). Although observed far
+less often in the infrared compared to the optical, flaring events can
+induce far stronger variability in M-dwarfs at amplitudes ranging
+from the subpercent level up to tens of percent (e.g. Tofflemire et al.
+2012; Goulding et al. 2012). Our measured variability amplitude for
+HD 1160 B is therefore also on the higher end of what has been ob-
+served for earlier spectral types such as late M- and early L-dwarfs,
+barring flares, although we again note the lack of literature studies
+of similar objects in the L-band.
+While the variability observed for HD 1160 B appears high,
+another point to consider is its orbital inclination, which the latest or-
+bital fits suggest is close to edge-on as viewed from Earth (92.0+8.7
+−9.3°,
+Bowler et al. 2020a). If the obliquity of HD 1160 B is aligned with
+its orbit such that we are viewing its rotation close to edge-on, the
+observed variability amplitude is likely to compose a much larger
+fraction of its true variability compared to if it were viewed face-on.
+Indeed, (Vos et al. 2017) demonstrated that the highest variability
+amplitudes are seen for targets with close to edge-on viewing angles.
+If we interpret the 3.24 h sinusoidal variation we observed as
+the true rotation period of HD 1160 B, we can further consider
+this within physical limitations. The breakup period of a rotating
+object is dependent on its radius, which is itself age-dependent, and
+mass. Both age and mass are poorly constrained for HD 1160 B:
+literature results place the system’s age in the 10-300 Myr range
+(Nielsen et al. 2012; Maire et al. 2016; Curtis et al. 2019), and
+mass estimates range from ∼20 MJup to 123 MJup (Curtis et al.
+2019; Mesa et al. 2020). Vos et al. (2020) calculated the breakup
+periods of brown dwarfs as a function of age by equating equatorial
+MNRAS 000, 1–24 (2022)
+
+15
+10
+ower
+P
+-2
+0
+10
+10
+10
+10
+PeriodExoplanet variability with vAPP coronagraphs
+17
+velocity with the escape velocity, accounting for radial contraction
+over time. When we compare our measured HD 1160 B variability
+period of 3.24 hours to their results (their Figure 13), we find that
+this is a physically feasible rotation period for most possible com-
+binations of mass and age from the literature, albeit very close to
+the breakup period in many cases. An alternative explanation is that
+the 3.24 hour variability signal that we see is produced by multiple
+features in the atmosphere of HD 1160 B, and that its rotation period
+is actually longer (Leggett et al. 2016). Additional observations of
+this variability over a longer baseline will help to further charac-
+terise its origin and confirm whether its periodicity reflects that of
+the companion’s rotation.
+The detrended differential light curves for each of the 30 indi-
+vidual wavelength channels over the 3.59-3.99 µm wavelength range
+that comprise the white-light curve (Figure 11) show increasing sta-
+tistical errors at longer wavelengths, as expected as our S/N is lower
+here. Using the RMS of each light curve as a metric to compare
+the scatter in each channel, we do see tentative evidence of increas-
+ing variability towards longer wavelengths beyond the increase of
+RMS expected from the S/N. However, as the total baseline of our
+observations is only a single night, we see too few repetitions to
+be confident of variability patterns in individual wavelength chan-
+nels. Additional spectrophotometric data will therefore be required
+to confirm this. Although we modelled the overall variability in our
+white-light curve with a single sinusoid, it is also possible that the
+phase and amplitude is different per wavelength channel as distinct
+atmospheric features at separate locations in the atmosphere of the
+companion may produce variability with a different wavelength de-
+pendence. Furthermore, although the overall wavelength range of
+these 30 channels is relatively small, different wavelengths probe
+different pressure levels in the companion’s atmosphere, and hence
+different layers of the atmosphere (e.g. Buenzli et al. 2012; Biller
+et al. 2013; Apai et al. 2017; Ge et al. 2019).
+7.2.2
+Light curve precision
+In Section 6, we found that after removing a single sinusoid from
+the data, we achieved a precision of 3.7% in the detrended dif-
+ferential light curve when it is binned to a bin size of 200 data
+points, corresponding to 11 bins of 18 minutes of integration time.
+There have been three previous studies searching for variability in
+substellar companion from the ground; Apai et al. (2016), Biller
+et al. (2021), and Wang et al. (2022) each conducted variability
+searches on the HR 8799 planets using satellite spots as photomet-
+ric references. In a pilot variability study, Apai et al. (2016) reach
+a ∼10% planet-to-planet photometric accuracy for SPHERE obser-
+vations of 25 minute cadence when data from different nights are
+combined for a total telescope time of 3.5 hours. Biller et al. (2021)
+goes further with SPHERE to conduct a longer (>4 hours) search,
+successfully constraining the sensitivity to variability to amplitudes
+>5% for HR 8799b and >25% for HR 8799c. More recently, Wang
+et al. (2022) used SCExAO/CHARIS to improve the variability con-
+straints of HR 8799c to the 10% level, and HR 8799d to the 30%
+level. They did this by combining the use of satellite spots with a
+spectrophotometric approach similar to the one we present in this
+paper, using the CHARIS IFS to disperse the light into individ-
+ual spectra before recombining the channels into wider bands. At
+first glance, the sensitivity that we achieve for HD 1160 B here
+may appear to compare favourably with these results. However, it is
+important to consider a number of caveats that make direct compar-
+ison unjustified. All three of the HR 8799 studies were conducted
+at near-infrared wavelengths with 8.2-m telescopes, while ours was
+in the mid-infrared with an 8.4-m telescope. More significantly, the
+HR 8799 planets are fainter than HD 1160 B, with contrasts of
+Δ𝐻 = 8 − 10 mag compared to their host star, which has a H-band
+magnitude of 5.28 mag (Marois et al. 2008, 2010). HD 1160 B is
+brighter, with a contrast of Δ𝐿′ = 6.35 mag compared to a host star
+with an 𝐿′-band magnitude of 7.06 mag (Nielsen et al. 2012). The
+lower sensitivity to variability achieved by these studies is therefore
+partially a reflection of the intrinsically lower fluxes of their targets,
+which leads to higher errors on their photometry.
+However, each of the HR 8799 variability studies also found
+that the satellite spots can demonstrate individual variations of their
+own and are often anti-correlated with each other. This means that
+they may not always serve as appropriate photometric references
+with which to detrend the light curve of a companion. Wang et al.
+(2022) found that the flux ratio of the SCExAO satellite spots shows
+time variation with a scatter of ∼3% across a night, and can show
+even larger variations on a shorter timescale, up to 10%. This poten-
+tially sets a limit to the precision that can be achieved using satellite
+spots, particularly on nights where observing conditions are less
+stable.
+A key advantage of the dgvAPP360 compared to satellite spots
+is its simplicity; the photometric reference it provides is simply an
+image of the host star, and so it does not suffer from the same cor-
+related systematics as the satellite spots. Differential photometry
+between the companion and the star can be carried out directly. It
+may be possible to reach an even deeper precision through differen-
+tial spectrophotometry with a dgvAPP360 than the 3.7% level that
+we achieve here. Indeed, if we compare the detrended light curve
+RMS as a function of bin size to the white noise expectation (Figure
+13), it continues to follow the trend of the white noise and does not
+plateau implying that we have not yet reached any noise floor. This
+means that in principle the precision of the differential light curve
+would improve further if more data from additional epochs was
+added. This also indicates that this technique should remain usable
+for companions with less favourable contrasts than HD 1160 B,
+such as those in the planetary-mass regime. However, more data
+per bin will be required to achieve the same precision for a fainter
+companion, so the time-sampling in the binned light curves may be
+less fine in these cases.
+Many transiting exoplanet studies make use of a region of
+the target light curve that is expected to be flat (i.e. an out-of-
+transit baseline) to test the degree to which systematics have been
+corrected. While we have shown here that key systematic trends are
+successfully removed in the detrended differential white-light curve
+of HD 1160 B, we do not have such a baseline to verify the level
+of impact of any remaining systematics. The possibility therefore
+remains that an unknown systematic could be present that has not
+been accounted for by any of the processes that we’ve applied here,
+and could be responsible for the variability that we see in the light
+curve of HD 1160 B. However, this is also inherently the case for
+any study that explores the variability of isolated brown dwarfs and
+planetary-mass objects, stellar variability due to star spots or other
+sources, or transiting exoplanet studies where exoplanets transit
+variable stars (e.g. Gelino et al. 2002; Rockenfeller et al. 2006;
+Biller et al. 2013, 2015, 2021; Girardin et al. 2013; Radigan et al.
+2014; Wilson et al. 2014; Naud et al. 2017; Eriksson et al. 2019;
+Vos et al. 2019; Manjavacas et al. 2021, 2022). A subsequent study
+is forthcoming in which we further investigate the precision that can
+be reached with this technique, and use injection-recovery tests to
+assess the extent to which known, simulated variability signals can
+be recovered (Sutlieff et al., in preparation).
+Nonetheless, ground-based differential spectrophotometry
+MNRAS 000, 1–24 (2022)
+
+18
+B. J. Sutlieff et al.
+with the vAPP is highly complementary and advantageous to space-
+based approaches for measuring the variability of high-contrast
+companions. There have been many successful space-based mea-
+surements of companion variability using HST, detecting variability
+with amplitudes down to the 1-2% level in some cases (e.g. Zhou
+et al. 2020a,b; Manjavacas et al. 2018, 2019b; Bowler et al. 2020b).
+Zhou et al. (2016) was further able to detect sub-percent variability
+using HST observations of planetary-mass companion 2M1207b,
+which lies at roughly the same angular separation as HD 1160 B,
+albeit with a more favourable contrast. Furthermore, the first vari-
+ability monitoring with JWST, which should reach an even greater
+precision, is currently underway as part of the Early Release Sci-
+ence Program (Hinkley et al. 2022). However, while JWST has the
+sensitivity to image fainter, lower mass companions and measure
+their variability with great precision, its ∼6.5 m mirror is smaller
+than those of the largest ground-based telescopes, and thus is can-
+not outperform large ground-based telescopes with extreme AO at
+small separations ≲ 0.5′′ at ∼3.5 µm (Girard et al. 2022). This
+means that companions at the closest angular separations such as
+Jupiter analogues are for now likely only accessible with ground-
+based monitoring techniques, for all but the nearest stars (Carter
+et al. 2021; Kammerer et al. 2022). Ground-based telescopes also
+uniquely provide access to higher resolution spectrographs, such
+that line profile variability could be used in Doppler imaging to
+create 2D global maps of features in exoplanet atmospheres such as
+storms similar to Jupiter’s Great Red Spot (e.g. Crossfield 2014).
+7.3
+Observing strategy
+During the observing sequence, we obtained 6 wavelength calibra-
+tions at intervals throughout the night to allow us to test whether
+differences in the wavelength calibration used would lead to dif-
+ferences in the differential light curve. In Figure 7 in Section 4.3,
+we found that the differential light curve is robust to changes in the
+wavelength calibration. It is therefore preferable to acquire wave-
+length calibrations at the start or end of future observations, perhaps
+along with a single precautionary wavelength calibration at high el-
+evation, and instead obtain additional data on target and minimise
+pixel offsets.
+We also used an on/off nodding pattern to enable background
+subtraction. However, future studies may wish to consider alter-
+native methods to remove the thermal background such that the
+amount of time spent on target can be doubled and the entire system
+stabilised. For example, Doelman et al. (2022) developed an ap-
+proach whereby 93% of the frames obtained were on-target and the
+thermal background was modelled and removed using the science
+frames themselves and a small number of background frames ob-
+tained before and after the observing sequence. As Figure 13 shows
+that we approach the photon noise limit, increased on-target time
+would therefore allow a greater differential light curve precision to
+be obtained in a single night of observations.
+Lastly, the absence of an instrument derotator in the LBTI
+architecture meant that the field of view was rotating throughout the
+observing sequence. In addition to the the drift due to lenslet flexure
+and the manual offsets applied to keep HD 1160 A centred in the field
+of view, this meant that HD 1160 A and B were not pixel-stabilised
+during our observations. However, the linear regression correlation
+coefficients of the positions of the star and the companion are small
+relative to those of airmass and air temperature. This suggests that
+that, when present, these factors dominate over any effect from
+HD 1160 A and B not being pixel-stabilised.
+Understanding whether the host star of a given target is itself
+varying is important when interpreting the trends in a differen-
+tial light curve. Most, if not all, potential targets for differential
+spectrophotometry will be present in at least the TESS full frame
+images available on the MAST archive. Even though this data will
+most likely not be contemporaneous with a particular set of observa-
+tions, the total baseline of the coverage should be relatively long and
+therefore sufficient to check a host star for variability at the required
+precision, especially if the target appears in multiple TESS sectors.
+We therefore recommend this method as a good way to verify the
+level of variation shown by the host star of a target for differential
+spectrophotometry, and hence whether it is stable enough to act
+as a simultaneous photometric reference without requiring further
+analysis to account for stellar variability.
+7.4
+Outlook
+In principle, the technique presented in this paper can be applied to
+any vAPP coronagraph used in combination with an IFS. Although
+ALES is currently the only IFS operating over the L and M bands
+(Doelman et al. 2021), a vAPP coronagraph is also available on
+SCExAO/CHARIS on the 8.2-m Subaru Telescope, offering R∼19
+spectrographic coverage over the J, H, and K bands (1.13-2.39 µm)
+(Groff et al. 2016; Doelman et al. 2017; Bos et al. 2019; Lozi et al.
+2020; Miller et al. 2021). There will also be two different vAPPs on
+the METIS instrument on the upcoming 39-m ELT, for which this
+work is a pathfinder. METIS will provide high spectral resolution
+spectroscopy (R∼100,000) over the L and M bands (Carlomagno
+et al. 2016; Brandl et al. 2018, 2021; Kenworthy et al. 2018b).
+Variability measurements using a vAPP may even be possible for
+broad-band imaging data where an IFS is unavailable, although
+sensitivity will be inherently more limited without the benefits of
+using differential spectrophotometry to reduce the effects of sys-
+tematics. There are several vAPPs currently available on such coro-
+nagraphic imagers, such as MagAO (Morzinski et al. 2016; Otten
+et al. 2017; Sutlieff et al. 2021) and MagAO-X (Miller et al. 2019;
+Close et al. 2020b) on the 6.5-m Magellan Clay Telescope, and
+the recently-commissioned ERIS instrument on the VLT (Boehle
+et al. 2018, 2021; Kenworthy et al. 2018a; Dubber et al. 2022), with
+more planned, including for GMagAO-X on the GMT (Close et al.
+2020a) and MICADO on the ELT (Clénet et al. 2018; Perrot et al.
+2018). Using the vAPPs that will be available on larger telescopes,
+variability monitoring through differential spectrophotometry will
+be possible for fainter companions at closer angular separations,
+including those in the exoplanet mass regime. While this will be in-
+herently more challenging for such companions at greater contrasts,
+these will remain accessible to this technique through the addition
+of data from multiple epochs as long as the systematic noise floor
+is not reached, albeit with a trade-off between light curve precision
+and time-sampling.
+In the era of extremely large telescopes, high-contrast imaging
+combined with high resolution spectroscopy will provide access
+to fainter companions at lower masses and older ages and allow
+their orbital velocities and spin to be measured (e.g. Snellen et al.
+2014, 2015; Schwarz et al. 2016; Birkby 2018; Wang et al. 2021b;
+Xuan et al. 2022). Measurements of how individual absorption lines
+change in depth and width as an exoplanet rotates will allow two-
+dimensional surface maps of exoplanet atmospheres and weather
+to be produced, through techniques such as Doppler imaging (e.g.
+Crossfield et al. 2014; Crossfield 2014; Luger et al. 2021; Plummer
+& Wang 2022). Further in the future, multi-wavelength variabil-
+ity measurements obtained in reflected light may even enable exo-
+cartography of directly imaged Earth-like exoplanets (e.g. Luger
+MNRAS 000, 1–24 (2022)
+
+Exoplanet variability with vAPP coronagraphs
+19
+et al. 2019, 2022; Kawahara 2020; Kuwata et al. 2022; Teinturier
+et al. 2022).
+A limitation of using the dgvAPP360 for variability measure-
+ments is that post-processing algorithms relying on angular diver-
+sity, such as ADI and PCA, cannot be used without also removing
+the central PSF of the star that we use as the simultaneous photomet-
+ric reference. Furthermore, the stellar PSF must remain unsaturated
+throughout the observing sequence. This potentially limits the sam-
+ple of targets with bright enough companions. Although HD 1160 B
+is bright enough that additional noise reduction techniques were not
+necessary to produce a detection of ample S/N, this may not be
+the case for fainter directly imaged companions in the exoplanet
+mass regime. However, it may be possible to use novel alternative
+algorithms to reach deeper contrasts.
+For example, the Temporal Reference Analysis of Planets
+(TRAP, Samland et al. 2021, Liu et al., submitted) algorithm instead
+relies on temporal diversity. TRAP reconstructs the systematics in
+a given region in the data using reference pixels that share the same
+underlying noise sources. By simultaneously fitting the model of
+a companion signal ‘transiting’ over detector pixels and the light
+curves of the reference pixels, TRAP can then remove these sys-
+tematics. It may be possible to leverage the information provided
+by TRAP to improve the companion S/N without removing the stel-
+lar PSF, or even to extract detrended light curves directly. Another
+option would be to use the gvAPP coronagraph, which is different
+from the dgvAPP360 in that it creates two images of the target star
+each with a 180° D-shaped dark hole on opposing sides, as well as
+an additional ‘leakage term’ positioned between the two (Snik et al.
+2012; Otten et al. 2014b; Sutlieff et al. 2021). The leakage term is an
+entirely separate PSF of the star that appears at a fraction of its full
+brightness, making it ideal as a simultaneous photometric reference
+and enabling observations of systems with host stars that would oth-
+erwise be too bright. Post-processing algorithms can be applied to
+the main PSFs to reach deeper contrasts without impacting the leak-
+age term, enabling differential variability measurements provided
+that the impact of the algorithms on the photometry of the com-
+panion can be characterised precisely. However, the gvAPP coro-
+nagraph can suffer from wavelength-dependent smearing, which
+would make such measurements more complex than those obtained
+with a dgvAPP360 (Otten et al. 2017). In addition to the leakage
+term, some vAPPs (such as the VLT/ERIS gvAPP) produce other
+faint reference spots specifically for use as photometric references
+in situations where the core of the target star PSF is saturated (Doel-
+man et al. 2021; Kravchenko et al. 2022).
+In addition to probing the intrinsic variability of high-contrast
+companions, differential spectrophotometry could also be used to
+observe the transits of satellites such as exomoons or binary plan-
+ets passing in front of these companions. Candidate satellites have
+been identified around transiting exoplanets, directly imaged com-
+panions, and isolated planetary-mass objects using a range of tech-
+niques, but none have yet been definitively detected (e.g. Teachey &
+Kipping 2018; Kreidberg et al. 2019; Lazzoni et al. 2020; Teachey
+et al. 2020; Limbach et al. 2021; Vanderburg & Rodriguez 2021;
+Kipping et al. 2022). For directly imaged companions, variability
+arising from transit events could be distinguished from that caused
+by inhomogeneous atmospheric features in similar ways to transit-
+ing exoplanets and star spots, by considering the companion light
+curves across the different wavelength channels. Transit signals are
+expected to be almost achromatic, while intrinsic variability is gen-
+erally wavelength-dependent (Manjavacas et al. 2019a; Limbach
+et al. 2021, 2022; Lazzoni et al. 2022). Lazzoni et al. (2022) found
+using simulations that although the probability of successfully de-
+tecting smaller exomoons around a directly imaged companion is
+very low with current instrumentation and techniques, detections of
+larger binary planets are already within reach. New techniques to
+improve differential light curve precision for directly imaged com-
+panions, including differential spectrophotometry with the dgvAPP,
+will help to increase these probabilities further and potentially en-
+able the first definitive detections of satellites around substellar
+companions.
+8
+CONCLUSIONS
+We present a novel, ground-based approach for constructing differ-
+ential light curves of high-contrast companions through direct dif-
+ferential spectrophotometric monitoring, using the double-grating
+360° vector Apodizing Phase Plate coronagraph and the ALES
+integral field spectrograph. The dgvAPP360 allows high-contrast
+companions to be detected while also providing an image of the
+host star, which crucially can be used as a simultaneous photomet-
+ric reference. We combine the dgvAPP360 with ALES to follow the
+highly successful technique of differential spectrophotometry used
+in exoplanet transmission spectroscopy, where light is spectrally-
+dispersed to reduce systematic effects that otherwise dominate the
+variability signal we aim to measure, and then recombined into
+white-light flux measurements.
+We demonstrated this approach using a full night of observa-
+tions of substellar companion HD 1160 B. The time-series fluxes
+of the companion and the star in each wavelength channel were
+extracted simultaneously using aperture photometry. We then pro-
+duced white-light measurements for both the companion and the
+star at each time frame by taking the median combination of the
+photometry in the wavelength dimension. The companion flux was
+then divided by that of the star to eliminate trends common to both,
+arising from Earth’s atmosphere and other systematics, producing
+a differential white-light curve that only contains non-shared varia-
+tions and covers a wavelength range of 3.59-3.99 µm. We find that
+the shape of the resulting light curve is robust against issues arising
+from instrumental flexure, as tested using calibration frames col-
+lected throughout the observation sequence. Using a multiple linear
+regression approach with eight decorrelation parameters, we mod-
+elled and removed non-shared trends from the differential white-
+light curve. We find that airmass and air temperature are the most
+correlated parameters with the light curve. We also analyse publicly
+available data from the TESS mission to check for variability in the
+host star HD 1160 A, and confirm that it is non-varying at the 0.03%
+level.
+We find that the detrended differential white-light curve of
+HD 1160 B shows sinusoidal-like variability over a short timescale.
+By fitting the unbinned light curve with a sinusoid, we identify
+that the variability has a semi-amplitude of ∼8.8% and a period
+of ∼3.24 hours. When binned to 18 minutes of integration time
+per bin, we achieve a light curve precision at the 3.7% level. After
+thorough investigation and rejection of systematic noise sources, we
+attribute this variability as likely due to heterogeneous features in
+the atmosphere of the companion, rotating in and out of view as
+it rotates. We find that if the period of this variability reflects the
+rotation period of HD 1160 B, physical limitations suggest that it is
+rotating at close to its breakup period. Alternatively, the short period
+variability in the light curve of HD 1160 B may arise from multi-
+ple periodic features in its atmosphere with different phase offsets.
+Furthermore, light curves in the 30 individual wavelength channels
+in the 3.59-3.99 µm range show tentative evidence of an increase in
+MNRAS 000, 1–24 (2022)
+
+20
+B. J. Sutlieff et al.
+variability amplitude at longer wavelengths. Further observations at
+additional epochs will help to confirm and characterise the variabil-
+ity of HD 1160 B and to determine its physical explanation.
+The precision that we achieve in the detrended differential
+white-light curve is the greatest achieved from ground-based stud-
+ies of sub-arcsecond high-contrast companions to date. However,
+direct comparisons to other ground-based studies that instead use
+satellite spots to search for variability in the light curves of high-
+contrast companions are not straightforward due to the different
+magnitude and contrast of the observed systems, with HD 1160 B
+having a more favourable contrast (Apai et al. 2016; Biller et al.
+2021; Wang et al. 2022). The RMS of the detrended differential
+light curve for HD 1160 B as a function of bin size follows the same
+trend as the theoretical white noise expectation with no evidence of
+yet approaching a noise floor. This indicated that the single night
+of data analysed here is not yet systematic-limited, and that further
+observations from additional epochs could enable greater sensitiv-
+ity to be reached. A deeper investigation of this type of data and its
+precision, including injection-recovery tests to test how effectively
+known variability signals can be recovered and which systematics
+have the greatest impact, is forthcoming (Sutlieff et al., in prepara-
+tion).
+While JWST will measure the variability of fainter, lower mass
+companions from space with unprecedented precision, its compar-
+atively smaller aperture means it cannot outperform the largest
+AO-equipped ground-based telescopes at separations ≲ 0.5′′ at
+∼3.5 µm (Girard et al. 2022), so companions such as Jupiter ana-
+logues at the closest angular separations, for all but the nearest stars,
+remain accessible only to ground-based monitoring techniques for
+the coming decade. Ground-based differential spectrophotometry
+with the vAPP is therefore highly complementary to space-based
+approaches for measuring the variability of high-contrast exoplanet
+and brown dwarf companions, and for searching for their transiting
+exomoons or binary planets. Moreover, ground-based telescopes
+can reach much higher spectral resolution, which then enables line
+profile variability studies to map atmospheric features, including
+storms and hurricanes like Jupiter’s Great Red Spot, via Doppler
+imaging. These results are promising for further variability studies
+using vAPP coronagraphs on current and upcoming instruments
+and telescopes, which include ERIS on the VLT and METIS on the
+ELT.
+ACKNOWLEDGEMENTS
+The authors would like to thank Frank Backs and Elisabeth C.
+Matthews for valuable discussions that improved this work. We also
+thank our anonymous referee whose comments helped us to im-
+prove this manuscript. BJS is fully supported by the Netherlands
+Research School for Astronomy (NOVA). JLB acknowledges fund-
+ing from the European Research Council (ERC) under the European
+Union’s Horizon 2020 research and innovation program under grant
+agreement No 805445. This paper is based on work funded by the
+United States NSF grants 1608834, 1614320, and 1614492. The
+research of DD and FS leading to these results has received fund-
+ing from the European Research Council under ERC Starting Grant
+agreement 678194 (FALCONER).
+We acknowledge the use of the Large Binocular Telescope
+Interferometer (LBTI) and the support from the LBTI team, specifi-
+cally from Emily Mailhot, Jared Carlson, Jennifer Power, Phil Hinz,
+Michael Skrutskie, Travis Barman, Ilya Ilyin, and Ji Wang. The LBT
+is an international collaboration among institutions in the United
+States, Italy and Germany. LBT Corporation partners are: The Uni-
+versity of Arizona on behalf of the Arizona Board of Regents; Isti-
+tuto Nazionale di Astrofisica, Italy; LBT Beteiligungsgesellschaft,
+Germany, representing the Max-Planck Society, The Leibniz In-
+stitute for Astrophysics Potsdam, and Heidelberg University; The
+Ohio State University, representing OSU, University of Notre Dame,
+University of Minnesota and University of Virginia. We gratefully
+acknowledge the use of Native land for our observations. LBT ob-
+servations were conducted on the stolen land of the Ndee/Nn¯e¯e,
+Chiricahua, Mescalero and San Carlos Apache tribes.
+This publication makes use of data products from the Two
+Micron All Sky Survey, which is a joint project of the Univer-
+sity of Massachusetts and the Infrared Processing and Analysis
+Center/California Institute of Technology, funded by the National
+Aeronautics and Space Administration and the National Science
+Foundation. This work has made use of data from the European
+Space Agency (ESA) mission Gaia (https://www.cosmos.esa.
+int/gaia), processed by the Gaia Data Processing and Anal-
+ysis Consortium (DPAC, https://www.cosmos.esa.int/web/
+gaia/dpac/consortium). Funding for the DPAC has been pro-
+vided by national institutions, in particular the institutions partic-
+ipating in the Gaia Multilateral Agreement. This paper includes
+data collected with the TESS mission, obtained from the Mikulski
+Archive for Space Telescopes (MAST) data archive at the Space
+Telescope Science Institute (STScI). Funding for the TESS mission
+is provided by the NASA Explorer Program. STScI is operated by
+the Association of Universities for Research in Astronomy, Inc.,
+under NASA contract NAS 5–26555. Support for MAST for non-
+HST data is provided by the NASA Office of Space Science via
+grant NNX13AC07G and by other grants and contracts. This re-
+search has made use of the NASA Exoplanet Archive, which is
+operated by the California Institute of Technology, under contract
+with the National Aeronautics and Space Administration under the
+Exoplanet Exploration Program. This research has made use of
+NASA’s Astrophysics Data System. This research has made use
+of the SIMBAD database, operated at CDS, Strasbourg, France
+(Wenger et al. 2000). This research made use of SAOImageDS9, a
+tool for data visualization supported by the Chandra X-ray Science
+Center (CXC) and the High Energy Astrophysics Science Archive
+Center (HEASARC) with support from the JWST Mission office
+at the Space Telescope Science Institute for 3D visualization (Joye
+& Mandel 2003). This work made use of the whereistheplanet3
+prediction tool (Wang et al. 2021a). This work makes use of the
+Python programming language4, in particular packages including
+NumPy (Harris et al. 2020), Astropy (Astropy Collaboration et al.
+2013, 2018), SciPy (Virtanen et al. 2020), HCIPy (Por et al. 2018),
+PynPoint (Amara & Quanz 2012; Stolker et al. 2019), scikit-image
+(van der Walt et al. 2014), scikit-learn (Pedregosa et al. 2011),
+statsmodels (Seabold & Perktold 2010), pandas (McKinney 2010;
+Reback et al. 2022), Photutils (Bradley et al. 2022), and Matplotlib
+(Hunter 2007).
+DATA AVAILABILITY
+The data from the LBT observations underlying this article are
+available in the Research Data Management Zenodo repository of
+the Anton Pannekoek Institute for Astronomy, at https://doi.
+org/10.5281/zenodo.5617572.
+3 http://whereistheplanet.com/
+4 Python Software Foundation; https://www.python.org/
+MNRAS 000, 1–24 (2022)
+
+Exoplanet variability with vAPP coronagraphs
+21
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+page_content='0  Measuring the variability of directly imaged exoplanets using vector  Apodizing Phase Plates combined with ground-based differential  spectrophotometry  Ben J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff,1,2★ Jayne L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Birkby,3,1 Jordan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Stone,4 David S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman,2  Matthew A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kenworthy,2 Vatsal Panwar,1,5 Alexander J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bohn,2 Steve Ertel,6,7  Frans Snik,2 Charles E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Woodward,8 Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Skemer,9 Jarron M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Leisenring,7  Klaus G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Strassmeier,10 and David Charbonneau11  1Anton Pannekoek Institute for Astronomy, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands  2Leiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Box 9513, 2300 RA Leiden, The Netherlands  3Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, United Kingdom  4Naval Research Laboratory, Remote Sensing Division, 4555 Overlook Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' SW, Washington, DC 20375, USA  5Department of Physics, University of Warwick, Coventry West Midlands, CV4 7AL, United Kingdom  6Large Binocular Telescope Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721, USA  7Steward Observatory and Department of Astronomy, University of Arizona, 933 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Cherry Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=', Tucson, AZ 85721, USA  8Minnesota Institute for Astrophysics, University of Minnesota, 116 Church Street SE, Minneapolis, MN 55455, USA  9Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High St, Santa Cruz, CA 95064, USA  10Leibniz-Institute for Astrophysics Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany  11Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' in original form ZZZ  ABSTRACT  Clouds and other features in exoplanet and brown dwarf atmospheres cause variations in bright-  ness as they rotate in and out of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ground-based instruments reach the high contrasts and  small inner working angles needed to monitor these faint companions, but their small fields-  of-view lack simultaneous photometric references to correct for non-astrophysical variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We present a novel approach for making ground-based light curves of directly imaged compan-  ions using high-cadence differential spectrophotometric monitoring, where the simultaneous  reference is provided by a double-grating 360° vector Apodizing Phase Plate (dgvAPP360)  coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The dgvAPP360 enables high-contrast companion detections without blocking  the host star, allowing it to be used as a simultaneous reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To further reduce systematic  noise, we emulate exoplanet transmission spectroscopy, where the light is spectrally-dispersed  and then recombined into white-light flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We do this by combining the dgvAPP360 with the  infrared ALES integral field spectrograph on the Large Binocular Telescope Interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To  demonstrate, we observed the red companion HD 1160 B (separation ∼780 mas) for one night,  and detect 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8% semi-amplitude sinusoidal variability with a ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='24 h period in its detrended  white-light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We achieve the greatest precision in ground-based high-contrast imaging  light curves of sub-arcsecond companions to date, reaching 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7% precision per 18-minute  bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Individual wavelength channels spanning 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm further show tentative evidence  of increasing variability with wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We find no evidence yet of a systematic noise floor,  hence additional observations can further improve the precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This is therefore a promis-  ing avenue for future work aiming to map storms or find transiting exomoons around giant  exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Key words: infrared: planetary systems – instrumentation: high angular resolution – planets  and satellites: detection – brown dwarfs – planets and satellites: atmospheres – techniques:  imaging spectroscopy  ★ E-mail: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='sutlieff@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='nl  1  INTRODUCTION  Planets do not have a homogeneous appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' When we look at  the planets in our own solar system, we see distinct cloud structures  © 2022 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='08689v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='EP]  20 Jan 2023  2  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' An on-sky example image of a 𝐾 ∼ 7 mag target observed with the dgvAPP360 coronagraph on LBT, copied and annotated on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  dgvAPP360 produces a dark hole of deep flux suppression, seen here surrounding the target star, allowing high-contrast companions to be detected in this  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The star itself remains unsaturated in the middle, allowing it to be used as a simultaneous reference PSF when making a differential light curve for a  companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The outer edge of the dark hole lies beyond the field of view in the LBT observations used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  and giant storms that show great diversity in size, shape, lifetime,  and brightness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Simon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Stauffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Coulter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These features rotate in and  out of view throughout the planet’s rotation period, modulating its  overall brightness and thus allowing us to map out its atmosphere  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kostov & Apai 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Karalidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Fletcher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Plummer & Wang 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Beyond the solar  system, variations in the light curves of stars deliver information on  the distribution of features such as star spots (Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Jeffers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Strassmeier 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Morales  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Herbst 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Thiemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' By measuring the pho-  tometric variability of exoplanets and brown dwarfs in the same  way, we can gain not only an insight into their visual appearance,  but key information on the distribution of condensate clouds that  strongly affect the infrared spectra of directly imaged companions,  allowing degeneracies between atmospheric models to be broken  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Rajan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Charnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Zhou  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Zhang 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ward-Duong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Tan & Showman  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Space-based photometric monitoring with the Hubble Space  Telescope (HST) has already shown that giant planetary-mass and  brown dwarf companions do exhibit such variability, at a range  of amplitudes and periods (Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Buenzli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014,  2015a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Manjavacas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Miles-Páez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These  results are in good agreement with observations of isolated brown  dwarfs and giant exoplanet analogues (Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2018, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ashraf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022), including a large  Spitzer survey by Metchev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2015) who found that photospheric  spots causing ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2% variability at 3-5 µm are ubiquitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Several  studies have identified objects with much stronger variability, at the  >10% level, with some even varying with peak-to-peak amplitudes  as high as 26% (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Radigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Eriksson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  (2022) further found that young, low-mass brown dwarfs with simi-  lar colours and spectra to directly imaged exoplanetary companions  are highly likely to display variability in the L2-T4 spectral type  range, with an enhancement in maximum amplitudes compared to  field dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The rotation periods of brown dwarf and planetary-mass com-  panions are consistent with those of the isolated low-mass brown  dwarf population, suggesting that they may share similar angular  momentum histories (Bryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These pe-  riods are generally short, ranging from ∼1 hour to ≳20 hours (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Radigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Metchev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Tannock  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021), within the range expected when evolutionary models  and the age- and mass-dependent breakup velocities are considered  (Leggett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These periods, derived from  photometric measurements, are complementary to measurements  of companion spin obtained from their spectra (Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018, 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Xuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' When combined, rotation period and spin mea-  surements can be used to constrain companion obliquities (Bryan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020a, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, the population of directly imaged  companions accessible to space-based facilities such as HST re-  mains small as most companions lie at close angular separations  within the inner working angles of these facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Equipped with coronagraphs and extreme adaptive optics (AO)  systems operating in the infrared, large ground-based observatories  have the resolution and photon collecting power needed to overcome  the glare of the host star and reach the high contrasts and close an-  gular separations of substellar companions currently inaccessible to  space telescopes (Bowler 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Hinkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Currie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although this provides the opportunity for variability studies  of such companions, precise photometric monitoring is difficult as  the companion light curve is contaminated by variability arising  from Earth’s atmosphere and other systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Therefore, a simul-  taneous, unsaturated photometric reference is required to remove  this contaminant variability from the companion light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For  non-coronagraphic, ground-based observations of isolated brown  dwarfs and planetary-mass objects, nonvariable stars present in the  field of view have often been used as photometric references to en-  able many successful measurements of variability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Gelino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Girardin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Radigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Naud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Eriksson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  MNRAS 000, 1–24 (2022)  Outer working  Unsaturated  angle  reference PSF  Dark holeExoplanet variability with vAPP coronagraphs  3  Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Manjavacas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, the typi-  cally narrow field of view of ground-based coronagraphic imagers  generally precludes the use of field stars as photometric references  for observations of companions, and widely used focal-plane coron-  agraphs block the host star to enable the detection of the companion  (Soummer 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Mawet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ruane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  One solution to this problem is to use off-axis satellite Point  Spread Functions (PSFs), or satellite spots, which can act as simul-  taneous photometric references even when a host star is blocked by  a coronagraph (Sivaramakrishnan & Oppenheimer 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Marois  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2006b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Satellite spots can be created by adding a periodic  modulation to the deformable mirror of an AO-equipped telescope  or by placing a square grid in the pupil plane to produce spots  through diffraction of starlight (Langlois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Jovanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The former approach has been used  by Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2016), Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2021), and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022)  for observations of the multi-planet HR 8799 system (Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2008, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The first two of these studies observed HR 8799 with  the Spectro-Polarimetric High-contrast imager for Exoplanets RE-  search (SPHERE, Beuzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019) instrument at the Very Large  Telescope (VLT), and the latter used the CHARIS integral field spec-  trograph in combination with the Subaru Coronagraphic Extreme  Adaptive Optics instrument at the Subaru Telescope (Jovanovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Groff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2021) used the satellite spots  with a broadband-H filter to successfully constrain their sensitivity  to variability to amplitudes >5% for HR 8799b for periods <10  hours, and amplitudes >25% for HR 8799c for similar periods, not-  ing that the observed amplitude of any variability would be muted  by the likely pole-on viewing angle of these planets (Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ruffio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' They also rule out non-shared  variability between HR 8799b and HR 8799c at the <10-20% level  over a 4-5 hour timescale by using one planet as a photometric refer-  ence for the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Using a spectrophotometric approach, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  (2022) further constrained the variability amplitudes of HR 8799c  to the 10% level, and HR 8799d to the 30% level, and found that  there was no significant variability in the planet’s colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However,  all three studies found that satellite spots are anti-correlated with  each other and can demonstrate individual variations of their own,  potentially setting a limit to the precision that can be achieved with  this technique (although Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2021) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022)  note that the satellite spot light curves can be flat in their most stable  epochs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  Ground-based differential spectrophotometry  In this paper we present a novel, alternative ground-based approach  for constructing light curves of high-contrast companions directly  through the technique of differential spectrophotometric monitor-  ing, akin to that used highly successfully to study exoplanet trans-  mission spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Arcangeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We use a double-grating 360° vector Apodizing Phase Plate (dg-  vAPP360) coronagraph (Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017, 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wagner  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020), which enables high-contrast companions to be detected  without blocking the host star, hence leaving an unsaturated image  of the host star available for use as a simultaneous reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  more widely used grating vector Apodizing Phase Plate (gvAPP)  coronagraph adjusts the phase of the incoming wavefront to modify  the PSFs of all objects in the field of view, creating two images of  the target star each with a 180° D-shaped ‘dark hole’, a region of  deep flux suppression in which high-contrast companions can be  observed, on opposing sides (Snik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Otten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014a,  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  dgvAPP360 instead creates a 360° dark hole surrounding each of  the two images of the target star, and then uses an additional grating  to overlap the images to produce a single image of the star (Doelman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' An example image obtained with the dgvAPP360 is  shown in Figure 1, with the target star in the centre and the dark  hole surrounding it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  In addition, we combine the dgvAPP360 with an integral field  spectrograph (IFS), enabling us to use differential spectrophotom-  etry for high-contrast directly imaged companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The incoming  light is first dispersed into individual spectra, and then recombined  into a single ‘white-light’ data point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This has the advantage of  smoothing out wavelength-dependent flat-fielding errors and allows  wavelength regions with instrumental absorption or highly variable  telluric bands to be excluded, meaning systematic effects can be  significantly reduced, thus yielding greater stability and precision  in the final white-light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  The HD 1160 system  We test this approach using observations of the HD 1160 system,  which is located at a distance of 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='6 pc (Gaia Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021) and consists of host  star HD 1160 A (spectral type A0V, Houk & Swift 1999) and two  high-contrast companions, HD 1160 B and C (Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  HD 1160 B and C lie at separations of ∼80 au (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='78′′) and ∼530 au  (∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1′′), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Several key properties of HD 1160 A are listed  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' HD 1160 A is bright (K = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='040±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='029 mag, Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2003), and the contrast ratio between HD 1160 A and HD 1160 B is  Δ𝐿′ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='12 mag (Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This makes it an ideal  target for demonstrating our technique, as a high signal-to-noise  detection of the companion allows high cadence monitoring and a  deep investigation into any residual systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' A-type stars  such as HD 1160 A generally vary below the millimagnitude level,  which corresponds to variability amplitudes comfortably below the  ∼1% level (Ciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We assess the variability of the host  star in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2 using observations from the Transiting Exoplanet  Survey Satellite (TESS) mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The age of the system is poorly  constrained, with estimates ranging from 50+50  −40 Myr (Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2012) to 100+200  −70 Myr (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The HD 1160 system may  be a member of the Pisces-Eridanus stellar stream, which would  place its age at ∼120 Myr, but this has yet to be confirmed (Curtis  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  HD 1160 B has a spectral type close to the brown dwarf/stellar  boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2012) found a spectral type of ∼L0, but  more recent papers suggest that it lies between M5-M7 (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  (2020) found its spectrum to be highly peculiar and that no spectral  model or template in current libraries can produce a satisfactory fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Although the cause of this peculiarity has not yet been explained,  Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2020) hypothesise that possible causes could include  a young system age, dust in the photosphere of HD 1160 B, or  ongoing evolutionary processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The mass of HD 1160 B also re-  mains unclear, primarily due to the poorly constrained age of the  system, with estimates ranging from that of a low mass brown dwarf  (∼20 MJup, Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020) to decisively in the stellar mass regime  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='01M⊙ ≈ 123MJup, Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019) if the system is in-  deed a member of the Pisces-Eridanus stellar stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' HD 1160 C is  a low-mass star (spectral type M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5), and its separation of ∼530 au  (∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1′′) places it beyond the field of view of most vAPPs currently  in use (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The observations carried out on the HD 1160 system are  MNRAS 000, 1–24 (2022)  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Properties of host star HD 1160 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Parameter  Value  Reference(s)  Spectral Type  A0V  (1)  Right Acension (J2000)  00:15:57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='32  (2)  Declination (J2000)  +04:15:03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='77  (2)  Age (Myr)  10-300  (3, 4)  Parallax (mas)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2721±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0354  (2)  Distance (pc)  120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='6  (2, 5)  Proper motion (RA, mas yr−1)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='150±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='040  (2)  Proper motion (Dec, mas yr−1)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='903±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='034  (2)  Mass (M⊙)  ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  (3)  Teff (K)  9011±85  (6)  log(L/L⊙)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='12±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='07  (6)  log(g) (dex)  ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5  (7)  [Fe/H]  ∼solar  (7)  V (mag)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='119±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='010  (8)  G (mag)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1248±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0004  (2)  J (mag)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='983±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='020  (9)  H (mag)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='013±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='023  (9)  K (mag)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='040±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='029  (9)  References: (1) Houk & Swift (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2) Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2016,  2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (3) Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (4) Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (5) Bailer-Jones  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (6) Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (7) Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (8) Tycho-2  (Høg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (9) 2MASS (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2006)  described in Section 2, and in Section 3 we describe the spec-  tral extraction and data reduction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In Section 4 we pro-  duce and present our differential spectrophotometric light curves of  HD 1160 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then examine various factors that may be correlated  with the light curves and detrend them in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These results  and their implications are then discussed in Sections 6 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lastly,  the conclusions of the work are summarised in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2  OBSERVATIONS  We observed the HD 1160 system on the night of 2020 September  25 (03:27:31 - 11:16:14 UT) with the double-grating 360° vector  Apodizing Phase Plate (dgvAPP360) coronagraph (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1)  and the Arizona Lenslets for Exoplanet Spectroscopy (ALES) inte-  gral field spectrograph (IFS) (Skemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Hinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' ALES is integrated inside the Large Binocu-  lar Telescope Interferometer (LBTI) (Hinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ertel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2020), which works in conjunction with the LBT mid-infrared cam-  era (LMIRcam), on the 2 x 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4-m LBT in Arizona (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Leisenring et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For these observations, ALES was in  single-sided mode and was therefore fed only by the left-side aper-  ture of LBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Atmospheric turbulence was corrected for by the LBTI  adaptive optics (AO) system (Bailey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Pinna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We used the ALES L-band prism, which covers a simulta-  neous wavelength range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2 µm with a spectral resolution of  R∼40 (Skemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The plate scale is ∼35 mas spaxel−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The other LBT aperture was used to feed the Potsdam Echelle Po-  larimetric and Spectroscopic Instrument (PEPSI), which obtained  R = 50,000 combined optical spectra of HD 1160 A and B in the  383-907nm wavelength range (Strassmeier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015, 2018), which  is subject to analysis in other forthcoming works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Conditions were exceptionally clear and stable throughout the  night, with no time lost to weather, and seeing ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We acquired 2210 on-target ALES frames, with an integration  time of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4 s per frame, giving a total on-target integration time of  11934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0 s (∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='32 h) spread over ∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='81 h once readout time, nod-  ding, and wavelength calibrations are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The integration time  was chosen such that the stellar PSF remained unsaturated in the  core so that it can be used as the photometric reference for the com-  panion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We used an on/off nodding pattern to enable background  subtraction, nodding to a position 5′′ away for the off-source nod  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As HD 1160 C is located at a similar separation (∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1′′),  we nodded in a direction away from this companion to prevent  it from contaminating the frames obtained in the off-source nod  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Beam-switching in this way is possible because of the  intrinsic stability provided by the dgvAPP360 coronagraph’s place-  ment in the pupil plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We obtained dark frames with the same  exposure time at the end of the night, and 6 wavelength calibrations  were acquired at irregular intervals during the night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' LBTI operates  in pupil-stabilized mode, such that the field of view was rotating  throughout the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The total field rotation across the ob-  serving sequence was 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The HD 1160 system was observed  from an elevation of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4° at the start of the night to a maximum  elevation of 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7°, and then back down to an elevation of 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  dgvAPP360 creates an annular dark hole around the target PSF,  with an inner radius close to the PSF core and an outer radius at the  edge of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2′′ field of view of the detector (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7 - 15 𝜆/D in ALES  mode) (Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For these observations, this  meant that HD 1160 B was located in the dark hole of HD 1160 A  across the entire wavelength range covered by the ALES L-band  prism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' HD 1160 C (separation ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1′′) remained beyond the ALES  field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  3  DATA REDUCTION  The raw ALES images contain the spectra that have been projected  onto the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ultimately, we are aiming to produce a light curve  for the companion that is made from the ‘white-light’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' combined  in the wavelength dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To do this, we must first extract the  spectra from the raw data along with bad pixel correction and flat-  fielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We can then extract the photometry of the star and the  companion at each wavelength before collapsing the data in the  wavelength dimension to obtain white-light fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' A light curve  for the companion is then obtained by dividing the companion flux  by the stellar flux to remove systematic trends shared by both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In  the following subsections we describe the methods used to carry  out each of these steps and obtain the white-light curve of the  companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  Spectral data cube extraction  Raw ALES data consists of a two-dimensional grid of 63×67 micro-  spectra over a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2′′x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2′′ field of view, which must be extracted  into three-dimensional data cubes of 𝑥-position, 𝑦-position, and  wavelength 𝜆 (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To do this, we first performed a  background subtraction using the sky frames obtained in the off-  source nod position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For each ALES image, we subtracted the me-  dian combination of the 100 sky frames closest in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then  extracted the micro-spectra into cubes using optimal extraction,  which is an inverse variance and spatial profile weighted extraction  approach (Horne 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Briesemeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The extraction weights were obtained by measuring the spatial pro-  file of each micro-spectrum in the dark-subtracted sky frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As  there is no significant change in the spatial profile as a function of  wavelength, we average the spatial profile of each micro-spectrum  MNRAS 000, 1–24 (2022)  Exoplanet variability with vAPP coronagraphs  5  over wavelength to obtain higher signal-to-noise (S/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The wave-  length calibration of the raw ALES data was then obtained using  four narrow-band photometric filters at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9 µm,  positioned upstream of the ALES optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These filters are each of  a higher spectral resolution  𝜆  Δ𝜆∼100 than ALES, so are therefore  unresolved and provide four single-wavelength fiducial spots with  which each micro-spectra can be calibrated (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  For each micro-spectrum, we performed this calibration by fitting a  second-order polynomial to the calculated pixel positions of these  four spots, therefore mapping pixel position to wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Each of  the 63×67 micro-spectra was thereby converted into a correspond-  ing spaxel in the three-dimensional data cube (Briesemeister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The resulting data cube contained 100  channels in the wavelength dimension ranging from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The primary wavelength calibration used to process this data  set was obtained at 08:25:00 UT on the night of observations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  4 hours 57 minutes into the observing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To test whether  a wavelength calibration obtained at a different point in the night  has a significant effect on the photometry of the target star and  the companion, we also separately processed the data using an  alternative wavelength calibration obtained at 04:54:00 UT, 1 hour  26 minutes into the observing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We compare and discuss  the results of the two wavelength calibrations in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Data processing  Once the spectra were extracted into background-subtracted three-  dimensional cubes of images for each exposure, we applied several  data reduction steps to remove systematics and improve the S/N at  the location of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We first removed 8 time frames from  the data in which the AO loop opened while the data was being col-  lected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Next, we identified bad pixels using a 6𝜎 filter and replaced  them with the mean of the neighbouring pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then applied  a flat-field correction to calibrate the data against the response of  the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For each wavelength channel, the corresponding flat  was a time-average of the frames obtained in the ‘off’ nod position,  which had then been corrected for bad pixels in the same way as  the science frames and smoothed over using a Gaussian filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These  flats were then divided by the maximum value in the frame such that  the value of every pixel was between zero and one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then divided  the science frames by these smoothed, normalised sky flats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This  flat-fielding process was also repeated separately using a median  filter instead of a Gaussian filter as a means to test the robustness  of this step in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We proceed with the Gaussian filter and  discuss the impact of the choice of flat frame on the photometry of  the star and companion in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022) previously identified that background-  subtracted, flat-fielded ALES images contain residual structure that  cannot be described by purely Gaussian noise, in the form of time-  varying row and column discontinuities (faintly visible in the top  panel of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Such discontinuities are expected and arise from  the way in which the micro-spectra lie across multiple channels  of the LMIRcam detector (Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We followed the  method of Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022) to characterise and remove these  discontinuities by fitting a third-order polynomial to each row and  column in each frame (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Removing these systematics is  important as they could impact the precision of our differential light  curve, or even generate a false variability signal if the target moves  over them throughout the observing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Prior to fitting, we  applied circular masks at the locations of the star and the compan-  ion in each frame such that their flux did not contaminate the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  To find the position of the star in each time frame, we selected a  wavelength channel with a high stellar flux per frame (channel 52,  𝜆 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='69 µm) and fit the PSF core with a 2D Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The position  of the companion in each time frame was then identified using its  separation and position angle relative to the star and accounting for  the effect of the field rotation over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then masked the star  and the companion across all wavelength channels using circular  masks with diameters of 18 pixels and 5 pixels, respectively, before  fitting the third-order polynomials to each column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The resulting  values were then subtracted from the data to remove the column  discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This process was then repeated for each row in the  resulting image to remove the row discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  In addition to removing these systematic discontinuities, this  process has the effect of removing residual background flux not  eliminated by earlier processing steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This is indicated by the his-  tograms in Figure 2, which show that the noise distribution of the  data was offset from zero prior to the removal of the discontinuities  (in blue) but is approximately consistent with zero after this process  has been applied (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then used the position of the star  in each frame, found when applying the masks in the previous step,  to spatially align the data such that the star was in the centre of each  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Finally, we rotationally aligned the images by applying an  anticlockwise rotation corresponding to their parallactic angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We did not use any further post-processing methods that reduce  quasistatic speckle noise through the subtraction of reference PSFs,  such as Angular Differential Imaging (ADI, Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2006a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Although the field rotation over the night of observations was suffi-  cient enough to use these to remove noise with minimal companion  self-subtraction, doing so would also remove the unsaturated stel-  lar reference PSF, which is required to eliminate systematics in the  companion photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' It would not be possible to use the host  star PSF prior to ADI subtraction as a photometric reference for the  companion PSF after ADI subtraction, as the two would no longer  share the same systematic trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Furthermore, HD 1160 B is suffi-  ciently bright that it can be detected at ample S/N for our purposes  without further noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3  Wavelength channel selection  Although the final processed cubes contain data from 100 wave-  length channels across the observed wavelength range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2 µm,  not all of these channels are suitable for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The first  3 and final 10 channels contain flux from the adjacent spaxel in the  dispersion direction as an oversized spectral length is required to  extract the spectrum at each position, so the extracted data overlaps  slightly in the wavelength dimension (Stone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Further-  more, the dgvAPP360 contains a glue layer that causes up to 100%  absorption between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='15 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='55 µm (Otten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As described in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1, one of the key advantages  of a spectrophotometric approach is the option to exclude channels  that are known to cause systematic variability in the ‘white-light’  curve, hence improving the companion S/N and stability in the com-  bined image when compared to combining all wavelength channels  without any selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This is key to reducing large systematic ef-  fects that may otherwise dominate the variability signals that we are  aiming to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For the purposes of this technique demonstra-  tion, we proceed using 30 sequential wavelength channels (45-74,  spanning 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm) which all have a high throughput and do  not lie in the regions affected by the dgvAPP360 glue absorption,  significant telluric absorption, or the overlapping spectral traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In  the left panel of Figure 3 we show the median combination of these  channels in both wavelength and time, while the centre and right  panels respectively show example images from the median com-  MNRAS 000, 1–24 (2022)  6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  0  20  40  60  Columns only  Rows only  Combined  0  20  40  60  Pixels  0  25  50  0  20  40  60  0  25  50  Pixels  0  25  50  100  0  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='015  100  0  100  Counts  100  0  100  125  100  75  50  25  0  25  50  75  100  125  150  175  Counts  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Demonstration of the process for removing systematic row and column discontinuities present in background-subtracted ALES images, as applied  to a single frame of data in the 52nd ALES wavelength channel (𝜆 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='69 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Top panels: input frame prior to the removal of the discontinuities, which are  faintly visible as a chequered pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' All three panels are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Both HD 1160 A and B are masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Second row: results of the third-order polynomial  fits individually to the columns (left panel) and rows (centre panel), and the combination of both (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The combination of both was produced by first  fitting and removing the column discontinuities, and then repeating the process on the resulting image to fit and remove the rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We show row and column fits  to the input frame separately here to highlight their individual contribution to the original systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The star and the companion were masked during this  process, and the values to be removed at their locations were found through interpolation of the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Third row: data frame with the discontinuities removed by  subtracting the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The histograms in the bottom row show the distributions of the counts in the unmasked regions of the third row images, with the original  noise distribution in blue and the noise distribution after the discontinuities were removed in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The bottom right panel shows the version used in the  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  bined cubes in the time and wavelength dimensions only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The images  shown are those processed using the flat frame that was smoothed  using a Gaussian filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' the equivalent images as processed using  the median-smoothed flat frame are visually indistinguishable from  these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  4  GENERATING DIFFERENTIAL  SPECTROPHOTOMETRIC LIGHT CURVES  Variability arising from instrumental systematics and the effects  of Earth’s atmosphere, such as airmass, seeing, and tellurics, con-  taminate the raw flux of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Simultaneous flux mea-  surements of a photometric reference are required to eliminate this  contaminant variability and produce a differential light curve of the  companion, relative to the photometric reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although suitable  photometric references are generally absent when using corona-  graphs (Section 1), the dgvAPP360 coronagraph uniquely provides  an image of the host star simultaneously to the companion, allowing  the star to be used as the photometric reference when it is not satu-  rated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Its placement in the pupil plane also makes it inherently stable  and insensitive to tip/tilt instabilities (Otten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  MNRAS 000, 1–24 (2022)  Exoplanet variability with vAPP coronagraphs  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  RA [arcsec]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  Dec [arcsec]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  RA [arcsec]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  RA [arcsec]  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Reduced images of the HD 1160 system obtained with LBT/ALES and the dgvAPP360 coronagraph after all data processing steps and wavelength  channel selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Left: the median combination, in both wavelength and time, of all wavelength channels in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm range, covering a total integration  time of 11891 s (∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='31 h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Centre: example image from the same data cube but median combined only in the time dimension, along the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='64 µm wavelength  channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Right: example time frame (integration time = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4 seconds) resulting from a median combination in wavelength over the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm wavelength  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' All three images use the same arbitrary logarithmic colour scale, and are aligned to north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' North is up, and east is to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  Aperture photometry  We used version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0 of the Photutils Python package (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2022) to simultaneously extract aperture photometry of HD 1160 A  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We carried out this process for every individual frame in  each of the 30 wavelength channels in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm range  chosen in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3, with the aim of then combining these in  the wavelength dimension to produce the white-light flux for each  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Circular apertures with radii of 9 pixels (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1 𝜆/D) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5  pixels (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9 𝜆/D) were used for the host star and the companion,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To estimate the background flux at the position of the  star, we also extracted photometry in a circular annulus centred on  the stellar location with inner and outer radii of 11 and 16 pixels,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' It was not possible to use this method to estimate the  background flux at the position of the companion as the companion  lies close to the edge of the field of view in some frames, limiting  the space available to place an annulus that would be statistically  wide enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We therefore instead followed an approach used by  Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2021), estimating the background at the companion  location by masking the companion and extracting flux in a circular  annulus centred on the host star, with a width of 6 pixels at the radial  separation of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As most of the residual background in  each frame was eliminated by the data processing steps in Section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2, these background values are close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These apertures and  annuli are shown in Figure 4, overlaid on a single processed time  frame of data in the 52nd ALES wavelength channel (𝜆 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='69 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We then removed the residual background from our stellar and  companion flux measurements by multiplying the mean flux per  pixel in the background annuli by the area of the corresponding  apertures and subtracting the resulting values from the aperture  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then produced single white-light measurements for  both the companion and the star at each time frame by taking the  median combination of the photometric measurements across the 30  wavelength channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These raw time series, uncorrected for shared  variations introduced by Earth’s atmosphere (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' before division),  are shown in grey in the top two panels of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The discrete  gaps in integration reflect time spent off-target due to the two-point  on/off nodding pattern used to enable background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We  also plot the data binned to 18 minutes of integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We  binned the data by taking the median value in each time bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  error on the binned fluxes are the Gaussian approximation of the root  mean square (RMS) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' median absolute deviation (MAD) × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='48  of the flux measurements inside each bin divided by  √  𝑁 − 1, where  𝑁 is the number of frames per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Next, we removed variability due  to Earth’s atmosphere and other systematics from the unbinned raw  flux of the companion using the unbinned raw flux of the host star,  which acts as a simultaneous photometric reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' By dividing the  unbinned companion flux by the unbinned stellar flux, we eliminate  trends common to both and produce a differential light curve that  only contains non-shared variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Assuming that the host star is  not itself varying (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2), the resulting differential light  curve reflects the intrinsic variability of the companion plus any  contamination arising from non-shared systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We show this  raw differential light curve in the third panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The bottom  panel shows a zoomed-in view of the binned version, with tighter  limits on the y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  In the following sections we examine a number of physical,  instrumental, and processing factors that may be correlated with  non-astrophysical features in the differential white-light curve, and  in Section 5 we model and remove non-shared variations arising  from some of these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  TESS light curves of host star HD 1160 A  Although the vast majority of A-type stars generally vary well below  the ∼1% level, a small fraction vary at a much higher level, showing  up to ∼15% variations (Ciardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To test our assumption  that the host star HD 1160 A is not varying at a level that impacts our  differential light curve, we used data from the TESS mission, which  is publicly available on the Mikulski Archive for Space Telescopes  (MAST) data archive1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The TESS mission observed HD 1160 A  for 25 days in Sector 42 (from 2021 August 21 to 2021 September  14) and for 26 days Sector 43 (from 2021 September 16 to 2021  1 MAST data archive portal: https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='edu/portal/  Mashup/Clients/Mast/Portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='html  MNRAS 000, 1–24 (2022)  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  RA [arcsec]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  Dec [arcsec]  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The apertures (continuous lines) and annuli (dashed lines) used to  extract photometry and background measurements for host star HD 1160 A  (in yellow) and companion HD 1160 B (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The image is a single time  frame from the 52nd ALES wavelength channel (𝜆 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='69 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' North is up,  and east is left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The orange aperture is placed at the location of HD 1160 B,  which is too faint to be visible in a single frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The companion was masked  when extracting photometry in the annulus for the companion background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  October 11), 51 days in total, with 2 minute cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The TESS de-  tector bandpass covers a broad-band wavelength range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='6-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0 µm  (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015), which does not overlap with our LBT/ALES  observations in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2 µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, as stars are generally  less variable in the infrared than in the optical regime, any variations  in the TESS light curve of HD 1160 A should represent an upper  limit for its variability at the wavelengths covered by ALES (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Solanki & Unruh 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Unruh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Fröhlich & Lean 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Davenport et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ermolli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Rackham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Each TESS pixel covers 21′′ on sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This  means that HD 1160 A, HD 1160 B (at a separation of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='78′′), and  HD 1160 C (at a separation of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1′′) are not resolved separately  in the TESS images and appear as a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, as both  HD 1160 B (Δ𝐽 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='10 mag, Δ𝐿′ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='12 mag) and C  (Δ𝐽 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='04 mag, Δ𝐿′ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='803 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='005 mag) are far fainter  than HD 1160 A, especially at shorter wavelengths, it is reasonable  to assume that flux of HD 1160 A will dominate in the TESS data  (Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We first masked out any bad quality exposures using the one-  hot encoded quality mask in the ‘QUALITY’ keyword in the header  of the light curve files provided by the TESS Science Processing  Operations Center (SPOC, Jenkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016) on MAST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then  used the ‘CROWDSAP’ keyword in the header to get an estimate  of the ratio of target flux to total flux in the optimal aperture used  for the PDC SAP (Pre-search Data Conditioning Simple Aperture  Photometry) flux (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The ‘CROWDSAP’  value for sector 42 and 43 indicates that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='17% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2% flux is  from dilution by nearby sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We subtract the estimated diluted  flux from each exposure in both the sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The resultant light  curves for both the sectors are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although the  TESS observations are not contemporaneous with our LBT/ALES  observations, we do not see variations above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='03% in the light  curve of HD 1160 A over the timescale covered by the two TESS  sectors (51 days).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As this is far smaller than the precision of our  differential light curve, we proceed with the assumption the host star  HD 1160 A is non-varying within the flux precision of our analysis  of the variations in the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3  Impact of wavelength calibration and flat-field smoothing  In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1, we described the process used to perform the wave-  length calibration of the raw data and to extract the micro-spectra  into a three-dimensional image cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This step was repeated sepa-  rately using the wavelength calibration that was the most divergent  of the 6 obtained throughout the observing sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9 µm  fiducial spots for this wavelength calibration were the most signifi-  cantly offset compared to the one that was originally used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Over the  course of the night, the projection of the micro-spectra onto the de-  tector drifts slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' If this drift is significant then a particular wave-  length calibration may not remain accurate for the entire observing  sequence, potentially producing a false variability signal when the  wrong part of the spectrum is assigned to a given channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Repeat-  ing our spectral extraction using a wavelength calibration obtained  at a different point during the observations allows us to test whether  this effect has a significant impact on the photometry of the target  star and the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' After extracting the micro-spectra using  the alternative wavelength calibration, we then processed the data  again in full to produce an alternative differential light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  original and alternative wavelength calibrations were obtained at 4  hours 57 minutes (08:25:00 UT) and 1 hour 26 minutes (04:54:00  UT) after the beginning of the observing sequence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We  plot the resulting alternative stellar and companion fluxes, and the  differential white-light curve (binned to 18 minutes) in Figure 7,  alongside the originals from Figure 5 for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The differ-  ential light curves in each case are consistent within 1𝜎, indicating  that the extracted photometry is sufficiently robust to changes in the  wavelength calibration and sub-pixel mis-registration of the spatial  profiles for each microspectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2, we described our method for applying a flat-  field correction to the data to calibrate for the non-uniform response  of the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Incorrect flat-fielding can lead to a false variability  signal if the companion moves over regions of the detector with a  non-uniform response that has not been properly calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To test  the robustness of our differential light curve to differences in the  flat used we processed the data in two separate streams, using a flat  that had been smoothed over using a Gaussian filter and a median  filter, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We plot the resulting differential light curves in  Figure 8 for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The two differential light curves are in  close agreement and every binned data point lies well within their  1𝜎 error bars, indicating that the method for producing the flat is  robust and does not significantly affect the final images or extracted  photometry of the star or companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  5  DETRENDING THROUGH LINEAR REGRESSION  Trends shared by the star and companion fluxes are removed in  the differential light curve (see bottom panel of Figure 5), but any  non-shared trends will still be present, including the intrinsic com-  panion variability signal that we aim to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To improve our  sensitivity to the companion’s variability, we needed to remove  the non-astrophysical residual trends in the differential light curve,  which can arise from both telluric and instrumental sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In  MNRAS 000, 1–24 (2022)  Exoplanet variability with vAPP coronagraphs  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  Raw white-light Star-only  Raw white-light Star-only (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  1  0  1  2  3  Raw white-light Companion-only  Raw white-light Companion-only (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  1  0  1  2  3  Raw white-light Companion/Star  Raw white-light Companion/Star (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  0  1  2  3  4  5  6  7  8  Time (hours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Raw white-light Companion/Star (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  Normalised Flux  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The top two panels show the normalised raw white-light flux (𝜆 =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm wavelength range) of host star HD 1160 A (in grey, top panel) and  companion HD 1160 B (in grey, second panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The blue and orange lines are the same fluxes of the star and companion, respectively, binned to 18 minutes  of integration time per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Each has been normalised by dividing by the mean value across the full sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The third panel shows, in grey, the resulting  differential white-light curve when the white-light of the companion is divided by the white-light of the star to remove trends shared by both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The same data  binned to 18 minutes of integration time per bin is shown in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The light curve in the bottom panel is the same as the third panel but zoomed in on the  y-axis with a dashed line at normalised flux = 1 for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Provided that there is no contamination from stellar variability, variations in this light curve are a  combination of any intrinsic companion variability and trends arising from non-shared systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The gaps in the data are due to the two-point on/off nodding  pattern used to collect sky frames for background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  MNRAS 000, 1–24 (2022)  10  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' TESS 2 minute cadence PDC SAP light curve of HD 1160 A  obtained from TESS sectors 42 (black points) and 43 (red points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Light  curves from both sectors have been normalized to the median count level for  the respective sector to remove the systematic change in the base flux level  from one sector to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Overplotted is the binned TESS light curve for  visual aid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The standard deviation of the stellar flux is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='027% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='03%  for sectors 42 and 43, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  the field of exoplanet transmission spectroscopy, such systematic  trends are generally modelled and removed from light curves using  either a polynomial model created by simultaneously fitting several  decorrelation parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' de Mooij & Snellen 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' de Mooij  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Brogi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Stevenson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Diamond-  Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018, 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Todorov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019), or a non-parametric  model produced using Gaussian processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012,  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Montet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Carter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Diamond-Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Un-  like traditional transmission spectroscopy observations, our target  is significantly fainter than the simultaneous reference that we use  for detrending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Furthermore, the target was not pixel-stabilised for  these observations and moved across the detector throughout the  night, so we might predict that the measured light curves could be  significantly correlated with the change in position of the compan-  ion and the star on the detector over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Knowing how to remove  these systematics is key to obtaining high-precision light curves in  future observations of directly imaged exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For space-based  observations the instrumentation is generally sufficiently stable such  that the systematics are repeatable over time, allowing them to be  well characterised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This is more challenging for ground-based ob-  servations, like those here, which are inherently less stable as Earth’s  atmosphere introduces systematics that can vary night by night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  As a basic demonstration of how to remove such residual trends  from ground-based differential light curves of directly imaged plan-  ets, we here used a multiple linear regression approach to simul-  taneously fit several possible sources of systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This is not  intended as a strictly rigorous statistical analysis of the trends in  the light curve, but is done to perform an initial investigation into  which parameters have the greatest impact and to illustrate an exam-  ple approach of how to do this for future observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As studies  move towards increasing precision to measure smaller amplitude  variability, one might instead consider approaches using Gaussian  processes or similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  An investigation of the LBT telemetry and white light im-  ages revealed eight physical and instrumental factors, shown plotted  against time in Figure 9, that varied notably during the observing  sequence and may be correlated with the residual trends in the dif-  ferential light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We therefore included these parameters in the  linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The first three of these were air temperature, wind  speed, and wind direction, shown in the three panels on the left-hand  side of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We also considered airmass, which is shown in  the top-right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' While the light from the companion and its host  star pass through almost identical airmass (maximum difference  ∼10−5), their significantly different colours mean that atmospheric  extinction due to absorption and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Rayleigh scattering can result  in a differing airmass dependence, even when such scattering effects  are reduced at our longer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2 µm wavelength range (Allen 1955;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Broeg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Croll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The remaining four parameters included in the linear regres-  sion were the x- and y- pixel positions of the star and the companion  in the images prior to spatial and rotational alignment, shown in the  centre-right and bottom-right panels of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The dgvAPP360  is located in the pupil plane, meaning that drifts in the locations  of the target PSFs on the detector do not affect its response and  performance as it applies the phase modification to every source in  the field of view (Otten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' How-  ever, systematics could still be introduced by such drifts if there are  variations in the instrumentation or detector response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' A number of  sharp discontinuities in the y-position, and a singular discontinuity  in the x-direction, can be seen in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These discontinuities  are the result of manual positional offsets applied during the ob-  servations to keep the star close to the centre of the small (∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2′′x  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2′′) ALES field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These offsets were always applied while  in the off-source nod position, and always along one axis at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The largest discontinuities in the x- and y-positions correspond to  shifts of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1′′ along the given axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Neglecting the discontinuities,  the drift of the stellar PSF in both the x- and y-directions follow  arcs with turn-overs approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5 hours into the observing se-  quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This slow drift is correlated with the pointing altitude of  the telescope and arises from flexure of the ALES lenslet array as  the telescope rotates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As the observations were obtained in pupil-  stabilized mode such that the field of view was rotating over time,  the change in position of the companion throughout the data has an  additional rotational component compared to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The drift aris-  ing from the flexure of the lenslet array therefore instead produces  an inflection point ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5 hours in the case of the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We used the linear regression tools in version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2 of the  scikit-learn Python package (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2011) to simultane-  ously fit these input parameters and produce a model fit to our  differential light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The coefficients of the linear model pro-  duced by the linear regression are shown in Table 2, and the model  itself is shown in green in the top panel of Figure 10, relative to the  raw differential white-light curve in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then divided the raw  differential light curve by this model to produce a detrended version,  shown in red in the bottom panel of Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We also overplot the  raw differential light curve from the bottom panel of Figure 5, prior  to detrending, in purple to allow the two to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We also  repeated this detrending process to produce detrended differential  light curves for each of the 30 individual wavelength channels over  the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm wavelength range that comprise the white-light  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These are shown in Figure 11, again binned to 18 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We note that while ALES has a resolution of R∼40, the raw data  were spectrally extracted into 100 wavelength channels and so there  is some correlation between wavelength channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  MNRAS 000, 1–24 (2022)  300  S  electrons/s  200  100  NMMMM  0  xn  100  200  300  HD 1160 A, Sector 42  400  HD 1160 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sector 43  450  460  480  470  490  500  Time[BiD]  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='459e6Exoplanet variability with vAPP coronagraphs  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='04  Raw white-light Star-only (original wavecal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=')  Raw white-light Star-only (alternate wavecal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Raw white-light Companion-only (original wavecal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=')  Raw white-light Companion-only (alternate wavecal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=')  0  1  2  3  4  5  6  7  8  Time (hours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Raw white-light Companion/Star (original wavecal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=')  Raw white-light Companion/Star (alternate wavecal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=')  Normalised Flux  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Impact of different wavelength calibrations on the star-only and companion-only fluxes, and the differential white-light curve, shown in 18 minute  binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The alternate wavelength calibration was chosen as the one that diverged most from the one originally used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The times at which the wavelength  calibrations were obtained are indicated by vertical dashed lines in the same colours as the corresponding light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  6  RESULTS  The detrended differential white-light curve (Figure 10) shows  sinusoidal-like variability over a timescale of a few hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In the  binned light curve, the normalised flux ranges from a minimum of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='91 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='874 hours to a maximum of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='13 at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='961 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' To better  allow us to estimate the differential precision that we achieve in our  light curve, we fitted the variability and removed it from our light  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As the variability signal appears periodic, we used the NASA  Exoplanet Archive periodogram service2 to apply the Lomb-Scargle  algorithm to the unbinned detrended differential white-light curve  2 https://exoplanetarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='edu/cgi-bin/  Pgram/nph-pgram  and thereby search for sinusoidal periodic signals (Lomb 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Scargle 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The strongest peak in the resulting periodogram (top panel,  Figure 12) is at a period of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='242 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then fit a sinusoid to the  light curve using this period as an initial guess, which returned a  function with the same period (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='239), a semi-amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='088, a  phase shift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='228, and a 𝑦-offset of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This sinusoid is shown  overplotted on the detrended light curve in the centre panel of Fig-  ure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The differential light curve was then divided by the fitted  sinusoid to remove the variability signal to the first order (centre  panel residuals, Figure 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The bottom panel of this figure is the  same as the panel above but with the data phase-folded to the period  of the fitted sinusoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Next, we followed the method of Kipping &  Bakos (2011) to assess the degree of “red” (correlated) noise in our  light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We binned our detrended differential white-light curve,  MNRAS 000, 1–24 (2022)  12  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  0  1  2  3  4  5  6  7  8  Time (hours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Raw white-light Companion/Star (Gaussian-smoothed flat)  Raw white-light Companion/Star (Median-smoothed flat)  Normalised Flux  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The raw differential companion/star white-light curves that are  produced when the flat-field correction uses a flat that was smoothed using  a median filter and a Gaussian filter, in turquoise and purple, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The latter light curve is the same as that in the bottom panel of Figure 5,  reproduced here for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The decorrelation parameters 𝑥𝑖 used for the linear regression and  the corresponding coefficients 𝑐𝑖 and intercept 𝑐0 of the resulting linear  model fit to the raw differential light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The linear model fit is then  given by 𝑦 = (�𝑛  𝑖=1 𝑐𝑖 𝑥𝑖) + 𝑐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Parameter (𝑥𝑖)  Value (𝑐𝑖)  Airmass  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='28782524  Air temperature  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='10690617  Star x-position  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='04819219  Star y-position  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='04368692  Companion x-position  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='02608288  Companion y-position  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='02129148  Wind speed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00059976  Wind direction  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00050927  Intercept (𝑐0)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='318553799  with the sinusoid removed, to a range of bin sizes, before normal-  ising and subtracting one to centre around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then measured  the RMS of each resulting binned light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These RMS values  are plotted against bin size in Figure 13, alongside the expectation  of independent random numbers as a function of bin size i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' the  white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For our chosen binning of 18 minutes of integration  time per bin, which has a bin size of 200 frames per bin, we find  an RMS of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For comparison, the RMS of the detrended dif-  ferential white-light curve prior to the removal of the sinusoid, but  with the same binning, is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We therefore conclude that the  light curve of HD 1160 B shows variations with a semi-amplitude  of ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8% or peak-to-peak amplitude of ∼17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='6%, and that the dif-  ferential precision achieved in the binned light curve is at the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7%  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The amplitude of the variations is therefore above the mea-  sured precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Furthermore, this estimate of the precision is likely  a conservative one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' the variability signal is unlikely to have been  perfectly removed by the sine fit and so the measured RMS values  may be higher than the true limiting precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' A caveat of this result  is that the baseline of our observations is only ∼7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='81 hours, so we  only cover ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4 periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Additional data is therefore needed to con-  firm the periodicity and amplitude of the variability of HD 1160 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We discuss these results further and compare the precision achieved  to similar studies in the literature in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  7  DISCUSSION  In this section we discuss the relative impact of the decorrelation  parameters on our results, and the physical explanations of the sys-  tematics they introduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We also compare the precision that we  achieve to other variability studies in the literature that use differ-  ent techniques, and discuss the potential application of differential  spectrophotometry in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  Impact of decorrelation parameters  Using the linear regression coefficients of each parameter from Ta-  ble 2, we can assess which parameters have the greatest impact on  the light curve of HD 1160 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The small angular separation of the  companion and star might suggest that the effect of airmass would  be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Airmass can be an important systematic for differential  spectrophotometric observations where other stars are used as the  simultaneous photometric reference as the angular separation be-  tween the target and the reference can be large, causing the light  from the two objects to pass through different atmospheric columns  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Broeg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, in our case  we use the companion’s host star as the photometric reference, so  the angular separation between the two is much smaller than is gen-  erally the case for observations using reference stars in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  However, there is a significant colour difference between the star  and the companion, which leads to different degrees of extinction at  a given airmass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Therefore, we expect an airmass dependence and  indeed it has the largest coefficient in the detrending model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Similar  extinction effects are often seen in studies of transiting exoplan-  ets, and can also exhibit a non-linear wavelength dependence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We also find the air temperature at LBT to be one of the  parameters that is most correlated with our differential light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  As the air temperature changes, this can potentially cause slight  changes in the optical path of the telescope or instrument that lead  to this correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We further predicted that the positions of the companion and  the star could introduce significant non-shared systematics to our  measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' as HD 1160 A was not pixel-stabilised during our  observations, we are sensitive to both intra- and interpixel varia-  tions as the star and the companion move across the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Both  the star and the companion changed position on the detector due  to flexure of the ALES lenslet array as the instrument moved and  positional offsets applied intentionally to keep the target close to  the centre of the field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We also nodded on and off of the  target to enable background subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' While the process of nod-  ding is itself relatively accurate, it is not repeatable at a sub-pixel  pointing precision, introducing a slight offset error between nods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Furthermore, LBTI data is always pupil-stabilized, so the field of  view was rotating throughout the night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although it may have been  optimal to fix the star and companion positions to the same detector  pixels for the duration of the observations, this was not possible due  to the effect of the lenslet array flexure and the lack of instrument  derotator in the LBTI architecture (Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However,  we find that these positional changes are not the most correlated  with our differential light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' It is possible that this is because  the light from the star and companion is spread out across multiple  detector pixels when spectrally dispersed, reducing the impact of  MNRAS 000, 1–24 (2022)  Exoplanet variability with vAPP coronagraphs  13  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='25  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='50  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='75  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='25  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='50  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='75  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00  Air Temperature (°C)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Airmass  0  1  2  3  4  5  Wind speed (m s  1)  30  32  34  36  38  Pixel position  Star y-position  Star x-position  0  1  2  3  4  5  6  7  8  Time (hours)  200  250  300  350  400  Wind direction (degs E of N)  0  1  2  3  4  5  6  7  8  Time (hours)  10  20  30  40  50  60  Pixel position  Companion y-position  Companion x-position  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The decorrelation parameters used when detrending the differential light curve via linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' From top to bottom, the left panels show the air  temperature (in ℃), wind speed (in m s−1), and wind direction (in degrees east of north) at the observatory as a function of time, as extracted from the FITS  headers of the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The panels on the right show airmass, the x- and y-positions of the star in pixels, and the x- and y-positions of the companion in pixels,  as a function of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The gaps in time reflect the on/off nodding pattern of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The companion and star positions are those in the images prior  to spatial and rotational alignment, and the sharp discontinuities in pixel position are due to manual offsets applied to keep the star close to the centre of the  small field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The stellar positions follow arc-shaped trends aside from these discontinuities, which correlate with the pointing altitude of the telescope  and arise from flexure of the ALES lenslet array as the telescope rotates throughout the night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The change in the companion position has an additional trend  due to the 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7° rotation of the field of view over the observing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  MNRAS 000, 1–24 (2022)  14  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  1  0  1  2  3  Raw white-light Companion/Star  Linear regression model  0  1  2  3  4  5  6  7  8  Time (hours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='25  Raw white-light Companion/Star (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  Companion/Star, model removed (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  Normalised Flux  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The model produced by the linear regression using the decorrelation parameter coefficients (in Table 2) is shown in the top panel in green alongside  the raw differential light curve in grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The bottom panel then shows in red the light curve produced when the raw differential light curve is divided by the  linear regression model to remove the modelled trends that are not shared by the stellar and companion fluxes, binned to 18 minutes of integration time per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The raw differential light curve (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' prior to detrending) is also shown faintly in purple, reproduced for comparison from the bottom panel of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  any systematic issues arising from any single pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The light of our  target is dispersed across wavelength, similar to a technique often  used for high-precision differential photometry whereby a telescope  is intentionally defocused to disperse starlight over the detector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  de Mooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2011, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Crossfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Croll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  This has the effect of reducing systematics due to intrapixel varia-  tions and minimises the impact of any residual flat-field errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  step of recombining our data into a white-light curve may therefore  have helped to reduce systematic trends that would otherwise have  been introduced by the positional movement of the targets through-  out the night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The use of a spectrograph also has the additional  benefit of allowing us to remove individual wavelength channels  that are found to contain defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For this dataset, this allowed us  to leave out channels affected by overlapping spectral traces, sig-  nificant telluric absorption, and absorption by the glue layer of the  dgvAPP360, as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Lastly, we find that the wind speed and direction are the least  correlated with our differential light curve, but note that the condi-  tions on the night were exceptionally stable and so cannot rule out  that these factors could have an impact in less optimal conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  For future observations, residual systematics in differential  light curves of directly imaged companions could be removed us-  ing more advanced methods from the exoplanet transmission spec-  troscopy and high-precision secondary eclipse literature such as  fitting the data using a Gaussian process regression (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Gibson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Nikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Diamond-  Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In general, methods used  to identify trends in transmission spectroscopy data also include a  model of the exoplanet transit itself, allowing the transit to be de-  tected even when the strength of the signal is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However,  this is possible because the expected shape of the signal is well un-  derstood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This is more difficult for searches for variability in directly  imaged companions, where the expected shape of the variability sig-  nal is not necessarily well-known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Furthermore, many  of these methods assume a linear relationship between systematics  and trends in the light curve, while the telluric and instrumental sys-  tematics present in time-series data can be complex and non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  In the future, an optimal approach should account for the functional  form of the correlated parameters (Panwar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  MNRAS 000, 1–24 (2022)  Exoplanet variability with vAPP coronagraphs  15  0  2  4  6  8  Time (hours)  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='221  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='605 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='13  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='62 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='15  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='635 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='15  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='65 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='167  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='664 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='158  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='679 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='103  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='693 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='149  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='708 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='23  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='722 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='122  Companion/Star, model removed (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  Normalised Flux  0  2  4  6  8  Time (hours)  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='736 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='163  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='75 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='163  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='764 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='247  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='778 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='259  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='791 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='209  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='805 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='312  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='819 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='357  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='832 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='12  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='846 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='123  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='859 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='176  Normalised Flux  0  2  4  6  8  Time (hours)  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='872 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='241  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='885 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='314  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='898 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='398  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='911 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='564  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='924 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='36  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='937 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='298  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='95 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='34  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='963 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='361  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='976 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='201  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='988 m, RMS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='335  Normalised Flux  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In red, we show the detrended versions of the individual differential light curves in each of the 30 wavelength channels that were combined to  produce the white-light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These wavelength channels cover 𝜆 =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm, and are binned to 18 minutes of integration time per bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' An offset factor of  2 has been applied between each light curve to separate them from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Overall variability appears to increase with longer wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Differential light curves  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  Variability interpretation  In Section 6 we found that HD 1160 B shows variations with a  semi-amplitude of ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8% (or peak-to-peak amplitude of ∼17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  To compare this result to literature observations of similar objects,  we must consider both spectral type and the wavelengths at which  variability has been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022) found that virtually  all L dwarfs are likely to be variable at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='05-3% range, and  several studies have measured higher variability, up to 26% (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Radigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Radigan 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, there is evidence that brown  dwarf variability amplitude may have a a strong wavelength depen-  dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For example, HST observations of highly variable L dwarf  companion VHS 1256-1257 b identified a large variability ampli-  tude of 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7% at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='27 µm, while Spitzer observations at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5 µm  found a far lower amplitude of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='76±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='04% (Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Miles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2020b) do note,  however, that the HST and Spitzer observations were not obtained  contemporaneously and so the atmosphere of VHS 1256-1257 b and  hence its variability properties are likely to have changed substan-  tially over the intervening timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Comparisons of large surveys  also suggest that variability amplitudes are lower in the mid-infrared  than in the near-infrared, although there is evidence for a weaker  wavelength dependence and enhanced mid-infrared variability am-  plitudes for the young isolated brown dwarfs most similar to sub-  stellar companions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Radigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Metchev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The amplitude of the L-band variability we measure for  HD 1160 B is quite extreme compared to literature results, but  it is important to note that there have been very few variability stud-  ies for substellar companions in the mid-infrared, making direct  comparison difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' High-amplitude variability in brown dwarfs is  generally attributed to heterogeneous surface features, such as spots  or clouds of varying thickness, rotating in and out of view as the  object rotates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Artigau 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Some light curves show more complex features that cannot be mod-  elled with a single atmospheric feature, or features that evolve over  short or long timescales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Artigau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Metchev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These features and time evolution may arise from changing  weather systems, or bands of clouds which rotate within the target’s  atmosphere and generate waves on a global scale (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Tan & Showman 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For HD 1160 B, observations over  MNRAS 000, 1–24 (2022)  16  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Time (hours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='30  Flux  Detrended Companion/Star  Detrended Companion/Star (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  Sinusoidal fit  0  1  2  3  4  5  6  7  Time (hours)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Flux  Phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='25  Flux  Detrended Companion/Star  Detrended Companion/Star (18 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' bins)  Sinusoidal fit  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  Phase  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Flux  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Top panel: the Lomb-Scargle periodogram for the unbinned  detrended differential white-light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The strongest peak is at a period  of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='242 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Centre panel: the same detrended differential white-light  curve from the bottom panel of Figure 10, unbinned in grey and binned  to 18 minutes of integration time per bin in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The blue line is the fitted  sinusoid with a semi-amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='088 and a phase shift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  residuals when the fitted sinusoid is divided out from the light curves are  shown underneath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bottom panel: the same as the panel above, but phase-  folded to a period of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  a longer baseline are required to be able to characterise any time  evolution in the variability signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  However, the spectral type of HD 1160 B is unclear as it has  a highly peculiar spectrum that cannot be satisfactorily fit with  spectral models or templates in current libraries, and some studies  suggest that it could instead be a late-M dwarf (Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' If HD 1160 B is an M dwarf,  its variability would most likely arise from cool star spots caused  by magnetic activity in its photosphere, which are common in M-  100  101  102  103  Frames per bin (bin size)  10  2  10  1  100  RMS  Light curve RMS after linear regression  and sinusoidal variability removed  White noise model  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The RMS of the binned detrended differential white-light curve,  after removing sinusoidal variability, is shown in orange as a function of  bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The black line shows the theoretical white noise model, or the  expectation of independent random numbers for a given bin size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  dwarfs and would rotate in and out of view in much the same way  as the cloud features of lower mass objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Barnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Frasca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Scholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ghosh  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Johnson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The properties of such variability  can be highly dependent on the spot distribution and fractional spot  coverage of a given object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' some M-dwarfs have a very high cov-  erage with multiple starspots covering as much as 20-50% of their  fractional surface area inhomogeneously (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' O’Neal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Morales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Irwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Jackson  & Jeffries 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Spot-induced variability amplitudes for M-dwarfs  generally range from the subpercent level up to around ∼5% (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Rockenfeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Charbonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Birkby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Nefs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' de Mooij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although observed far  less often in the infrared compared to the optical, flaring events can  induce far stronger variability in M-dwarfs at amplitudes ranging  from the subpercent level up to tens of percent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Tofflemire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Goulding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Our measured variability amplitude for  HD 1160 B is therefore also on the higher end of what has been ob-  served for earlier spectral types such as late M- and early L-dwarfs,  barring flares, although we again note the lack of literature studies  of similar objects in the L-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  While the variability observed for HD 1160 B appears high,  another point to consider is its orbital inclination, which the latest or-  bital fits suggest is close to edge-on as viewed from Earth (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='0+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3°,  Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' If the obliquity of HD 1160 B is aligned with  its orbit such that we are viewing its rotation close to edge-on, the  observed variability amplitude is likely to compose a much larger  fraction of its true variability compared to if it were viewed face-on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Indeed, (Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017) demonstrated that the highest variability  amplitudes are seen for targets with close to edge-on viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  If we interpret the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='24 h sinusoidal variation we observed as  the true rotation period of HD 1160 B, we can further consider  this within physical limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The breakup period of a rotating  object is dependent on its radius, which is itself age-dependent, and  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Both age and mass are poorly constrained for HD 1160 B:  literature results place the system’s age in the 10-300 Myr range  (Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Maire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019), and  mass estimates range from ∼20 MJup to 123 MJup (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Mesa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2020) calculated the breakup  periods of brown dwarfs as a function of age by equating equatorial  MNRAS 000, 1–24 (2022)  15  10  ower  P  2  0  10  10  10  10  PeriodExoplanet variability with vAPP coronagraphs  17  velocity with the escape velocity, accounting for radial contraction  over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' When we compare our measured HD 1160 B variability  period of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='24 hours to their results (their Figure 13), we find that  this is a physically feasible rotation period for most possible com-  binations of mass and age from the literature, albeit very close to  the breakup period in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' An alternative explanation is that  the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='24 hour variability signal that we see is produced by multiple  features in the atmosphere of HD 1160 B, and that its rotation period  is actually longer (Leggett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Additional observations of  this variability over a longer baseline will help to further charac-  terise its origin and confirm whether its periodicity reflects that of  the companion’s rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The detrended differential light curves for each of the 30 indi-  vidual wavelength channels over the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm wavelength range  that comprise the white-light curve (Figure 11) show increasing sta-  tistical errors at longer wavelengths, as expected as our S/N is lower  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Using the RMS of each light curve as a metric to compare  the scatter in each channel, we do see tentative evidence of increas-  ing variability towards longer wavelengths beyond the increase of  RMS expected from the S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, as the total baseline of our  observations is only a single night, we see too few repetitions to  be confident of variability patterns in individual wavelength chan-  nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Additional spectrophotometric data will therefore be required  to confirm this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although we modelled the overall variability in our  white-light curve with a single sinusoid, it is also possible that the  phase and amplitude is different per wavelength channel as distinct  atmospheric features at separate locations in the atmosphere of the  companion may produce variability with a different wavelength de-  pendence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Furthermore, although the overall wavelength range of  these 30 channels is relatively small, different wavelengths probe  different pressure levels in the companion’s atmosphere, and hence  different layers of the atmosphere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Buenzli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2  Light curve precision  In Section 6, we found that after removing a single sinusoid from  the data, we achieved a precision of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7% in the detrended dif-  ferential light curve when it is binned to a bin size of 200 data  points, corresponding to 11 bins of 18 minutes of integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  There have been three previous studies searching for variability in  substellar companion from the ground;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2016), Biller  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2021), and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022) each conducted variability  searches on the HR 8799 planets using satellite spots as photomet-  ric references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In a pilot variability study, Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2016) reach  a ∼10% planet-to-planet photometric accuracy for SPHERE obser-  vations of 25 minute cadence when data from different nights are  combined for a total telescope time of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2021)  goes further with SPHERE to conduct a longer (>4 hours) search,  successfully constraining the sensitivity to variability to amplitudes  >5% for HR 8799b and >25% for HR 8799c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' More recently, Wang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022) used SCExAO/CHARIS to improve the variability con-  straints of HR 8799c to the 10% level, and HR 8799d to the 30%  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' They did this by combining the use of satellite spots with a  spectrophotometric approach similar to the one we present in this  paper, using the CHARIS IFS to disperse the light into individ-  ual spectra before recombining the channels into wider bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' At  first glance, the sensitivity that we achieve for HD 1160 B here  may appear to compare favourably with these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, it is  important to consider a number of caveats that make direct compar-  ison unjustified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' All three of the HR 8799 studies were conducted  at near-infrared wavelengths with 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2-m telescopes, while ours was  in the mid-infrared with an 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4-m telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' More significantly, the  HR 8799 planets are fainter than HD 1160 B, with contrasts of  Δ𝐻 = 8 − 10 mag compared to their host star, which has a H-band  magnitude of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='28 mag (Marois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2008, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' HD 1160 B is  brighter, with a contrast of Δ𝐿′ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='35 mag compared to a host star  with an 𝐿′-band magnitude of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='06 mag (Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  lower sensitivity to variability achieved by these studies is therefore  partially a reflection of the intrinsically lower fluxes of their targets,  which leads to higher errors on their photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  However, each of the HR 8799 variability studies also found  that the satellite spots can demonstrate individual variations of their  own and are often anti-correlated with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This means that  they may not always serve as appropriate photometric references  with which to detrend the light curve of a companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  (2022) found that the flux ratio of the SCExAO satellite spots shows  time variation with a scatter of ∼3% across a night, and can show  even larger variations on a shorter timescale, up to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This poten-  tially sets a limit to the precision that can be achieved using satellite  spots, particularly on nights where observing conditions are less  stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  A key advantage of the dgvAPP360 compared to satellite spots  is its simplicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' the photometric reference it provides is simply an  image of the host star, and so it does not suffer from the same cor-  related systematics as the satellite spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Differential photometry  between the companion and the star can be carried out directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' It  may be possible to reach an even deeper precision through differen-  tial spectrophotometry with a dgvAPP360 than the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7% level that  we achieve here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Indeed, if we compare the detrended light curve  RMS as a function of bin size to the white noise expectation (Figure  13), it continues to follow the trend of the white noise and does not  plateau implying that we have not yet reached any noise floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This  means that in principle the precision of the differential light curve  would improve further if more data from additional epochs was  added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This also indicates that this technique should remain usable  for companions with less favourable contrasts than HD 1160 B,  such as those in the planetary-mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, more data  per bin will be required to achieve the same precision for a fainter  companion, so the time-sampling in the binned light curves may be  less fine in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Many transiting exoplanet studies make use of a region of  the target light curve that is expected to be flat (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' an out-of-  transit baseline) to test the degree to which systematics have been  corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' While we have shown here that key systematic trends are  successfully removed in the detrended differential white-light curve  of HD 1160 B, we do not have such a baseline to verify the level  of impact of any remaining systematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The possibility therefore  remains that an unknown systematic could be present that has not  been accounted for by any of the processes that we’ve applied here,  and could be responsible for the variability that we see in the light  curve of HD 1160 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, this is also inherently the case for  any study that explores the variability of isolated brown dwarfs and  planetary-mass objects, stellar variability due to star spots or other  sources, or transiting exoplanet studies where exoplanets transit  variable stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Gelino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Rockenfeller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013, 2015, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Girardin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Radigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Naud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Eriksson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Vos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Manjavacas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' A subsequent study  is forthcoming in which we further investigate the precision that can  be reached with this technique, and use injection-recovery tests to  assess the extent to which known, simulated variability signals can  be recovered (Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=', in preparation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Nonetheless, ground-based differential spectrophotometry  MNRAS 000, 1–24 (2022)  18  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  with the vAPP is highly complementary and advantageous to space-  based approaches for measuring the variability of high-contrast  companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' There have been many successful space-based mea-  surements of companion variability using HST, detecting variability  with amplitudes down to the 1-2% level in some cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Zhou  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Manjavacas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bowler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2016) was further able to detect sub-percent variability  using HST observations of planetary-mass companion 2M1207b,  which lies at roughly the same angular separation as HD 1160 B,  albeit with a more favourable contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Furthermore, the first vari-  ability monitoring with JWST, which should reach an even greater  precision, is currently underway as part of the Early Release Sci-  ence Program (Hinkley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, while JWST has the  sensitivity to image fainter, lower mass companions and measure  their variability with great precision, its ∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5 m mirror is smaller  than those of the largest ground-based telescopes, and thus is can-  not outperform large ground-based telescopes with extreme AO at  small separations ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5′′ at ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5 µm (Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This  means that companions at the closest angular separations such as  Jupiter analogues are for now likely only accessible with ground-  based monitoring techniques, for all but the nearest stars (Carter  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kammerer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ground-based telescopes also  uniquely provide access to higher resolution spectrographs, such  that line profile variability could be used in Doppler imaging to  create 2D global maps of features in exoplanet atmospheres such as  storms similar to Jupiter’s Great Red Spot (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Crossfield 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3  Observing strategy  During the observing sequence, we obtained 6 wavelength calibra-  tions at intervals throughout the night to allow us to test whether  differences in the wavelength calibration used would lead to dif-  ferences in the differential light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In Figure 7 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='3,  we found that the differential light curve is robust to changes in the  wavelength calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' It is therefore preferable to acquire wave-  length calibrations at the start or end of future observations, perhaps  along with a single precautionary wavelength calibration at high el-  evation, and instead obtain additional data on target and minimise  pixel offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We also used an on/off nodding pattern to enable background  subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, future studies may wish to consider alter-  native methods to remove the thermal background such that the  amount of time spent on target can be doubled and the entire system  stabilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For example, Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022) developed an ap-  proach whereby 93% of the frames obtained were on-target and the  thermal background was modelled and removed using the science  frames themselves and a small number of background frames ob-  tained before and after the observing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' As Figure 13 shows  that we approach the photon noise limit, increased on-target time  would therefore allow a greater differential light curve precision to  be obtained in a single night of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Lastly, the absence of an instrument derotator in the LBTI  architecture meant that the field of view was rotating throughout the  observing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In addition to the the drift due to lenslet flexure  and the manual offsets applied to keep HD 1160 A centred in the field  of view, this meant that HD 1160 A and B were not pixel-stabilised  during our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, the linear regression correlation  coefficients of the positions of the star and the companion are small  relative to those of airmass and air temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This suggests that  that, when present, these factors dominate over any effect from  HD 1160 A and B not being pixel-stabilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Understanding whether the host star of a given target is itself  varying is important when interpreting the trends in a differen-  tial light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Most, if not all, potential targets for differential  spectrophotometry will be present in at least the TESS full frame  images available on the MAST archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Even though this data will  most likely not be contemporaneous with a particular set of observa-  tions, the total baseline of the coverage should be relatively long and  therefore sufficient to check a host star for variability at the required  precision, especially if the target appears in multiple TESS sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We therefore recommend this method as a good way to verify the  level of variation shown by the host star of a target for differential  spectrophotometry, and hence whether it is stable enough to act  as a simultaneous photometric reference without requiring further  analysis to account for stellar variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='4  Outlook  In principle, the technique presented in this paper can be applied to  any vAPP coronagraph used in combination with an IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although  ALES is currently the only IFS operating over the L and M bands  (Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021), a vAPP coronagraph is also available on  SCExAO/CHARIS on the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='2-m Subaru Telescope, offering R∼19  spectrographic coverage over the J, H, and K bands (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='13-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='39 µm)  (Groff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Doelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Bos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' There will also be two different vAPPs on  the METIS instrument on the upcoming 39-m ELT, for which this  work is a pathfinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' METIS will provide high spectral resolution  spectroscopy (R∼100,000) over the L and M bands (Carlomagno  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Brandl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kenworthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Variability measurements using a vAPP may even be possible for  broad-band imaging data where an IFS is unavailable, although  sensitivity will be inherently more limited without the benefits of  using differential spectrophotometry to reduce the effects of sys-  tematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' There are several vAPPs currently available on such coro-  nagraphic imagers, such as MagAO (Morzinski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Otten  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021) and MagAO-X (Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Close et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020b) on the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5-m Magellan Clay Telescope, and  the recently-commissioned ERIS instrument on the VLT (Boehle  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kenworthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Dubber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022), with  more planned, including for GMagAO-X on the GMT (Close et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2020a) and MICADO on the ELT (Clénet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Perrot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Using the vAPPs that will be available on larger telescopes,  variability monitoring through differential spectrophotometry will  be possible for fainter companions at closer angular separations,  including those in the exoplanet mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' While this will be in-  herently more challenging for such companions at greater contrasts,  these will remain accessible to this technique through the addition  of data from multiple epochs as long as the systematic noise floor  is not reached, albeit with a trade-off between light curve precision  and time-sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  In the era of extremely large telescopes, high-contrast imaging  combined with high resolution spectroscopy will provide access  to fainter companions at lower masses and older ages and allow  their orbital velocities and spin to be measured (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Snellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Birkby 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Xuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Measurements of how individual absorption lines  change in depth and width as an exoplanet rotates will allow two-  dimensional surface maps of exoplanet atmospheres and weather  to be produced, through techniques such as Doppler imaging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Crossfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Crossfield 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Luger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Plummer  & Wang 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Further in the future, multi-wavelength variabil-  ity measurements obtained in reflected light may even enable exo-  cartography of directly imaged Earth-like exoplanets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Luger  MNRAS 000, 1–24 (2022)  Exoplanet variability with vAPP coronagraphs  19  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kawahara 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kuwata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Teinturier  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  A limitation of using the dgvAPP360 for variability measure-  ments is that post-processing algorithms relying on angular diver-  sity, such as ADI and PCA, cannot be used without also removing  the central PSF of the star that we use as the simultaneous photomet-  ric reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Furthermore, the stellar PSF must remain unsaturated  throughout the observing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This potentially limits the sam-  ple of targets with bright enough companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Although HD 1160 B  is bright enough that additional noise reduction techniques were not  necessary to produce a detection of ample S/N, this may not be  the case for fainter directly imaged companions in the exoplanet  mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, it may be possible to use novel alternative  algorithms to reach deeper contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  For example, the Temporal Reference Analysis of Planets  (TRAP, Samland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=', submitted) algorithm instead  relies on temporal diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' TRAP reconstructs the systematics in  a given region in the data using reference pixels that share the same  underlying noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' By simultaneously fitting the model of  a companion signal ‘transiting’ over detector pixels and the light  curves of the reference pixels, TRAP can then remove these sys-  tematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' It may be possible to leverage the information provided  by TRAP to improve the companion S/N without removing the stel-  lar PSF, or even to extract detrended light curves directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Another  option would be to use the gvAPP coronagraph, which is different  from the dgvAPP360 in that it creates two images of the target star  each with a 180° D-shaped dark hole on opposing sides, as well as  an additional ‘leakage term’ positioned between the two (Snik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Otten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The leakage term is an  entirely separate PSF of the star that appears at a fraction of its full  brightness, making it ideal as a simultaneous photometric reference  and enabling observations of systems with host stars that would oth-  erwise be too bright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Post-processing algorithms can be applied to  the main PSFs to reach deeper contrasts without impacting the leak-  age term, enabling differential variability measurements provided  that the impact of the algorithms on the photometry of the com-  panion can be characterised precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However, the gvAPP coro-  nagraph can suffer from wavelength-dependent smearing, which  would make such measurements more complex than those obtained  with a dgvAPP360 (Otten et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' In addition to the leakage  term, some vAPPs (such as the VLT/ERIS gvAPP) produce other  faint reference spots specifically for use as photometric references  in situations where the core of the target star PSF is saturated (Doel-  man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kravchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  In addition to probing the intrinsic variability of high-contrast  companions, differential spectrophotometry could also be used to  observe the transits of satellites such as exomoons or binary plan-  ets passing in front of these companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Candidate satellites have  been identified around transiting exoplanets, directly imaged com-  panions, and isolated planetary-mass objects using a range of tech-  niques, but none have yet been definitively detected (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Teachey &  Kipping 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Kreidberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lazzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Teachey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Limbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Vanderburg & Rodriguez 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Kipping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' For directly imaged companions, variability  arising from transit events could be distinguished from that caused  by inhomogeneous atmospheric features in similar ways to transit-  ing exoplanets and star spots, by considering the companion light  curves across the different wavelength channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Transit signals are  expected to be almost achromatic, while intrinsic variability is gen-  erally wavelength-dependent (Manjavacas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Limbach  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lazzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Lazzoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' (2022) found  using simulations that although the probability of successfully de-  tecting smaller exomoons around a directly imaged companion is  very low with current instrumentation and techniques, detections of  larger binary planets are already within reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' New techniques to  improve differential light curve precision for directly imaged com-  panions, including differential spectrophotometry with the dgvAPP,  will help to increase these probabilities further and potentially en-  able the first definitive detections of satellites around substellar  companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  8  CONCLUSIONS  We present a novel, ground-based approach for constructing differ-  ential light curves of high-contrast companions through direct dif-  ferential spectrophotometric monitoring, using the double-grating  360° vector Apodizing Phase Plate coronagraph and the ALES  integral field spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The dgvAPP360 allows high-contrast  companions to be detected while also providing an image of the  host star, which crucially can be used as a simultaneous photomet-  ric reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We combine the dgvAPP360 with ALES to follow the  highly successful technique of differential spectrophotometry used  in exoplanet transmission spectroscopy, where light is spectrally-  dispersed to reduce systematic effects that otherwise dominate the  variability signal we aim to measure, and then recombined into  white-light flux measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We demonstrated this approach using a full night of observa-  tions of substellar companion HD 1160 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The time-series fluxes  of the companion and the star in each wavelength channel were  extracted simultaneously using aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We then pro-  duced white-light measurements for both the companion and the  star at each time frame by taking the median combination of the  photometry in the wavelength dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The companion flux was  then divided by that of the star to eliminate trends common to both,  arising from Earth’s atmosphere and other systematics, producing  a differential white-light curve that only contains non-shared varia-  tions and covers a wavelength range of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We find that  the shape of the resulting light curve is robust against issues arising  from instrumental flexure, as tested using calibration frames col-  lected throughout the observation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Using a multiple linear  regression approach with eight decorrelation parameters, we mod-  elled and removed non-shared trends from the differential white-  light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We find that airmass and air temperature are the most  correlated parameters with the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We also analyse publicly  available data from the TESS mission to check for variability in the  host star HD 1160 A, and confirm that it is non-varying at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='03%  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We find that the detrended differential white-light curve of  HD 1160 B shows sinusoidal-like variability over a short timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  By fitting the unbinned light curve with a sinusoid, we identify  that the variability has a semi-amplitude of ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='8% and a period  of ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' When binned to 18 minutes of integration time  per bin, we achieve a light curve precision at the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='7% level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' After  thorough investigation and rejection of systematic noise sources, we  attribute this variability as likely due to heterogeneous features in  the atmosphere of the companion, rotating in and out of view as  it rotates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We find that if the period of this variability reflects the  rotation period of HD 1160 B, physical limitations suggest that it is  rotating at close to its breakup period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Alternatively, the short period  variability in the light curve of HD 1160 B may arise from multi-  ple periodic features in its atmosphere with different phase offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Furthermore, light curves in the 30 individual wavelength channels  in the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='59-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='99 µm range show tentative evidence of an increase in  MNRAS 000, 1–24 (2022)  20  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  variability amplitude at longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Further observations at  additional epochs will help to confirm and characterise the variabil-  ity of HD 1160 B and to determine its physical explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  The precision that we achieve in the detrended differential  white-light curve is the greatest achieved from ground-based stud-  ies of sub-arcsecond high-contrast companions to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' However,  direct comparisons to other ground-based studies that instead use  satellite spots to search for variability in the light curves of high-  contrast companions are not straightforward due to the different  magnitude and contrast of the observed systems, with HD 1160 B  having a more favourable contrast (Apai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Biller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The RMS of the detrended differential  light curve for HD 1160 B as a function of bin size follows the same  trend as the theoretical white noise expectation with no evidence of  yet approaching a noise floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This indicated that the single night  of data analysed here is not yet systematic-limited, and that further  observations from additional epochs could enable greater sensitiv-  ity to be reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' A deeper investigation of this type of data and its  precision, including injection-recovery tests to test how effectively  known variability signals can be recovered and which systematics  have the greatest impact, is forthcoming (Sutlieff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=', in prepara-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  While JWST will measure the variability of fainter, lower mass  companions from space with unprecedented precision, its compar-  atively smaller aperture means it cannot outperform the largest  AO-equipped ground-based telescopes at separations ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5′′ at  ∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5 µm (Girard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2022), so companions such as Jupiter ana-  logues at the closest angular separations, for all but the nearest stars,  remain accessible only to ground-based monitoring techniques for  the coming decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Ground-based differential spectrophotometry  with the vAPP is therefore highly complementary to space-based  approaches for measuring the variability of high-contrast exoplanet  and brown dwarf companions, and for searching for their transiting  exomoons or binary planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Moreover, ground-based telescopes  can reach much higher spectral resolution, which then enables line  profile variability studies to map atmospheric features, including  storms and hurricanes like Jupiter’s Great Red Spot, via Doppler  imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' These results are promising for further variability studies  using vAPP coronagraphs on current and upcoming instruments  and telescopes, which include ERIS on the VLT and METIS on the  ELT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The authors would like to thank Frank Backs and Elisabeth C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Matthews for valuable discussions that improved this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We also  thank our anonymous referee whose comments helped us to im-  prove this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' BJS is fully supported by the Netherlands  Research School for Astronomy (NOVA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' JLB acknowledges fund-  ing from the European Research Council (ERC) under the European  Union’s Horizon 2020 research and innovation program under grant  agreement No 805445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This paper is based on work funded by the  United States NSF grants 1608834, 1614320, and 1614492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  research of DD and FS leading to these results has received fund-  ing from the European Research Council under ERC Starting Grant  agreement 678194 (FALCONER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  We acknowledge the use of the Large Binocular Telescope  Interferometer (LBTI) and the support from the LBTI team, specifi-  cally from Emily Mailhot, Jared Carlson, Jennifer Power, Phil Hinz,  Michael Skrutskie, Travis Barman, Ilya Ilyin, and Ji Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The LBT  is an international collaboration among institutions in the United  States, Italy and Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' LBT Corporation partners are: The Uni-  versity of Arizona on behalf of the Arizona Board of Regents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Isti-  tuto Nazionale di Astrofisica, Italy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' LBT Beteiligungsgesellschaft,  Germany, representing the Max-Planck Society, The Leibniz In-  stitute for Astrophysics Potsdam, and Heidelberg University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' The  Ohio State University, representing OSU, University of Notre Dame,  University of Minnesota and University of Virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' We gratefully  acknowledge the use of Native land for our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' LBT ob-  servations were conducted on the stolen land of the Ndee/Nn¯e¯e,  Chiricahua, Mescalero and San Carlos Apache tribes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  This publication makes use of data products from the Two  Micron All Sky Survey, which is a joint project of the Univer-  sity of Massachusetts and the Infrared Processing and Analysis  Center/California Institute of Technology, funded by the National  Aeronautics and Space Administration and the National Science  Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This work has made use of data from the European  Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  int/gaia), processed by the Gaia Data Processing and Anal-  ysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
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+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='int/web/  gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Funding for the DPAC has been pro-  vided by national institutions, in particular the institutions partic-  ipating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This paper includes  data collected with the TESS mission, obtained from the Mikulski  Archive for Space Telescopes (MAST) data archive at the Space  Telescope Science Institute (STScI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Funding for the TESS mission  is provided by the NASA Explorer Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' STScI is operated by  the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=',  under NASA contract NAS 5–26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Support for MAST for non-  HST data is provided by the NASA Office of Space Science via  grant NNX13AC07G and by other grants and contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This re-  search has made use of the NASA Exoplanet Archive, which is  operated by the California Institute of Technology, under contract  with the National Aeronautics and Space Administration under the  Exoplanet Exploration Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This research has made use of  NASA’s Astrophysics Data System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This research has made use  of the SIMBAD database, operated at CDS, Strasbourg, France  (Wenger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This research made use of SAOImageDS9, a  tool for data visualization supported by the Chandra X-ray Science  Center (CXC) and the High Energy Astrophysics Science Archive  Center (HEASARC) with support from the JWST Mission office  at the Space Telescope Science Institute for 3D visualization (Joye  & Mandel 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This work made use of the whereistheplanet3  prediction tool (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' This work makes use of the  Python programming language4, in particular packages including  NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020), Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  2013, 2018), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2020), HCIPy (Por et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2018),  PynPoint (Amara & Quanz 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' Stolker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2019), scikit-image  (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2014), scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content=' 2011),  statsmodels (Seabold & Perktold 2010), pandas (McKinney 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  Reback et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
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+page_content=' 2022), and Matplotlib  (Hunter 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  DATA AVAILABILITY  The data from the LBT observations underlying this article are  available in the Research Data Management Zenodo repository of  the Anton Pannekoek Institute for Astronomy, at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='  org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
+page_content='5617572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNFAT4oBgHgl3EQfxx6m/content/2301.08689v1.pdf'}
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+1 
+ 
+Deep Learning-Driven Nonlinear Reduced-Order Models for Predicting 
+Wave- Structure Interaction  
+ 
+R. Halder*, M. Damodaran** and B.C. Khoo# 
+National University of Singapore, Temasek Laboratories 9 Engineering Drive 1, Singapore 117575 
+(Dated 31 July 2022) 
+ 
+                                                                            Abstract 
+Long Short-Term Memory (LSTM) network-driven Non-Intrusive Reduced Order Model 
+(NROM) for predicting the dynamics of a floating box on the water surface in a wavemaker 
+basin is addressed in this study. The ground truth or actual data for these wave-structure 
+interactions (WSI) problems, namely box displacements and hydrodynamic forces and 
+moments acting on the box due to wave interaction corresponding to a particular wave 
+profile, are computed using the Smoothed Particle Hydrodynamics (SPH). The 
+dimensionality of the system is first reduced using the Discrete Empirical Interpolation 
+Method (DEIM) and the LSTM is applied to the reduced system resulting in a DEIM-LSTM 
+network for developing a surrogate for prediction. The network is further enhanced by 
+incorporating the physics information into the loss function resulting in a physics-informed 
+LSTM (LSTM-PINN) for predicting the rigid body dynamics of box motion. The 
+performance of predictions for these networks is assessed for the two-dimensional wave 
+basin WSI problem as a proof-of-concept demonstration. 
+ 
+1. Introduction 
+During the past two decades, several Reduced Order Model (ROM) methods outlined in Carlberg et al. 
+[1] have been instrumental in expediting the efficient computational modeling of dynamical systems 
+implying significant benefits for flow prediction in engineered systems. The conventional Computational 
+Fluid Dynamics (CFD) methods offer solutions to well-posed flow problems with proper initial and 
+boundary conditions. Recent efforts in deep learning (DL) appear to offer efficient and effective solutions 
+easily for ill-posed problems such as inverse, regression, and classification problems in diverse fields as 
+outlined in Schmidhuber et al. [2] have gained significant attention in a wide variety of fluid mechanics 
+applications such as the closure model for turbulence using machine learning as in Maulik and San [3] 
+and Duraisamy et al. [4], the combined POD and LSTM network for incompressible flows with spectral 
+proper orthogonal decomposition (SPOD) of Wang et al. [5] and the application of a dimensionality 
+reduction method with DL networks for learning feature dynamics from noisy data sets in Lui and Wolf  
+________________________________ 
+*Research Scientist; e0010762@u.nus.edu; Currently Research Fellow University of Michigan, USA 
+** Senior Research Scientist; tslmura@nus.edu.sg 
+# Professor of Mechanical Engineering and Director Temasek Laboratories; tslhead@nus.edu.sg  
+ 
+
+2 
+ 
+ 
+ 
+[6]. The Physics Informed Neural Network (PINN) introduced by Raissi et al. [7] in computational 
+continuum mechanics offers a trade-off between the availability of the data and the accuracy of the 
+solution and the possibilities of training DL networks with low or no data to moderate and big data for 
+effective prediction of flow problems. The application of machine learning in fluid mechanics as outlined 
+in Brunton et al. [8] has motivated the present study in which the LSTM and LSTM-PINN network is 
+coupled with the DEIM algorithm as the nonlinear model order reduction method outlined in 
+Chaturantabut and Sorensen [9] to develop a surrogate for predicting a couple of wave-structure 
+interaction (WSI) problems. The wave structure interaction (WSI) possesses different numerical 
+challenges because of strong nonlinearities in the dynamical system and its multi-disciplinary nature as 
+outlined by Chakrabarti et al. [10].  Although SPH is originally developed for astrophysical applications 
+by Monaghan [11] and Monaghan and Lattanzio [12], it has been used  for modelling wave generation 
+and WSI problems in several studies such as Wen et al. [13] and Altomare et al. [14] which uses the 
+implementation of the SPH in the open-source platform DualSPHysics outlined in Dominique et al. [15] 
+which is the platform used in the present study to generate necessary data for the proposed nonlinear 
+reduced-order model DEIM-LSTM and LSTM-PINN algorithms and also to compare the predicted result 
+with the ground truth (or  actual data).  Figure 1 shows the schematic of the wave-basin setup for 
+predicting the nonlinear wave interaction and the unsteady motion dynamics of a freely floating box on 
+the surface of a wave in a two-dimensional wave basin as a proof-of-concept demonstration of the 
+proposed LSTM-DEIM/LSTM-PINN-DEIM surrogates. The freely floating box considered has a length 
+of lb of 0.3m, a height of hb of 0.2m, and a unit width in the spanwise direction and is assumed to have a 
+density of 500kg/m3 which will float in water which has twice its density. This problem has been 
+simulated using a single phase SPH by Ren et al. [16] and Domínguez et al. [17].  The movement of the 
+paddle on the left will set up wave flumes in the wave basin and the wave heights are measured relative 
+to the datum line (black solid line at the bottom of the wave basin) which corresponds to the undisturbed 
+water level in the wave basin. 
+ 
+ 
+Figure 1: Schematic of the initial set-up and initial location for a freely floating box in the 2D 
+wave basin.  
+ 
+In this study, the problem of the hydrodynamic wave interaction with a freely floating rectangular box is 
+
+....Bufferl
+Buffer2
+p=500kg/m
+d.
+Paddle
+hs
+d
+1:4
+S
+4m
+Wall
+6m3 
+ 
+considered in the context of a single-phase SPH.  A freely floating box is initially assumed to be still on 
+the surface of the water. The box is excited by a wave generated from a wake maker. The wave profile 
+has a period of T of 1.2s and a wave height H0 of 0.1m and its interaction with the floating box create 
+box motions, namely surge in the direction and heave perpendicular to the direction of wave propagation 
+and pitching rotation (pitch) about the center of mass of the box. The motion of the piston creates a 
+second-order Stokes wave which is absorbed in the 1:4-sloped dissipative beach at the end. The no-slip 
+boundary conditions are implemented at the bottom wall and the dissipative beach head. Buffers 1 and 2 
+denote the open boundaries.  As this is a two-dimensional problem there will be two translational motions 
+of the box center of mass, namely, surge and heave motions along with the x and y directions respectively, 
+and a pitching motion about the center of mass. The dynamics of the floating box is defined 
+mathematically as: 
+ 
+                          
+d
+1
+(1
+)
+d
+d
+d
+d(
+)
+d
+y
+y
+r
+x
+x
+s
+M
+M
+t
+M
+t
+t
+
+=
++
+−
+=
+
+=
+U
+F
+g
+U
+F
+J ω
+T
+                                                              (1) 
+where M is the mass of the box  assumed to be 30 kg, J is the moment of inertia of the box about its 
+center of mass assumed to be 0.325 kg-m2, ρr is the specific density of the box material  assumed to be 
+0.5 and 
+x
+F , 
+y
+F and T the forces in the surge x, and heave y directions and torque about the box center 
+of mass respectively.  A second-order Stokes wave is generated by the wavemaker on the left end of the 
+basin and the waves are absorbed in the 1:4-sloped dissipative beach head on the right end of the basin. 
+No-slip boundary conditions are implemented on the bottom wall and the dissipative beach while Buffers 
+1 and 2 are treated as open boundaries. The total number of particles used for the single-phase SPH is 
+70023. A resolution of 0.01 m proposed in Ren et al. [16] is used in the present study. The box motion 
+dynamics of Eqn. (1) in a matrix form is as follows: 
+
+   
+M
+X = F  
+(2.1) 
+ 
+ 
+ 
+0
+0
+M
+0
+0
+, X
+, F
+0
+0
+where,
+1
+1
+x
+y
+r
+F
+M
+S
+M
+h
+F
+M
+J
+
+
+=
+=
+=
++
+
+
+
+
+ 
+
+
+
+
+
+
+ 
+
+
+−
+
+
+
+
+ 
+
+
+
+
+
+
+ 
+
+
+ 
+
+
+
+
+g
+T
+ where ,
+S h and   are the surge, 
+heave, and pitch. The first order time derivatives of these kinematic variables, i.e., ,  and 
+S h
+  are the 
+surge, heave and pitch rates and g are the acceleration due to gravity. Since 
+0
+X
+X
+−
+=
+ it follows that 
+
+4 
+ 
+ 
+ 
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+1
+0
+[
+]
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+X
+X
+M
+F
+X
+M F
+X
+X
+−
+−
+•
+−
+−
+−
+=
++
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ 
+
+ 
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+  
+                  (2.2)                               
+ 
+II. Nonlinear Reduced Order Model: 
+ 
+Formulation of the nonlinear reduced order models (NROMs) proposed for the possibility of developing 
+digital twins for the wave-structure interactions are outlined in this section. Section II(a) discusses the 
+basic architecture of the LSTM and LSTM PINN networks and outlines the elements of the application 
+of LSTM-PINN network for rigid body dynamics equation governing the box motion while Section II(b) 
+addresses the coupling of the LSTM with the dimensionality reduction approach of DEIM. Although the 
+PINN network is computationally expensive than conventional LSTM network and numerical solution 
+of the benchmark rigid body dynamics equation, the objective of the current work is to present a proof-
+of-concept model for the realization of a LSTM-PINN model which can be used for predicting the 
+response of a large-scale application using sparse or no data to assess if the potential of the LSTM-PINN 
+network can be realized. The DEIM-LSTM and LSTM-PINN codes are available in 
+https://github.com/rahulhalderAERO/Hydrodynamic_LSTM_ROM  
+ 
+ II(a) LSTM and LSTM-PINN network: 
+ 
+In this current section, the formulation of functional relationship between input and output are first 
+explained with a simpler ANN network. A single layer ANN network consisting of few neurons mapping 
+input data, namely the hydrodynamic forces acting on a floating box computed from SPH and the output 
+data consisting of the floating box kinematic variables computed from SPH, which form the dataset for 
+training the surrogate for predicting the dynamics of the floating box in a wave basin. The mapping can 
+be expressed as follows: 
+                                  
+
+2
+1
+1
+2
+(
+)
+(
+)
+x
+y
+f W W
+b
+b
+S
+h
+S
+h
+F
+F
+T
+
+ =
++
++
+
+
+
+
+                  (3) 
+where 
+
+6
+tn
+S
+h
+S
+h
+
+
+
+
+and 
+3
+t
+x
+n
+y
+F
+F
+T
+
+
+
+
+
+
+are output and input data, W1  and W2  
+are the weight matrices, b1 and b2 are the bias vectors respectively and f  is the nonlinear activation 
+function which in this study uses the tanh function. The structure of a simple LSTM network is different 
+from that of the ANN architecture, and it will be demonstrated for the same input and output datasets 
+associated with the box dynamics floating on the hydrodynamics waves. The input and output structure 
+of the LSTM network includes all the variables 
+
+
+T
+o
+S
+h
+S
+h
+p
+
+
+=
+as the output and 
+
+5 
+ 
+
+
+x
+T
+i
+y
+F
+F
+T
+p =
+as the input with the LSTM input-output mapping defined as follows: 
+                                                           
+
+
+
+1
+(
+)
+n
+n
+n
+l
+i
+o
+p
+f W p
+Uh
+b
++ =
++
++
+                                                    (4) 
+where the additional matrix U which is absent in the ANN architecture associates input from the previous 
+time steps and introduces the memory effect in the ANN architecture leading to the Recurrent Neural 
+Network (RNN) outlined in Lipton et al. [18]. For the simplest RNN, 
+lh is the predicted output value at 
+the nth time step that is 
+
+n
+op
+. In the current work, this network is implemented in the Keras platform 
+[19]. The input and output structure of such a network is shown in Fig. 2(a) and 2(b) respectively. The 
+input structure is a tensor which contains the first dimension indicating total time steps
+tn , the second 
+dimension indicating the backpropagation length (memory effect or previous time steps data) in case of 
+input and number of neurons in case of output, and the third one indicating the type of inputs or outputs 
+(i.e., plunge, pitch, forces, plunge velocity, pitch velocity, forces, and moments, etc.).  
+ 
+ 
+(a) 
+(b) 
+Figure 2: (a) Input and (b) Output data structure 
+ 
+The limitation of the RNN layer is that it creates instances of vanishing or exploding gradient problems 
+because of the long-term dependencies over the time-sequential dataset (i.e., multiplication of gradient 
+terms over total timestep 
+tn  in a sequence prediction). Hochreiter et al. [20] introduced the LSTM 
+network to mitigate this problem of vanishing or exploding gradients whereby unlike the conventional 
+RNN which keeps its contents of previous time steps while computing the gradients, the LSTM 
+introduces four distinct operations in each RNN cell: memory cell, input gate, forget gate, and output 
+gates. Details of the Memory cell, Input gate, Forget gate and Output gate, and the hidden layer operations 
+in connection with the LSTM network are defined as follows: 
+
+Back Propagation Length
+of
+Xi
+xNumberofNeurons
+-
+16 
+ 
+ 
+                            
+n
+n-1
+i
+i
+n
+n-1
+n
+n-1
+o
+o
+n
+n
+n-1
+n
+n-1
+c
+c
+n
+n
+n
+Input Gate:
+i = σ(Wh
++b )
+Forget Gate:        f = σ(W h
++b )
+Output Gate:        o  
+σ(W h
++b )
+Cell State:             c = i
+c
+i
+tanh(W h
+b )
+Hidden Layer:        h
+o
+tanh(c ).
+f
+f
+•
+•
+•
+=
+•
++
++
+•
+=
+                              (5)                                                                                                                                      
+where 
+is the Hadamard product, input forces are fed to the LSTM network at each step of the input 
+sequence defined as n, i is the Input gate and f is the Forget gate which will choose the necessary 
+information to be passed through or forget through cell state c, Wf and Wc are the corresponding weight 
+matrices, 
+ib , b f  and bc  are the bias vectors for the Input gate, Forget gate and Cell state, respectively. 
+The Output gate, o, on the other hand, decides the control of the flow of the information from the cell 
+state to the next hidden layer. These operations help the LSTM cells to keep only the necessary memories 
+thereby mitigating the occurrence of exploding and vanishing gradient problems in the back-propagation 
+algorithm for each iteration step of the gradient descent optimization. Figure 3(a) shows a schematic of 
+the internal architecture of a single LSTM cell. Figure 3(b) shows the stacking of LSTM cells to enhance 
+the network size. The output dimension of the LSTM layer 
+tn
+N
+
+
+where N is the number of neurons, 
+the final layer output must be sent through a dense ANN layer to obtain the required output size which 
+is 
+3
+tn  here (
+
+T
+S
+h
+S
+h
+
+
+. Figure 3(b) also shows the stacked LSTM layers sent to the 
+dense ANN layer to form the final LSTM/LSTM-PINN network. 
+ 
+ 
+(a) 
+(b) 
+ 
+Figure 3: (a) Structure of a Single LSTM cell (b) Stacked LSTM cells forming LSTM/LSTM-PINN  
+ 
+The loss function associated with the LSTM network, i.e.,
+LSTM
+MSE
+is computed using input and output 
+values of the LSTM network. The outputs from the LSTM network are used in the governing equations 
+
+Xr,yt-l
+LSTMCellMSEo.=MSELsTM+MSEpi
+yt-1
+yi-n+1
+4
+LaerL+DenseLaer
+...
+yt-1
+yt
+t-n+1
+LSTM Cell
+LSTMCell
+LSTMCell
+Laverl
+Y,-n+2 X,-n+1
+4.
+y-2x,-1
+y-1
+x7 
+ 
+to estimate the physics informed loss
+PINN
+MSE
+. The total loss for a LSTM-PINN network is then 
+expressed as 
+1
+2
+*
+LSTM
+PINN
+LSTM
+PINN
+MSE
+w
+MSE
+w
+MSE
+−
+=
++
+
+                                                (6) 
+where 
+2
+1
+1
+(
+)
+n
+LSTM
+pred
+actual
+i
+MSE
+y
+y
+n
+=
+=
+−
+
+ and 
+                 
+2
+1
+1
+1
+1
+1
+1
+1
+0
+0
+1
+0
+0
+3
+4
+0
+0
+0
+1
+1
+0
+0
+0
+2
+0
+0
+n
+n
+n
+n
+n
+n
+i
+PINN
+MSE
+X
+X
+X
+X
+X
+X
+X
+M
+K
+X
+n
+dt
+M
+F
++
+−
++
++
+−
+=
+−
+=
+
+
+−
+
+
+
+
+
+
+
+
+
+
+−
++
+
+
+
+
+
+
+
+
+
+
+−
+
+
+
+
+
+
+
+
+
+
+
+
+
+
++
+−
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+                (7) 
+are the losses associated with the LSTM network and the loss associated with the physics of the floating 
+box dynamics respectively.  w1 and w2 are the weightage values associated with the data driven and 
+physics-based loss function. The input of the LSTM-PINN network is set as 
+[
+]
+i
+x
+y
+P
+F F
+T
+=
+  whereas 
+the output of the LSTM-PINN network consists of the dynamic responses and their time derivatives, i.e. 
+
+
+T
+S
+h
+S
+h
+
+
+ of the floating box at the 
+,
+1,
+2
+th
+th
+th
+t
+t
+t
+n
+n
+n
+−
+−
+ time steps. Therefore, the size 
+of the input of the network is 
+3
+t
+N
+
+whereas size of the output of the network is 
+18
+t
+N
+
+. The 
+tn is any 
+time instant in total time steps of 
+t
+N .  
+ 
+II.(b) DEIM-LSTM network 
+The manifold hypothesis outlined by Bengio et al. [21] forms the basis of dimensionality reduction which 
+states that the solution of a high dimensional dynamical system lies near a low dimensional manifold S 
+embedded in a large dataset of size Rn. The Discrete Empirical Interpolation Method (DEIM) is a data 
+compression method for reconstructing the full data set from the information of the flow variables 
+specified at a few sensor locations as shown in Halder et al. [23]. For example, if the dataset D is a 
+function of time t or any parameter 𝜇, then D can be expressed as 𝐷(𝑡) = Ф𝑐(𝑡)  where Ф defines the 
+DEIM modes and c(t) is the coefficient vector which is computed from an index matrix or mask matrix 
+P used for indexing sensor locations in the computational domain. The index matrix is defined as P = 
+[𝑒𝜌1, … … … . , 𝑒𝜌𝑙], P ϵ ℝ𝑛×𝑙 where each column is defined as  𝑒𝜌𝑖 = [0, … 0. .1 … 0]𝑇 which implies a 
+value of 1 at location ρi, which is decided from an error minimization-based algorithm as shown in 
+Chaturantabut and Sorensen [9]. Hence, the full dataset can be approximated as: 
+        
+-1
+( , )
+(
+)
+( , )
+T
+T
+D t
+U P U
+P D t
+
+
+=
+                                                       (8) 
+where U(PTU)-1 is considered as the DEIM modes Ф and PTD(t) indicates the sensor locations values 
+which form coefficient vector c. For the large datasets, the CUR approximate matrix decomposition using 
+the DEIM algorithm outlined in Sorensen and Embree [22] can also be used to compute the control points 
+
+8 
+ 
+efficiently. The basic construct of the non-intrusive reduced order model (NIROM), results in a 
+dimensional reduction from Rn to Rm. The size of an actual dataset is n, and the size of the reduced dataset 
+is m where m << n. This reduced dataset is fed to a deep learning algorithm for exploring the feature 
+dynamics of the. A series of snapshots, each having the size of Rn is reduced to a set of snapshots of the 
+size of Rm before feeding into the deep-learning network. 
+For the prediction of the wave-structure interaction, Discrete Empirical Interpolation Method (DEIM) is 
+coupled with the LSTM network for the floating box problem with three degrees of freedom (pitch, 
+plunge ad surge), and the reader is referred to details of this step in Halder et al. [23]. Figure 4 shows the 
+schematic of the application of the DEIM-LSTM to predict the dynamics of a floating box in an unsteady   
+hydrodynamic wave. First, the unsteady wave elevation distribution data is generated from the 
+DualSPHysics simulation by exciting the wavemaker so that the computed data of hydrodynamic loads 
+on the box and the kinematics of the floating box forms the ground truth or the actual data. The DEIM 
+algorithm is then applied to the spatio-temporal wave height distribution and a few control points are 
+selected for determining the DEIM modes. For the training phase of the LSTM/LSTM-PINN network, 
+LSTM is applied to the wave heights at these control points and the predicted box motion from actual 
+data. After the LSTM network has been trained sufficiently, it can be used to predict the box motion 
+under any arbitrary wavemaker excitation profiles.  
+ 
+ 
+Figure 4: Schematic of the DEIM-LSTM Network. 
+ 
+ 
+ 
+
+0.06
+0.04
+h(m)
+0.02
+DEIM
+0.02
+-0.04
+1
+20
+N
+15
+10
+x(m)
+5
+t(s)
+0
+0
+DIEM ControlPoints
+FloatingBoxResponseUnder
+LSTM
+anyWaveElevation
+Network
+Distribution(Only Control
+E0.6
+0.4
+PointsValuesReqd.)
+0.2
+15
+(s
+BoxResponse(Output)9 
+ 
+III. Results and Discussion: 
+ 
+In this section, the prediction of the floating box dynamics due to the surface wave elevations in the wave 
+basin is discussed. First, the DEIM-LSTM is used for predicting the dynamics of the floating box where 
+the wave elevation is the input. In the next section, the governing equations for the dynamics of the 
+floating box are coupled with the data-driven loss function in the LSTM network.  
+Figure 5 compares the temporal dynamical motion of the floating box responses of computed surge, 
+heave and pitching with the experimental data of Ren et al. [16] corresponding to the paddle in the 
+wavemaker in the wave-basin being excited with a sinusoidal function of period 1.2 s and amplitude of 
+0.1 m to validate the SPH model predictions with experimental data. This is the ground truth for the 
+current study. 
+ 
+ 
+ 
+(a) 
+(b) 
+ 
+(c) 
+Figure 5: Comparison of the box response (a) surge (b) heave and (c) pitch motion using experiments 
+and current numerical model  
+ 
+III. a DEIM-LSTM Prediction of Wave-Floating Box Interaction Dynamics 
+The paddle in the wavemaker is excited with a sinusoidal function of period 1.2 s and amplitude of 0.1 
+m for the training of the DEIM-LSTM network and a sinusoidal function of period 1.3 s and amplitude 
+of 0.12 m as a test function. The temporal-spatial distribution of the wave elevations corresponding to 
+these scenarios is shown in Figures 6(a) and 6(b).  
+
+1
+sph
+0.8
+Experimental
+0.6
+(m)
+S
+0.4
+0.2
+0
+0
+5
+10
+15
+20
+t(s)0.1
+sph
+Experimental
+0.05
+h(m)
+-0.05
+0
+5
+10
+15
+20
+t(s)20
+sph
+Experimental
+10
+e
+p
+-10
+-20
+0
+5
+10
+15
+20
+t(s)10 
+ 
+ 
+(a) 
+ 
+ 
+(b) 
+Figure 6: (a) Training and (b) Test elevation of the wave surface considered as an input to the LSTM 
+network. 
+The training and test wave elevation data is of size 
+x
+t
+N
+N
+
+where
+x
+N is the 300 spatial locations and 
+t
+N is 
+the 2001 temporal locations for this study. DEIM is then applied to dimensionally reduce this large input 
+dataset of Fig. 6 from 
+x
+t
+N
+N
+
+ to 
+x
+t
+N
+m 
+ (
+x
+m  << 
+x
+N ). The Deep Learning-based approach is considered 
+at this point to learn the temporal dynamics. Since the LSTM approach is already a computationally 
+efficient approach as compared to the SPH model, the dimensionality reduction is considered only in the 
+spatial direction. The Singular Value Decomposition (SVD) is first applied to the input dataset to decide 
+on an optimal number of modes to represent the temporal-spatial distribution of the wave elevation 
+considered as an input to the LSTM network as shown in Fig.7. Any dynamical system can be represented 
+
+0.06
+0.04
+h(m)
+0.02
+0
+-0.02
+-0.04
+4
+3
+20
+15
+2
+10
+1
+x(m)
+5
+t(s)
+0
+00.1
+0.05
+(u)
+0
+-0.05
+4
+3
+20
+15
+2
+10
+x(m)
+1
+5
+t(s)
+0
+011 
+ 
+as a combination of few modes and the singular values are associated with the energy contained by those 
+modes. Figure 6 shows the variation of the scaled singular values (scaled with the largest singular value, 
+max
+Sv
+) which indicate the number of modes that contain most of the energy of the dynamic system and 
+after a few modes, the singular values drop to a very small value which is not important for system 
+dynamics.  
+ 
+ 
+Figure 7: Normalized Singular Values after the application of the SVD on the input data. 
+ 
+Figure 8 shows a selection of DEIM control points for the reconstruction of instantaneous wave profile. 
+Among the 300 spatial points, only 30 points indicated as 
+1
+2
+,
+,.... n
+x x
+x  are chosen for all the temporals.  
+ 
+ 
+Figure 8:  DEIM Control points marked as black dots at a time instant. 
+
+0.8
+0.6
+0.4
+0.2
+0
+0
+10
+20
+30
+40
+50
+Modes0.06
+0.04
+h(m)
+0.02
+0
+A
+-0.02
+-0.04
+4
+3
+20
+15
+2
+10
+x(m)
+5
+t(s)
+0
+0
+DIEM ControlPoints12 
+ 
+locations to significantly reduce the size of the input data to
+30 2000
+
+noting that 
+h
+ is the difference 
+between the instantaneous wave elevation and initial wave elevation. The points 
+1
+2
+,
+,.... n
+x x
+x are termed 
+as DEIM control points and the DEIM modes termed as  in Section II. B contains the inherent features 
+of the wave dynamics. Among the 30 DEIM modes corresponding to these control points, the spatial 
+variation of the first 4 modes is shown in Fig.9(a). If the values of the DEIM control points for the training 
+and test wave distribution as shown in Fig.8 are known, then the entire spatio-temporal distribution of 
+the wave can be reconstructed as shown in Fig. 9(b). The zoomed view of region marked by the red box 
+in Fig.9(b) is shown in Fig.9(c) to showcase the accuracy of the prediction of the test wave elevation at 
+the 12 s instant by comparing the predictions using different numbers of DEIM modes ranging from 10 
+to 30 in steps of 10 with the computed ground truth or actual wave height (computed using SPH) 
+illustrating the adequacy of the selected number of DEIM control points. It shows that if 10 DEIM modes 
+are selected the reconstructed signal deviates from the actual wave height whereas, after 20 modes, the 
+change in the reconstructed wave height is insignificant. Figure 10(a) to 10(c) show that the error 
+distribution drastically drops when the number of selected modes is increased from 10 to 30. 
+ 
+ 
+ 
+(a) 
+(b) 
+ 
+(c) 
+Figure 9: (a) First four DEIM Modes and (b) comparison of predicted wave elevation using 
+different numbers of the DEIM modes (c) zoomed view of the red box in (b) compared to 
+
+2
+Mode 1
+1.5
+Mode 2
+-Mode 3
+Mode 4
+DEIM Modes
+1
+0.5
+0
+-0.5
+.1
+1
+2
+3
+4
+x(m)0.1
+-·Actual
+Mode 30
+·Mode20
+Mode10
+0.05
+△h(m)
+0
+-0.05
+1
+1.5
+2
+2.5
+3
+3.5
+x(m)*- ·Actual
+Mode 30
+0.08
+- ·Mode 20
+Mode10
+△h(m)
+0.075
+0.07
+2.55
+2.6
+2.65
+2.7
+2.75
+2.8
+(w)x13 
+ 
+computed wave elevation at the 12 s instant (ground truth or actual data). 
+ 
+ 
+ 
+(a) 
+(b) 
+ 
+(c) 
+Figure 10: Reconstruction error distribution of the test elevation using (a) 10 (b) 20 and (c) 30 DEIM 
+modes of the training wave elevation.   
+ 
+Figures 11 (a)-(c) compares the prediction of temporal variation of heave, surge, and pitch angle 
+dynamics of the floating box respectively for both training and test wave elevation distributions shown 
+respectively in Fig. 6(a) and 6(b) and which will be used for DEIM-LSTM predictions. 
+ 
+ 
+ 
+(a) 
+(b) 
+ 
+
+0.02
+0
+-0.02
+4
+20
+2
+10
+0
+0
+(w)x
+t(s)0.01
+0
+-0.01
+4
+20
+2
+10
+0
+0
+(w)x
+t(s)2
+0
+-2
+4
+20
+2
+10
+0
+0
+x(m)
+t(s)0.1
+test
+train
+0.05
+h(m)
+-0.05
+0
+5
+10
+15
+t(s)1
+test
+train
+0.8
+0.6
+S
+0.4
+0.2
+0
+0
+5
+10
+15
+t(s)14 
+ 
+ 
+(c) 
+Figure 11: Temporal variation of (a) heave (b)surge and (c) pitch dynamics of the floating box 
+computed using SPH used as training and test data for DEIM-LSTM predictions. 
+ 
+A LSTM network consisting of 25 input time sequence termed as n in Fig.3 of the Section II.a, 4 hidden 
+layers, 10 neurons in each layer, and tanh activation function is considered for prediction.  Figure 12(a)-
+(c) compares the temporal variation of floating box dynamics of heave, surge, and pitching predicted 
+using DEIM-LSTM with actual data computed from SPH when the training and test wave elevations and 
+the corresponding box motions are the same. Since the training and test dataset are the same, the predicted 
+results have matched significantly well with the high-fidelity simulation results.   
+ 
+ 
+ 
+               (a) 
+          (b) 
+ 
+
+20
+test
+10
+train
+0
+H
+-10
+-20
+-30
+0
+5
+10
+15
+t(s)0.15
+Predict
+ ·Actual
+0.1
+h(m)
+0.05
+-0.05
+0
+5
+10
+15
+t(s)Predict
+0.8
+-Actual
+0.6
+(m
+S
+0.4
+0.2
+0
+0
+5
+10
+15
+t(s)15 
+ 
+ 
+          (c) 
+ 
+Figure 12: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of floating 
+box from DEIM-LSTM prediction with SPH predictions for the case when the training and test dataset 
+are the same. 
+ 
+Next the prediction of the floating box dynamics using DEIM-LSTM when the training and test datasets 
+used are different is considered and the impact of some of the pertinent network hyperparameters, namely 
+the number of epochs reached during training, the size of the backpropagation length (termed here as 
+input sequence) and the number of neurons in the hidden layers on the prediction is assessed. Figures 
+13(a)-(b) compares the DEIM-LSTM prediction of the heave, surge, and pitch motion of the floating box 
+with the actual data from SPH simulation for number of epochs ranging from 150-250-500 for which the 
+training time ranges from 344s, 599 s and 1264s respectively. Mean Absolute Errors (MAE) associated 
+with the surge, heave, and pitch response corresponding to epoch 150 are 0.0251, 0.0164, and 8.29 
+respectively whereas for 250 and 500 epochs they are 0.0241, 0.0152, 8.54, and 0.0218, 0.0153 and 
+8.3089 respectively.  Although the training time increases almost linearly with the number of epochs, the 
+prediction accuracy remains the same and compares well with the actual data. Figure 13(d) shows the 
+mean squared error variation with the number of epochs. It shows that the training error does not further 
+decay after 300 epochs.   
+The effect of varying the size of the backpropagation length or the training input sequence on the DEIM-
+LSTM prediction capability is shown in Fig. 14 for the same data sets. For the input sequence of 15, 25, 
+and 35, the training time is 378s, 599s, and 775s respectively. For all three cases, the predicted result 
+matches well with the actual data. The MAE associated with the input sequence of 15 and 35 are 0.0224, 
+0.0145, 7.808, and 0.0248, 0.0171, 8.26 respectively for the surge, heave and pitch motion while the 
+MAE of the input sequence 25 is indicated earlier for the epoch variation of 250. It shows that the 
+prediction accuracy is better at the lower input sequence.     
+ 
+
+20
+0
+(Sop)
+-20
+-Predict
+-40
+- ·Actual
+0
+5
+10
+15
+t(s)16 
+ 
+ 
+ 
+(a) 
+(b) 
+ 
+ 
+(c) 
+(d) 
+ 
+Figure 13: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of 
+floating box from DEIM-LSTM prediction with SPH predictions for the case when the training 
+and test dataset are different for different epoch length. (d) Variation of the mean squared loss 
+history. 
+ 
+ 
+ 
+(a) 
+(b) 
+
+0.15
+Epoch 150
+Epoch 250
+0.1
+Epoch 500
+Actual
+0.05
+0
+-0.05
+0
+5
+10
+15
+t(s)1
+Epoch 150
+Epoch 250
+0.8
+Epoch 500
+Actual
+(m)
+0.6
+S
+0.4
+0.2
+0
+0
+5
+10
+15
+t(s)20
+0
+(sop)6
+-20
+Epoch150
+Epoch 250
+Epoch 500
+-40
+Actual
+0
+5
+10
+15
+t(s)10
+Loss
+0
+100
+200
+300
+400
+500
+Epochs0.15
+-·Seq 15
+Seq 25
+0.1
+Seq 35
+Actual
+0.05
+h
+0
+-0.05
+0
+5
+10
+15
+t(s)Seq 15
+Seq 25
+0.8
+Seq 35
+Actual
+0.6
+(w)
+S
+0.4
+0.2
+0
+0
+5
+10
+15
+t(s)17 
+ 
+ 
+(c) 
+ 
+ 
+Figure 14: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of 
+floating box from DEIM-LSTM prediction with SPH predictions for the case when the training 
+and test dataset are different for different input sequence. 
+ 
+The effect of varying the number of neurons in the hidden layers on the prediction capability of the 
+DEIM-LSTM is shown in Fig. 15(a)-(c) for the same data sets.  
+ 
+ 
+ 
+(a) 
+(b) 
+ 
+(c) 
+ 
+Figure 15: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of floating 
+box from DEIM-LSTM prediction with SPH predictions for the case when the training and test 
+dataset are different for different input sequence. 
+
+20
+0
+(sop)
+-20
+·Seq 15
+Seq 25
+Seq 35
+-40
+Actual
+0
+5
+10
+15
+t(s)0.15
+- - ·Neurons 10
+Neurons50
+0.1
+Neurons100
+Actual
+(m)
+0.05
+-0.05
+0
+5
+10
+15
+t(s)1
+ -·Neurons 10
+Neurons 50
+0.8
+Neurons 100
+Actual
+0.6
+0.4
+0.2
+0
+0
+5
+10
+15
+t(s)20
+0
+d
+-20
+--·Neurons 10
+-Neurons 50
+Neurons 100
+-40
+Actual
+0
+5
+10
+15
+t(s)18 
+ 
+As the number of neurons in each layer is varied from 10 to 50 to 100, and the training time vary from 
+times 599s, 583s to 1637s. The training time is not linearly scaled like the previous two cases of epoch 
+numbers and input sequence variation. The MAE error associated with the 50 neurons and 100 
+neurons are 0.0223, 0.0169 8.0429 and 0.0240, 0.0167 7.95 respectively for the surge, heave, and 
+pitch motion whereas the MAE error of 10 neurons is indicated earlier for the epoch variation of 250 
+and input sequence of 25. The deviation of the predicted pitch response from the actual data decreases 
+with the increase in the number of neurons.  
+ 
+For all three variations such as the number of epochs, input sequence, and the number of neurons, the 
+pitch response shows a constant phase shift from the benchmark result which is also evident from the 
+MAE associated with the pitch response as well. If the neural network is trained with the input wave 
+elevation and corresponding box motion generated by a multi-harmonic paddle excitation instead of 
+a single frequency signal, it may be able to capture the inherent nonlinearity in the flow physics and 
+this deviation of the predicted result from the actual data is expected to get minimized.   
+ 
+III.b LSTM-PINN Prediction of Wave-Floating Box Interaction Dynamics: 
+ 
+In this section the rigid body dynamics i.e., Eqn. 1 is incorporated to enhance the LSTM network loss 
+function to construct a physics informed network (LSTM-PINN) and to assess the effect of training data 
+size on the quality of the predictions.  The SPH computed hydrodynamic forces are applied to the box 
+motion which are considered as an input of the LSTM network and box motion i.e., pitch, plunge, and 
+surge motion are taken as output. The high frequency noise content in the surge, heave and pitch 
+responses of the box motion is first smoothed out and the acceleration signals based on the smoothed 
+displacements are computed using the second-order backward Euler method. The noisy dataset generates 
+spurious residual in the optimization process of the learning algorithm which often gets stuck at a local 
+minimum and thereby results in poor prediction of the overall LSTM-PINN method. Figure 16(a)-(c) 
+compares the smoothed accelerations in the surge, heave, and pitch axes with the actual computed 
+accelerations from SPH and it shows that even if the displacement signal is smoothened by removing the 
+high-frequency numerical noise, obtained acceleration matches the mean value of the actual acceleration. 
+After the initial transients die out (~ 5s), the actual acceleration coincides with the smoothened signal. 
+As the aim of the present effort is to show a proof of concept of the proposed surrogates, a force signal 
+based on the smoothed displacement is used instead of the actual force obtained from the SPH simulation. 
+ 
+Both the training and the test dataset for this network are the same as that shown in Fig. 6(a) and the 
+training box responses in Fig.11. A percentage of the training input and output is considered as the 
+training dataset of the network, and the aim is to predict the entire output response using the partial data 
+used for training.  
+ 
+ 
+
+19 
+ 
+ 
+ 
+(a) 
+(b) 
+ 
+(c) 
+ 
+ 
+Figure 16: Smoothening of the acceleration signal in (a) heave (b)surge (c) pitch direction 
+ 
+ 
+The implication of using LSTM-PINN is that the size of the training data can be reduced since the 
+incorporation of the physics will correct the weights and biases accordingly. Figure 17(a)-(c) and 
+Fig.18(a)-(c) show the LSTM and LSTM PINN prediction of heave, surge and pitch responses using 
+59.87 % and 0.2% data. For both the cases, the heave and pitch motions are predicted well using the 
+LSTM-PINN network whereas the predicted surge motion using 0.2% data does not fit well with the 
+benchmark result. 
+ 
+ 
+Fig 17: Comparison of DEIM-LSTM and DEIM-LSTM-PINN (trained using 59.87% of the data 
+prediction of the temporal variation of (a) heave (b) surge (c) pitch response of floating box with actual 
+data. 
+ 
+
+2
+-·Actual
+Smoothened
+1
+/
+0
+-1
+-2
+0
+5
+10
+15
+t(s)10
+:Actual
+Smoothened
+0
+S
+-10
+ay
+-20
+-30
+0
+5
+10
+15
+t(s)·Actual
+20
+Smoothened
+10
+/pu)?
+0
+-10
+-20
+0
+5
+10
+15
+t(s)10
+0.05
+0.8
+LSTM
+LSTM-PINN
+0.6
+--Actual
+-10
+S
+0.4
+0.2
+-0.05
+-20
+LSTM
+LSTM
+LSTM-PINN
+-30
+LSTM-PINN
+·Actual
+- -Actual
+-0.2
+5
+10
+15
+-0.1
+0
+5
+10
+15
+0
+5
+10
+15
+t(s)
+t(s)
+t(s)
+(a)
+(b)
+(c)20 
+ 
+ 
+Fig 18: Comparison of DEIM-LSTM and DEIM-LSTM-PINN (trained using 0.2% of the data) 
+prediction of the temporal variation of (a) heave (b) surge (c) pitch response of floating box with actual 
+data computed using SPH. 
+ 
+ 
+Table 1 compares the L2 error of the prediction accuracy using DEIM-LSTM and DEIM-LSTM-PINN 
+based on the percentage of the training data (ground truth or actual data computed using SPH) used. 
+Table 1 shows that for sparse data (~0.2%) the LSTM-PINN offers a better prediction of pitch and heave 
+motion. If the training data used in large, LSTM predictions appear to be better than LSTM-PINN thereby 
+suggesting that with the use of big data physics may not have significant influence on the prediction.  
+ 
+Table 1: L2 Error in percentage for Surge, heave, and pitch motion for DEIM-LSTM and 
+DEIM-LSTM-PINN prediction as compared to the benchmark data. 
+ 
+Percentage of Data 
+Used 
+LSTM 
+(% L2 error) 
+ 
+LSTM-PINN 
+           (% L2 error) 
+ 
+ 
+ 
+S 
+~100% 
+0.0013 
+0.0085 
+~59.87% 
+0.0048 
+0.0098 
+~0.20% 
+8.661 
+13.4368 
+ 
+ 
+ 
+h 
+~100% 
+0.0060 
+0.0233 
+~59.87% 
+0.0293 
+0.0654 
+~0.20% 
+57.67 
+24.81 
+ 
+ 
+ 
+α 
+~100% 
+0.0044 
+0.0051 
+~59.87% 
+0.0103 
+0.0214 
+~0.20% 
+42.2455 
+12.86 
+ 
+ 
+From Table 1 the LSTM network outperforms the LSTM-PINN network despite the introduction of 
+physics loss in the total loss function. Figure 19 compares the variation of the mean squared error-based 
+loss function as shown in Eqn.6. It appears that the loss associated with the LSTM prediction decays 
+faster than that of the LSTM-PINN prediction when the complete dataset is used. The additional physics-
+based loss function increases the residual values, and if the parameters of the optimization algorithms are 
+not set properly the residual values often get stuck in a local minimum point thereby deteriorating the 
+prediction accuracy as compared to the LSTM algorithm.  Therefore, different weightages (w1 and w2 
+in Eqn.6) of the loss functions are considered and their effect on the decay in mean squared error (MSE) 
+
+LSTM
+0.05
+15
+-LSTM
+0.8
+LSTM-PINN
+10
+LSTM-PINN
+--·Actual
+·Actual
+0.6
+5
+S
+0.4
+a
+-5
+0.2
+-0.05
+-10
+LSTM
+LSTM-PINN
+-15
+-Actual
+-20
+-0.2
+-0.1
+5
+10
+15
+0
+5
+10
+15
+0
+5
+10
+15
+t(s)
+t(s)
+t(s)
+(a)
+(b)
+(c)21 
+ 
+are compared in Fig.20. The weightage matrix of the loss function w2 are as follows: 
+ 
+                      
+
+
+
+
+
+
+2
+1 1 1 1 1 1 ,
+2
+1 1 1
+0.001
+0.001
+0.001 ,
+2
+0.01
+0.01
+0.01
+0.001
+0.001
+0.001 ,
+w a
+w b
+w c
+=
+=
+=
+                                (9) 
+w1 and w2 are associated with the data-based loss function 
+LSTM
+MSE
+ and 
+PINN
+MSE
+ respectively. The 
+LSTM-PINN algorithms show better accuracy only for the w2c which also indicates the 
+minimization of the residuals and the MSE values are decaying faster than LSTM approach. With 
+the w2a and w2b weightages the loss functions do not decay beyond a certain value thereby resulting in 
+a very poor prediction of the box responses even though the governing equations-based loss functions 
+are considered.    
+ 
+ 
+Figure 19: Loss history for full dataset with LSTM and LSTM-PINN algorithm  
+ 
+ 
+Figure 20: Decay of mese values with different weightage of the loss function.        
+ 
+IV. Conclusion:  
+ A novel non-intrusive reduced order model is introduced for the prediction of the temporal variation of 
+the surge, heave and pitching dynamics of a freely floating box on the wavy surface of a water wave 
+maker basin. A dimensionality reduction approach DEIM is coupled with a LSTM-based deep learning 
+approach for the reduction of the computational training time. The neural network parameters are varied 
+
+LSTM-PINN
+LSTM
+100
+mse
+10-2
+0
+50
+100
+150
+200
+250
+Epochsw2a
+w2b
+100
+w2c
+Only Data
+e
+mse
+10
+5
+0
+100
+200
+300
+400
+500
+Epochs22 
+ 
+to assess their influence on the prediction accuracy. It is noticed that there is a constant phase shift of the 
+predicted pitch motion from the benchmark result, and it is expected that introduction of a mixed 
+harmonic paddle excitation will learn the inherent nonlinearity present in the hydrodynamics and the 
+prediction accuracy can be improved. Finally, the governing rigid body dynamics equation is coupled 
+with the LSTM network in the discrete form to create the LSTM-PINN network. The addition of the 
+physics-based loss function increases the magnitude of the residual of the LSTM network, resulting in a 
+tendency for the loos function to get stuck at a local minimum frequently if the optimization algorithm 
+in the network is not tuned. In the current work, the LSTM-PINN network is applied to the simple 
+structural dynamical system which can be very well learned by the LSTM network. The introduction of 
+the physics-based governing equation in the loss function did not make any significant improvement in 
+the prediction accuracy. On, the other hand, the use of a larger dataset, and the introduction of the physics-
+based loss function appear to deteriorate the performance by increasing the magnitude of the residual. 
+The potential of the LSTM-PINN will be realized if the LSTM and LSTM-PINN network is also be 
+applied to the SPH equations to form an LSTM/LSTM-PINN network mapping between spatiotemporal 
+coordinates and unsteady flow variables such as velocities and pressure which the authors will address 
+and also explore the coupling of the networks for fluid flow and structural dynamics as a possible set of 
+digital twins for box dynamics in their future work. 
+  
+Acknowledgment: The authors acknowledge the efforts of Ouyang Zhenyu and Tang Xiaoyang for their 
+prompt generation of computational data sets using DualSPHysics which serves as the ground truth 
+(actual) data and training data for this work. 
+ 
+References:  
+[1] Carlberg, K., Amsallem, D., Avery, P., Zahr, M. and Farhat, C., “The GNAT Nonlinear Model 
+Reduction Method and its Application to Fluid Dynamics Problems”, AIAA 2011-3112 in  Proceedings 
+of 6th AIAA Theoretical Fluid Mechanics Conference, 27 - 30 June 2011, Honolulu, Hawaii, USA. 
+ 
+[2] Schmidhuber, J.” Deep Learning in Neural Networks: An Overview”, Neural Netw. 61, 85–117 
+(2015). 
+ 
+[3] Maulik, R., San, O., Rasheed, A., Vedula, P.,” Subgrid Modelling for Two-Dimensional Turbulence 
+using Neural Networks”, Journal of Fluid Mechanics, 858(1), pp. 122-144, Jan 2019.  
+ 
+[4] Parish, E. J., Duraisamy K.,” A Paradigm for Data-driven Predictive Modeling using Field Inversion 
+and Machine Learning”, Journal of Computational Physics, 305(1), pp. 758-774, Jan 2016. 
+ 
+[5] Wang, Z., Xiao, D., Fang, F., Govindan, R., Pain, C.C. and Guo, Y. “Model Identification of Reduced 
+Order Fluid Dynamics Systems Using Deep Learning”, Int. J. Numer. Meth. Fluids. 86(4), 255–268, July 
+2017. 
+
+23 
+ 
+ 
+[6] Lui, H.F.S., and Wolf, W.R.,” Construction of Reduced-Order Models for Fluid Flows using Deep 
+Feedforward Neural Networks”, Journal of Fluid Mechanics, 872(6), pp. 963–994, Jun 2019. 
+ 
+[7] Raissi, M., “Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential 
+Equations”, The Journal of Machine Learning Research,19(1), pp. 932-955, January 2018. 
+ 
+[8] Brunton, S. L., Noack, B.R., and Koumoutsakos, P.,” Machine Learning for Fluid Mechanics”, 
+Annual Review of Fluid Mechanics, 52(1), pp 477-508, Jan 2020.  
+ 
+[9] Chaturantabut, S. and Sorensen, D.C.,” Nonlinear Reduced Order Modelling Via Discrete Empirical 
+Interpolation”, SIAM J. Sci. Comput.,32(5), pp. 2737-2764, May 2010. 
+ 
+[10] Chakrabarti, S. K., Numerical Models in Fluid-Structure Interaction Advances in Fluid Mechanics. 
+Vol. 42, WIT Press (2005). 
+ 
+[11] Monaghan, J. J. Smoothed Particle Hydrodynamics. Annual Review of Astronomy and Astrophysics, 
+30(1), 543-574 (1992). 
+ 
+[12] Monaghan, J. J., & Lattanzio, J. C. “A Refined Particle Method for Astrophysical Problems”, 
+Astronomy and astrophysics, 149, 135-143 (1985). 
+ 
+[13] Wen, H., Ren, B., Dong, P., & Wang, Y. (2016). “A SPH Numerical Wave Basin for Modeling Wave-
+Structure Interactions”, Applied Ocean Research, 59, 366-377. 
+ 
+[14] Altomare, C., Domínguez, J., Crespo, A., González-Cao, J., Suzuki, T., Gómez-Gesteira, M., & 
+Troch, P. Long-Crested Wave Generation and Absorption for SPH-Based DualSPHysics Model. Coastal 
+Engineering, 127, 37-54. (2017). 
+ 
+[15] Domínguez JM, Fourtakas G, Altomare C, Canelas RB, Tafuni A, García-Feal O, Martínez-Estévez 
+I, Mokos A, Vacondio R, Crespo AJC, Rogers BD, Stansby PK, Gómez-Gesteira M. (2021) 
+DualSPHysics: from Fluid Dynamics to Multiphysics Problems. Computational Particle Mechanics. 
+doi:10.1007/s40571-021-00404-2. 
+ 
+[16] Ren, B., He, M., Dong, P., & Wen, H. (2015). Nonlinear Simulations of Wave-Induced Motions of 
+a Freely Floating Body using WCSPH Method. Applied Ocean Research, 50, 1-12.  
+ DOI: https://doi.org/10.1016/j.apor.2014.12.003 
+ 
+[17] Domínguez, J. M., Crespo, A. J., Hall, M., Altomare, C., Wu, M., Stratigaki, V., Troch, P., 
+
+24 
+ 
+Cappietti, L., & Gómez-Gesteira, M. (2019). SPH simulation of floating structures with moorings. 
+Coastal Engineering, 153, 103560. DOI: https://doi.org/10.1016/j.coastaleng.2019.103560 
+ 
+[18] Keras Platform at https://keras.io/guides/ and https://github.com/keras-team/keras  
+ 
+[19] Lipton, Z.C., Berkowitz, J., and Elkan, J.,” A Critical Review of Recurrent Neural Networks for 
+Sequence Learning”, Proceedings of the ACM International Conference on Multimedia - MM '14, pp. 
+675-678, 2014. 
+ 
+[20] Hochreiter, S. and Schmidhuber, J., “Long Short-Term Memory”, Neural Computation, 9 (8), 
+pp. 1735-1780, August 1997. 
+ 
+[21] Bengio, Y., Courville, A., and Vincent, P., “Representation Learning: A Review and New 
+Perspectives,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 35, No. 8, Aug. 
+2013, pp. 1798–1828. 
+ 
+[22] Sorensen, D. C., and Embree, “A DEIM Induced CUR Factorization,” SIAM Journal of Scientific 
+Computing, Vol. 38, No. 3, March 2016, pp. 1454–1482. 
+ 
+[23] R. Halder, M. Damodaran and B.C. Khoo, “A Deep-Learning Based Nonlinear Reduced Order 
+Model for Airfoil Gust and Aileron Buzz Response", AIAA Journal, 58:10, pp. 4304-4321, Oct 2020.  
+DOI: https://doi.org/10.2514/1.J059027 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
diff --git a/PtFKT4oBgHgl3EQfgy6A/content/tmp_files/load_file.txt b/PtFKT4oBgHgl3EQfgy6A/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf,len=822
+page_content='1  Deep Learning-Driven Nonlinear Reduced-Order Models for Predicting  Wave- Structure Interaction  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Halder*, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Damodaran** and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Khoo#  National University of Singapore, Temasek Laboratories 9 Engineering Drive 1, Singapore 117575  (Dated 31 July 2022)  Abstract  Long Short-Term Memory (LSTM) network-driven Non-Intrusive Reduced Order Model  (NROM) for predicting the dynamics of a floating box on the water surface in a wavemaker  basin is addressed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The ground truth or actual data for these wave-structure  interactions (WSI) problems, namely box displacements and hydrodynamic forces and  moments acting on the box due to wave interaction corresponding to a particular wave  profile, are computed using the Smoothed Particle Hydrodynamics (SPH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The  dimensionality of the system is first reduced using the Discrete Empirical Interpolation  Method (DEIM) and the LSTM is applied to the reduced system resulting in a DEIM-LSTM  network for developing a surrogate for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The network is further enhanced by  incorporating the physics information into the loss function resulting in a physics-informed  LSTM (LSTM-PINN) for predicting the rigid body dynamics of box motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The  performance of predictions for these networks is assessed for the two-dimensional wave  basin WSI problem as a proof-of-concept demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Introduction  During the past two decades, several Reduced Order Model (ROM) methods outlined in Carlberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  [1] have been instrumental in expediting the efficient computational modeling of dynamical systems  implying significant benefits for flow prediction in engineered systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The conventional Computational  Fluid Dynamics (CFD) methods offer solutions to well-posed flow problems with proper initial and  boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Recent efforts in deep learning (DL) appear to offer efficient and effective solutions  easily for ill-posed problems such as inverse, regression, and classification problems in diverse fields as  outlined in Schmidhuber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [2] have gained significant attention in a wide variety of fluid mechanics  applications such as the closure model for turbulence using machine learning as in Maulik and San [3]  and Duraisamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [4], the combined POD and LSTM network for incompressible flows with spectral  proper orthogonal decomposition (SPOD) of Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [5] and the application of a dimensionality  reduction method with DL networks for learning feature dynamics from noisy data sets in Lui and Wolf  ________________________________  Research Scientist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' e0010762@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Currently Research Fellow University of Michigan, USA  ** Senior Research Scientist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' tslmura@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='sg  # Professor of Mechanical Engineering and Director Temasek Laboratories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' tslhead@nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='sg  2  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The Physics Informed Neural Network (PINN) introduced by Raissi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [7] in computational  continuum mechanics offers a trade-off between the availability of the data and the accuracy of the  solution and the possibilities of training DL networks with low or no data to moderate and big data for  effective prediction of flow problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The application of machine learning in fluid mechanics as outlined  in Brunton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [8] has motivated the present study in which the LSTM and LSTM-PINN network is  coupled with the DEIM algorithm as the nonlinear model order reduction method outlined in  Chaturantabut and Sorensen [9] to develop a surrogate for predicting a couple of wave-structure  interaction (WSI) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The wave structure interaction (WSI) possesses different numerical  challenges because of strong nonlinearities in the dynamical system and its multi-disciplinary nature as  outlined by Chakrabarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Although SPH is originally developed for astrophysical applications  by Monaghan [11] and Monaghan and Lattanzio [12], it has been used  for modelling wave generation  and WSI problems in several studies such as Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [13] and Altomare et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [14] which uses the  implementation of the SPH in the open-source platform DualSPHysics outlined in Dominique et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [15]  which is the platform used in the present study to generate necessary data for the proposed nonlinear  reduced-order model DEIM-LSTM and LSTM-PINN algorithms and also to compare the predicted result  with the ground truth (or  actual data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 1 shows the schematic of the wave-basin setup for  predicting the nonlinear wave interaction and the unsteady motion dynamics of a freely floating box on  the surface of a wave in a two-dimensional wave basin as a proof-of-concept demonstration of the  proposed LSTM-DEIM/LSTM-PINN-DEIM surrogates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The freely floating box considered has a length  of lb of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='3m, a height of hb of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2m, and a unit width in the spanwise direction and is assumed to have a  density of 500kg/m3 which will float in water which has twice its density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' This problem has been  simulated using a single phase SPH by Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [16] and Domínguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The movement of the  paddle on the left will set up wave flumes in the wave basin and the wave heights are measured relative  to the datum line (black solid line at the bottom of the wave basin) which corresponds to the undisturbed  water level in the wave basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figure 1: Schematic of the initial set-up and initial location for a freely floating box in the 2D  wave basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  In this study, the problem of the hydrodynamic wave interaction with a freely floating rectangular box is  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='.Bufferl  Buffer2  p=500kg/m  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Paddle  hs  d  1:4  S  4m  Wall  6m3  considered in the context of a single-phase SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' A freely floating box is initially assumed to be still on  the surface of the water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The box is excited by a wave generated from a wake maker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The wave profile  has a period of T of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2s and a wave height H0 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1m and its interaction with the floating box create  box motions, namely surge in the direction and heave perpendicular to the direction of wave propagation  and pitching rotation (pitch) about the center of mass of the box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The motion of the piston creates a  second-order Stokes wave which is absorbed in the 1:4-sloped dissipative beach at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The no-slip  boundary conditions are implemented at the bottom wall and the dissipative beach head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Buffers 1 and 2  denote the open boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' As this is a two-dimensional problem there will be two translational motions  of the box center of mass, namely, surge and heave motions along with the x and y directions respectively,  and a pitching motion about the center of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The dynamics of the floating box is defined  mathematically as:  d  1  (1  )  d  d  d  d(  )  d  y  y  r  x  x  s  M  M  t  M  t  t  \uf072  =  +  −  =  \uf0d7  =  U  F  g  U  F  J ω  T  (1)  where M is the mass of the box  assumed to be 30 kg, J is the moment of inertia of the box about its  center of mass assumed to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='325 kg-m2, ρr is the specific density of the box material  assumed to be  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5 and  x  F ,  y  F and T the forces in the surge x, and heave y directions and torque about the box center  of mass respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' A second-order Stokes wave is generated by the wavemaker on the left end of the  basin and the waves are absorbed in the 1:4-sloped dissipative beach head on the right end of the basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  No-slip boundary conditions are implemented on the bottom wall and the dissipative beach while Buffers  1 and 2 are treated as open boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The total number of particles used for the single-phase SPH is  70023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' A resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='01 m proposed in Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [16] is used in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The box motion  dynamics of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' (1) in a matrix form is as follows:  \uf05b  \uf05d\uf07b \uf07d \uf07b \uf07d  M  X = F  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1)  \uf05b \uf05d  \uf07b \uf07d  \uf07b \uf07d  0  0  M  0  0  , X  , F  0  0  where,  1  1  x  y  r  F  M  S  M  h  F  M  J  \uf061  \uf072  =  =  =  +  \uf0e9  \uf0f9  \uf0e9  \uf0f9  \uf0e9 \uf0f9  \uf0ea  \uf0fa  \uf0e6  \uf0f6  \uf0ea  \uf0fa  \uf0ea \uf0fa  \uf0ea  \uf0fa  −  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0ea \uf0fa  \uf0ea  \uf0fa  \uf0e8  \uf0f8  \uf0ea  \uf0fa  \uf0ea \uf0fa  \uf0eb  \uf0fb  \uf0eb \uf0fb  \uf0ea  \uf0fa  \uf0eb  \uf0fb  g  T  where ,  S h and \uf061  are the surge,  heave, and pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The first order time derivatives of these kinematic variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', ,  and  S h  \uf061  are the  surge, heave and pitch rates and g are the acceleration due to gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Since  0  X  X  −  =  it follows that  4  \uf07b \uf07d  \uf07b \uf07d  1  1  0  0  0  1  0  0  0  0  0  0  1  0  0  0  0  0  0  0  1  0  [  ]  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  X  X  M  F  X  M F  X  X  −  − −  −  −  =  +  =  \uf0e9  \uf0f9  \uf0ea  \uf0fa  \uf0ec  \uf0fc  \uf0ea  \uf0fa  \uf0ef  \uf0ef  \uf0ea  \uf0fa  \uf0e9  \uf0f9  \uf0ef  \uf0ef  \uf0ec \uf0fc  \uf0de  \uf0ed \uf0fd  \uf0ed  \uf0fd  \uf0ea  \uf0fa  \uf0ea  \uf0fa  \uf0ee \uf0fe  \uf0eb  \uf0fb  \uf0ef  \uf0ef  \uf0ea  \uf0fa  \uf0ef  \uf0ef  \uf0ea  \uf0fa  \uf0ee  \uf0fe  \uf0ea  \uf0fa  \uf0eb  \uf0fb  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2)  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Nonlinear Reduced Order Model:  Formulation of the nonlinear reduced order models (NROMs) proposed for the possibility of developing  digital twins for the wave-structure interactions are outlined in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Section II(a) discusses the  basic architecture of the LSTM and LSTM PINN networks and outlines the elements of the application  of LSTM-PINN network for rigid body dynamics equation governing the box motion while Section II(b)  addresses the coupling of the LSTM with the dimensionality reduction approach of DEIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Although the  PINN network is computationally expensive than conventional LSTM network and numerical solution  of the benchmark rigid body dynamics equation, the objective of the current work is to present a proof-  of-concept model for the realization of a LSTM-PINN model which can be used for predicting the  response of a large-scale application using sparse or no data to assess if the potential of the LSTM-PINN  network can be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The DEIM-LSTM and LSTM-PINN codes are available in  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='com/rahulhalderAERO/Hydrodynamic_LSTM_ROM  II(a) LSTM and LSTM-PINN network:  In this current section, the formulation of functional relationship between input and output are first  explained with a simpler ANN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' A single layer ANN network consisting of few neurons mapping  input data, namely the hydrodynamic forces acting on a floating box computed from SPH and the output  data consisting of the floating box kinematic variables computed from SPH, which form the dataset for  training the surrogate for predicting the dynamics of the floating box in a wave basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The mapping can  be expressed as follows:  \uf07b  \uf07d  2  1  1  2  (  )  (  )  x  y  f W W  b  b  S  h  S  h  F  F  T  \uf061  \uf061 =  +  +  \uf0e9  \uf0f9  \uf0eb  \uf0fb  (3)  where \uf07b  \uf07d  6  tn  S  h  S  h  \uf061  \uf061  \uf0b4  \uf0ce  and  3  t  x  n  y  F  F  T  \uf0b4  \uf0ce  \uf0e9  \uf0f9  \uf0eb  \uf0fb  are output and input data, W1  and W2  are the weight matrices, b1 and b2 are the bias vectors respectively and f  is the nonlinear activation  function which in this study uses the tanh function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The structure of a simple LSTM network is different  from that of the ANN architecture, and it will be demonstrated for the same input and output datasets  associated with the box dynamics floating on the hydrodynamics waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The input and output structure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='of the LSTM network includes all the variables  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf07b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf07d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+page_content='\uf061  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+page_content='as the output and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf05b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf05d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='x  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+page_content='F  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+page_content='p =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='as the input with the LSTM input-output mapping defined as follows:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf05b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf05d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf05b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf05d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+page_content='f W p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='Uh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='b  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='+ =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='(4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='where the additional matrix U which is absent in the ANN architecture associates input from the previous  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='time steps and introduces the memory effect in the ANN architecture leading to the Recurrent Neural  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='Network (RNN) outlined in Lipton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' For the simplest RNN,  lh is the predicted output value at  the nth time step that is \uf07b  \uf07d  n  op  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' In the current work, this network is implemented in the Keras platform  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The input and output structure of such a network is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 2(a) and 2(b) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The  input structure is a tensor which contains the first dimension indicating total time steps  tn , the second  dimension indicating the backpropagation length (memory effect or previous time steps data) in case of  input and number of neurons in case of output, and the third one indicating the type of inputs or outputs  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', plunge, pitch, forces, plunge velocity, pitch velocity, forces, and moments, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  Figure 2: (a) Input and (b) Output data structure  The limitation of the RNN layer is that it creates instances of vanishing or exploding gradient problems  because of the long-term dependencies over the time-sequential dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', multiplication of gradient  terms over total timestep  tn  in a sequence prediction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Hochreiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [20] introduced the LSTM  network to mitigate this problem of vanishing or exploding gradients whereby unlike the conventional  RNN which keeps its contents of previous time steps while computing the gradients, the LSTM  introduces four distinct operations in each RNN cell: memory cell, input gate, forget gate, and output  gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Details of the Memory cell, Input gate, Forget gate and Output gate, and the hidden layer operations  in connection with the LSTM network are defined as follows:  Back Propagation Length  of  Xi  xNumberofNeurons 16  n  n-1  i  i  n  n-1  n  n-1  o  o  n  n  n-1  n  n-1  c  c  n  n  n  Input Gate:  i = σ(Wh  +b )  Forget Gate:        f = σ(W h  +b )  Output Gate:        o  σ(W h  +b )  Cell State:             c = i  c  i  tanh(W h  b )  Hidden Layer:        h  o  tanh(c ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  f  f = +  + =  (5)  where  is the Hadamard product, input forces are fed to the LSTM network at each step of the input  sequence defined as n, i is the Input gate and f is the Forget gate which will choose the necessary  information to be passed through or forget through cell state c, Wf and Wc are the corresponding weight  matrices,  ib , b f  and bc  are the bias vectors for the Input gate, Forget gate and Cell state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  The Output gate, o, on the other hand, decides the control of the flow of the information from the cell  state to the next hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' These operations help the LSTM cells to keep only the necessary memories  thereby mitigating the occurrence of exploding and vanishing gradient problems in the back-propagation  algorithm for each iteration step of the gradient descent optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 3(a) shows a schematic of  the internal architecture of a single LSTM cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 3(b) shows the stacking of LSTM cells to enhance  the network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The output dimension of the LSTM layer  tn  N  \uf0b4  \uf0ce  where N is the number of neurons,  the final layer output must be sent through a dense ANN layer to obtain the required output size which  is  3  tn \uf0b4 here (\uf07b  \uf07d  T  S  h  S  h  \uf061  \uf061  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 3(b) also shows the stacked LSTM layers sent to the  dense ANN layer to form the final LSTM/LSTM-PINN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  Figure 3: (a) Structure of a Single LSTM cell (b) Stacked LSTM cells forming LSTM/LSTM-PINN  The loss function associated with the LSTM network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=',  LSTM  MSE  is computed using input and output  values of the LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The outputs from the LSTM network are used in the governing equations  Xr,yt-l  LSTMCellMSEo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='=MSELsTM+MSEpi  yt-1  yi-n+1  4  LaerL+DenseLaer  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+page_content='\uf0f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='\uf0e5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='(7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='are the losses associated with the LSTM network and the loss associated with the physics of the floating  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='box dynamics respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' w1 and w2 are the weightage values associated with the data driven and  physics-based loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The input of the LSTM-PINN network is set as  [  ]  i  x  y  P  F F  T  =  whereas  the output of the LSTM-PINN network consists of the dynamic responses and their time derivatives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  \uf07b  \uf07d  T  S  h  S  h  \uf061  \uf061  of the floating box at the  ,  1,  2  th  th  th  t  t  t  n  n  n  −  −  time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Therefore, the size  of the input of the network is  3  t  N  \uf0b4  whereas size of the output of the network is  18  t  N  \uf0b4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The  tn is any  time instant in total time steps of  t  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' (b) DEIM-LSTM network  The manifold hypothesis outlined by Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [21] forms the basis of dimensionality reduction which  states that the solution of a high dimensional dynamical system lies near a low dimensional manifold S  embedded in a large dataset of size Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The Discrete Empirical Interpolation Method (DEIM) is a data  compression method for reconstructing the full data set from the information of the flow variables  specified at a few sensor locations as shown in Halder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' For example, if the dataset D is a  function of time t or any parameter 𝜇, then D can be expressed as 𝐷(𝑡) = Ф𝑐(𝑡)  where Ф defines the  DEIM modes and c(t) is the coefficient vector which is computed from an index matrix or mask matrix  P used for indexing sensor locations in the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The index matrix is defined as P =  [𝑒𝜌1, … … … .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' , 𝑒𝜌𝑙], P ϵ ℝ𝑛×𝑙 where each column is defined as  𝑒𝜌𝑖 = [0, … 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1 … 0]𝑇 which implies a  value of 1 at location ρi, which is decided from an error minimization-based algorithm as shown in  Chaturantabut and Sorensen [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Hence, the full dataset can be approximated as:  1  ( , )  (  )  ( , )  T  T  D t  U P U  P D t  \uf06d  \uf06d  =  (8)  where U(PTU)-1 is considered as the DEIM modes Ф and PTD(t) indicates the sensor locations values  which form coefficient vector c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' For the large datasets, the CUR approximate matrix decomposition using  the DEIM algorithm outlined in Sorensen and Embree [22] can also be used to compute the control points  8  efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The basic construct of the non-intrusive reduced order model (NIROM), results in a  dimensional reduction from Rn to Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The size of an actual dataset is n, and the size of the reduced dataset  is m where m << n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' This reduced dataset is fed to a deep learning algorithm for exploring the feature  dynamics of the.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' A series of snapshots, each having the size of Rn is reduced to a set of snapshots of the  size of Rm before feeding into the deep-learning network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  For the prediction of the wave-structure interaction, Discrete Empirical Interpolation Method (DEIM) is  coupled with the LSTM network for the floating box problem with three degrees of freedom (pitch,  plunge ad surge), and the reader is referred to details of this step in Halder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 4 shows the  schematic of the application of the DEIM-LSTM to predict the dynamics of a floating box in an unsteady  hydrodynamic wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' First, the unsteady wave elevation distribution data is generated from the  DualSPHysics simulation by exciting the wavemaker so that the computed data of hydrodynamic loads  on the box and the kinematics of the floating box forms the ground truth or the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The DEIM  algorithm is then applied to the spatio-temporal wave height distribution and a few control points are  selected for determining the DEIM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' For the training phase of the LSTM/LSTM-PINN network,  LSTM is applied to the wave heights at these control points and the predicted box motion from actual  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' After the LSTM network has been trained sufficiently, it can be used to predict the box motion  under any arbitrary wavemaker excitation profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figure 4: Schematic of the DEIM-LSTM Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='04  h(m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  DEIM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='04  1  20  N  15  10  x(m)  5  t(s)  0  0  DIEM ControlPoints  FloatingBoxResponseUnder  LSTM  anyWaveElevation  Network  Distribution(Only Control  E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  PointsValuesReqd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  15  (s  BoxResponse(Output)9  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Results and Discussion:  In this section, the prediction of the floating box dynamics due to the surface wave elevations in the wave  basin is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' First, the DEIM-LSTM is used for predicting the dynamics of the floating box where  the wave elevation is the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' In the next section, the governing equations for the dynamics of the  floating box are coupled with the data-driven loss function in the LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figure 5 compares the temporal dynamical motion of the floating box responses of computed surge,  heave and pitching with the experimental data of Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' [16] corresponding to the paddle in the  wavemaker in the wave-basin being excited with a sinusoidal function of period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2 s and amplitude of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1 m to validate the SPH model predictions with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' This is the ground truth for the  current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  (c)  Figure 5: Comparison of the box response (a) surge (b) heave and (c) pitch motion using experiments  and current numerical model  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' a DEIM-LSTM Prediction of Wave-Floating Box Interaction Dynamics  The paddle in the wavemaker is excited with a sinusoidal function of period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2 s and amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  m for the training of the DEIM-LSTM network and a sinusoidal function of period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='3 s and amplitude  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='12 m as a test function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The temporal-spatial distribution of the wave elevations corresponding to  these scenarios is shown in Figures 6(a) and 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  1  sph  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  Experimental  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  (m)  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0  0  5  10  15  20  t(s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  sph  Experimental  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  h(m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0  5  10  15  20  t(s)20  sph  Experimental  10  e  p  10  20  0  5  10  15  20  t(s)10  (a)  (b)  Figure 6: (a) Training and (b) Test elevation of the wave surface considered as an input to the LSTM  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  The training and test wave elevation data is of size  x  t  N  N  \uf0b4  where  x  N is the 300 spatial locations and  t  N is  the 2001 temporal locations for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' DEIM is then applied to dimensionally reduce this large input  dataset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 6 from  x  t  N  N  \uf0b4  to  x  t  N  m \uf0b4  (  x  m  <<  x  N ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The Deep Learning-based approach is considered  at this point to learn the temporal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Since the LSTM approach is already a computationally  efficient approach as compared to the SPH model, the dimensionality reduction is considered only in the  spatial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The Singular Value Decomposition (SVD) is first applied to the input dataset to decide  on an optimal number of modes to represent the temporal-spatial distribution of the wave elevation  considered as an input to the LSTM network as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Any dynamical system can be represented  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='04  h(m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='04  4  3  20  15  2  10  1  x(m)  5  t(s)  0  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  (u)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  4  3  20  15  2  10  x(m)  1  5  t(s)  0  011  as a combination of few modes and the singular values are associated with the energy contained by those  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 6 shows the variation of the scaled singular values (scaled with the largest singular value,  max  Sv  ) which indicate the number of modes that contain most of the energy of the dynamic system and  after a few modes, the singular values drop to a very small value which is not important for system  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figure 7: Normalized Singular Values after the application of the SVD on the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figure 8 shows a selection of DEIM control points for the reconstruction of instantaneous wave profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Among the 300 spatial points, only 30 points indicated as  1  2  ,  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='. n  x x  x  are chosen for all the temporals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figure 8:  DEIM Control points marked as black dots at a time instant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0  0  10  20  30  40  50  Modes0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='04  h(m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  0  A  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='04  4  3  20  15  2  10  x(m)  5  t(s)  0  0  DIEM ControlPoints12  locations to significantly reduce the size of the input data to  30 2000  \uf0b4  noting that  h  \uf044 is the difference  between the instantaneous wave elevation and initial wave elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The points  1  2  ,  ,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='. n  x x  x are termed  as DEIM control points and the DEIM modes termed as \uf046 in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' B contains the inherent features  of the wave dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Among the 30 DEIM modes corresponding to these control points, the spatial  variation of the first 4 modes is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' If the values of the DEIM control points for the training  and test wave distribution as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8 are known, then the entire spatio-temporal distribution of  the wave can be reconstructed as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The zoomed view of region marked by the red box  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='9(b) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='9(c) to showcase the accuracy of the prediction of the test wave elevation at  the 12 s instant by comparing the predictions using different numbers of DEIM modes ranging from 10  to 30 in steps of 10 with the computed ground truth or actual wave height (computed using SPH)  illustrating the adequacy of the selected number of DEIM control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' It shows that if 10 DEIM modes  are selected the reconstructed signal deviates from the actual wave height whereas, after 20 modes, the  change in the reconstructed wave height is insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 10(a) to 10(c) show that the error  distribution drastically drops when the number of selected modes is increased from 10 to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  (c)  Figure 9: (a) First four DEIM Modes and (b) comparison of predicted wave elevation using  different numbers of the DEIM modes (c) zoomed view of the red box in (b) compared to  2  Mode 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5  Mode 2  Mode 3  Mode 4  DEIM Modes  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  1  2  3  4  x(m)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  -·Actual  Mode 30  Mode20  Mode10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  △h(m)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='5  x(m)*- ·Actual  Mode 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='08  ·Mode 20  Mode10  △h(m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='07  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  (w)x13  computed wave elevation at the 12 s instant (ground truth or actual data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  (c)  Figure 10: Reconstruction error distribution of the test elevation using (a) 10 (b) 20 and (c) 30 DEIM  modes of the training wave elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figures 11 (a)-(c) compares the prediction of temporal variation of heave, surge, and pitch angle  dynamics of the floating box respectively for both training and test wave elevation distributions shown  respectively in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 6(a) and 6(b) and which will be used for DEIM-LSTM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='02  4  20  2  10  0  0  (w)x  t(s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='01  4  20  2  10  0  0  (w)x  t(s)2  0  2  4  20  2  10  0  0  x(m)  t(s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  test  train  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  h(m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0  5  10  15  t(s)1  test  train  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0  0  5  10  15  t(s)14  (c)  Figure 11: Temporal variation of (a) heave (b)surge and (c) pitch dynamics of the floating box  computed using SPH used as training and test data for DEIM-LSTM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  A LSTM network consisting of 25 input time sequence termed as n in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='3 of the Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='a, 4 hidden  layers, 10 neurons in each layer, and tanh activation function is considered for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 12(a)-  (c) compares the temporal variation of floating box dynamics of heave, surge, and pitching predicted  using DEIM-LSTM with actual data computed from SPH when the training and test wave elevations and  the corresponding box motions are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Since the training and test dataset are the same, the predicted  results have matched significantly well with the high-fidelity simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  20  test  10  train  0  H  10  20  30  0  5  10  15  t(s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='15  Predict  Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  h(m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0  5  10  15  t(s)Predict  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  (m  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0  0  5  10  15  t(s)15  (c)  Figure 12: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of floating  box from DEIM-LSTM prediction with SPH predictions for the case when the training and test dataset  are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Next the prediction of the floating box dynamics using DEIM-LSTM when the training and test datasets  used are different is considered and the impact of some of the pertinent network hyperparameters, namely  the number of epochs reached during training, the size of the backpropagation length (termed here as  input sequence) and the number of neurons in the hidden layers on the prediction is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figures  13(a)-(b) compares the DEIM-LSTM prediction of the heave, surge, and pitch motion of the floating box  with the actual data from SPH simulation for number of epochs ranging from 150-250-500 for which the  training time ranges from 344s, 599 s and 1264s respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Mean Absolute Errors (MAE) associated  with the surge, heave, and pitch response corresponding to epoch 150 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0251, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0164, and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='29  respectively whereas for 250 and 500 epochs they are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0241, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0152, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='54, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0218, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0153 and  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='3089 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Although the training time increases almost linearly with the number of epochs, the  prediction accuracy remains the same and compares well with the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 13(d) shows the  mean squared error variation with the number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' It shows that the training error does not further  decay after 300 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  The effect of varying the size of the backpropagation length or the training input sequence on the DEIM-  LSTM prediction capability is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 14 for the same data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' For the input sequence of 15, 25,  and 35, the training time is 378s, 599s, and 775s respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' For all three cases, the predicted result  matches well with the actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The MAE associated with the input sequence of 15 and 35 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0224,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0145, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='808, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0248, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0171, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='26 respectively for the surge, heave and pitch motion while the  MAE of the input sequence 25 is indicated earlier for the epoch variation of 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' It shows that the  prediction accuracy is better at the lower input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  20  0  (Sop)  20  Predict  40  ·Actual  0  5  10  15  t(s)16  (a)  (b)  (c)  (d)  Figure 13: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of  floating box from DEIM-LSTM prediction with SPH predictions for the case when the training  and test dataset are different for different epoch length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' (d) Variation of the mean squared loss  history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='15  Epoch 150  Epoch 250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  Epoch 500  Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0  5  10  15  t(s)1  Epoch 150  Epoch 250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  Epoch 500  Actual  (m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0  0  5  10  15  t(s)20  0  (sop)6  20  Epoch150  Epoch 250  Epoch 500  40  Actual  0  5  10  15  t(s)10  Loss  0  100  200  300  400  500  Epochs0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='15  -·Seq 15  Seq 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  Seq 35  Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  h  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0  5  10  15  t(s)Seq 15  Seq 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  Seq 35  Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  (w)  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0  0  5  10  15  t(s)17  (c)  Figure 14: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of  floating box from DEIM-LSTM prediction with SPH predictions for the case when the training  and test dataset are different for different input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  The effect of varying the number of neurons in the hidden layers on the prediction capability of the  DEIM-LSTM is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 15(a)-(c) for the same data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  (a)  (b)  (c)  Figure 15: Comparison of the temporal variation of (a) heave (b)surge (c) pitch response of floating  box from DEIM-LSTM prediction with SPH predictions for the case when the training and test  dataset are different for different input sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  20  0  (sop)  20  Seq 15  Seq 25  Seq 35  40  Actual  0  5  10  15  t(s)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='15  - ·Neurons 10  Neurons50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  Neurons100  Actual  (m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0  5  10  15  t(s)1  -·Neurons 10  Neurons 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  Neurons 100  Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0  0  5  10  15  t(s)20  0  d  20  --·Neurons 10  Neurons 50  Neurons 100  40  Actual  0  5  10  15  t(s)18  As the number of neurons in each layer is varied from 10 to 50 to 100, and the training time vary from  times 599s, 583s to 1637s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The training time is not linearly scaled like the previous two cases of epoch  numbers and input sequence variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The MAE error associated with the 50 neurons and 100  neurons are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0223, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0169 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0429 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0240, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0167 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='95 respectively for the surge, heave, and  pitch motion whereas the MAE error of 10 neurons is indicated earlier for the epoch variation of 250  and input sequence of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The deviation of the predicted pitch response from the actual data decreases  with the increase in the number of neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  For all three variations such as the number of epochs, input sequence, and the number of neurons, the  pitch response shows a constant phase shift from the benchmark result which is also evident from the  MAE associated with the pitch response as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' If the neural network is trained with the input wave  elevation and corresponding box motion generated by a multi-harmonic paddle excitation instead of  a single frequency signal, it may be able to capture the inherent nonlinearity in the flow physics and  this deviation of the predicted result from the actual data is expected to get minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='b LSTM-PINN Prediction of Wave-Floating Box Interaction Dynamics:  In this section the rigid body dynamics i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 1 is incorporated to enhance the LSTM network loss  function to construct a physics informed network (LSTM-PINN) and to assess the effect of training data  size on the quality of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The SPH computed hydrodynamic forces are applied to the box  motion which are considered as an input of the LSTM network and box motion i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', pitch, plunge, and  surge motion are taken as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The high frequency noise content in the surge, heave and pitch  responses of the box motion is first smoothed out and the acceleration signals based on the smoothed  displacements are computed using the second-order backward Euler method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The noisy dataset generates  spurious residual in the optimization process of the learning algorithm which often gets stuck at a local  minimum and thereby results in poor prediction of the overall LSTM-PINN method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 16(a)-(c)  compares the smoothed accelerations in the surge, heave, and pitch axes with the actual computed  accelerations from SPH and it shows that even if the displacement signal is smoothened by removing the  high-frequency numerical noise, obtained acceleration matches the mean value of the actual acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  After the initial transients die out (~ 5s), the actual acceleration coincides with the smoothened signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  As the aim of the present effort is to show a proof of concept of the proposed surrogates, a force signal  based on the smoothed displacement is used instead of the actual force obtained from the SPH simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Both the training and the test dataset for this network are the same as that shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 6(a) and the  training box responses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' A percentage of the training input and output is considered as the  training dataset of the network, and the aim is to predict the entire output response using the partial data  used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  19  (a)  (b)  (c)  Figure 16: Smoothening of the acceleration signal in (a) heave (b)surge (c) pitch direction  The implication of using LSTM-PINN is that the size of the training data can be reduced since the  incorporation of the physics will correct the weights and biases accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 17(a)-(c) and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='18(a)-(c) show the LSTM and LSTM PINN prediction of heave, surge and pitch responses using  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='87 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2% data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' For both the cases, the heave and pitch motions are predicted well using the  LSTM-PINN network whereas the predicted surge motion using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2% data does not fit well with the  benchmark result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Fig 17: Comparison of DEIM-LSTM and DEIM-LSTM-PINN (trained using 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='87% of the data  prediction of the temporal variation of (a) heave (b) surge (c) pitch response of floating box with actual  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  2  -·Actual  Smoothened  1  /  0  1  2  0  5  10  15  t(s)10  :Actual  Smoothened  0  S  10  ay  20  30  0  5  10  15  t(s)·Actual  20  Smoothened  10  /pu)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  0  10  20  0  5  10  15  t(s)10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  LSTM  LSTM-PINN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  --Actual  10  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  20  LSTM  LSTM  LSTM-PINN  30  LSTM-PINN  Actual  -Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  5  10  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  0  5  10  15  0  5  10  15  t(s)  t(s)  t(s)  (a)  (b)  (c)20  Fig 18: Comparison of DEIM-LSTM and DEIM-LSTM-PINN (trained using 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2% of the data)  prediction of the temporal variation of (a) heave (b) surge (c) pitch response of floating box with actual  data computed using SPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Table 1 compares the L2 error of the prediction accuracy using DEIM-LSTM and DEIM-LSTM-PINN  based on the percentage of the training data (ground truth or actual data computed using SPH) used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Table 1 shows that for sparse data (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2%) the LSTM-PINN offers a better prediction of pitch and heave  motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' If the training data used in large, LSTM predictions appear to be better than LSTM-PINN thereby  suggesting that with the use of big data physics may not have significant influence on the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Table 1: L2 Error in percentage for Surge, heave, and pitch motion for DEIM-LSTM and  DEIM-LSTM-PINN prediction as compared to the benchmark data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Percentage of Data  Used  LSTM  (% L2 error)  LSTM-PINN  (% L2 error)  S  ~100%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0013  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0085  ~59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='87%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0048  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0098  ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='20%  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='661  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4368  h  ~100%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0060  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0233  ~59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='87%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0293  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0654  ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='20%  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='67  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='81  α  ~100%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0051  ~59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='87%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='0214  ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='20%  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2455  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='86  From Table 1 the LSTM network outperforms the LSTM-PINN network despite the introduction of  physics loss in the total loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Figure 19 compares the variation of the mean squared error-based  loss function as shown in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' It appears that the loss associated with the LSTM prediction decays  faster than that of the LSTM-PINN prediction when the complete dataset is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The additional physics-  based loss function increases the residual values, and if the parameters of the optimization algorithms are  not set properly the residual values often get stuck in a local minimum point thereby deteriorating the  prediction accuracy as compared to the LSTM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Therefore, different weightages (w1 and w2  in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6) of the loss functions are considered and their effect on the decay in mean squared error (MSE)  LSTM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  15  LSTM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='8  LSTM-PINN  10  LSTM-PINN  --·Actual  Actual  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='6  5  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='4  a  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='05  10  LSTM  LSTM-PINN  15  Actual  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='1  5  10  15  0  5  10  15  0  5  10  15  t(s)  t(s)  t(s)  (a)  (b)  (c)21  are compared in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The weightage matrix of the loss function w2 are as follows:  \uf05b  \uf05d  \uf05b  \uf05d  \uf05b  \uf05d  2  1 1 1 1 1 1 ,  2  1 1 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='001 ,  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='001 ,  w a  w b  w c  =  =  =  (9)  w1 and w2 are associated with the data-based loss function  LSTM  MSE  and  PINN  MSE  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The  LSTM-PINN algorithms show better accuracy only for the w2c which also indicates the  minimization of the residuals and the MSE values are decaying faster than LSTM approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' With  the w2a and w2b weightages the loss functions do not decay beyond a certain value thereby resulting in  a very poor prediction of the box responses even though the governing equations-based loss functions  are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Figure 19: Loss history for full dataset with LSTM and LSTM-PINN algorithm  Figure 20: Decay of mese values with different weightage of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Conclusion:  A novel non-intrusive reduced order model is introduced for the prediction of the temporal variation of  the surge, heave and pitching dynamics of a freely floating box on the wavy surface of a water wave  maker basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' A dimensionality reduction approach DEIM is coupled with a LSTM-based deep learning  approach for the reduction of the computational training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The neural network parameters are varied  LSTM-PINN  LSTM  100  mse  10-2  0  50  100  150  200  250  Epochsw2a  w2b  100  w2c  Only Data  e  mse  10  5  0  100  200  300  400  500  Epochs22  to assess their influence on the prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' It is noticed that there is a constant phase shift of the  predicted pitch motion from the benchmark result, and it is expected that introduction of a mixed  harmonic paddle excitation will learn the inherent nonlinearity present in the hydrodynamics and the  prediction accuracy can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' Finally, the governing rigid body dynamics equation is coupled  with the LSTM network in the discrete form to create the LSTM-PINN network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The addition of the  physics-based loss function increases the magnitude of the residual of the LSTM network, resulting in a  tendency for the loos function to get stuck at a local minimum frequently if the optimization algorithm  in the network is not tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' In the current work, the LSTM-PINN network is applied to the simple  structural dynamical system which can be very well learned by the LSTM network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' The introduction of  the physics-based governing equation in the loss function did not make any significant improvement in  the prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' On, the other hand, the use of a larger dataset, and the introduction of the physics-  based loss function appear to deteriorate the performance by increasing the magnitude of the residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  The potential of the LSTM-PINN will be realized if the LSTM and LSTM-PINN network is also be  applied to the SPH equations to form an LSTM/LSTM-PINN network mapping between spatiotemporal  coordinates and unsteady flow variables such as velocities and pressure which the authors will address  and also explore the coupling of the networks for fluid flow and structural dynamics as a possible set of  digital twins for box dynamics in their future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  Acknowledgment: The authors acknowledge the efforts of Ouyang Zhenyu and Tang Xiaoyang for their  prompt generation of computational data sets using DualSPHysics which serves as the ground truth  (actual) data and training data for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  References:  [1] Carlberg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', Amsallem, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', Avery, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', Zahr, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' and Farhat, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', “The GNAT Nonlinear Model  Reduction Method and its Application to Fluid Dynamics Problems”, AIAA 2011-3112 in  Proceedings  of 6th AIAA Theoretical Fluid Mechanics Conference, 27 - 30 June 2011, Honolulu, Hawaii, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  [2] Schmidhuber, J.” Deep Learning in Neural Networks: An Overview”, Neural Netw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 61, 85–117  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content='  [3] Maulik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', San, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', Rasheed, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=', Vedula, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=',” Subgrid Modelling for Two-Dimensional Turbulence  using Neural Networks”, Journal of Fluid Mechanics, 858(1), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
+page_content=' 122-144, Jan 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFKT4oBgHgl3EQfgy6A/content/2301.11835v1.pdf'}
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+ZU-TH 01/23, TIF-UNIMI-2023-1
+Determination of the W-boson mass at hadron colliders
+Luca Rottoli,1, ∗ Paolo Torrielli,2, † and Alessandro Vicini3, ‡
+1Physik Institut, Universit¨at Z¨urich, CH-8057 Z¨urich, Switzerland
+2Dipartimento di Fisica, Universit`a di Torino and INFN, Sezione di Torino, I-10125 Torino, Italy
+3Dipartimento di Fisica, Universit`a di Milano and INFN, Sezione di Milano, I-20133 Milano, Italy
+We introduce an observable relevant for the determination of the W-boson mass mW at hadron
+colliders. This observable is defined as an asymmetry around the jacobian peak of the charged-
+lepton transverse-momentum distribution in the charged-current Drell-Yan process. We discuss the
+observable’s theoretical prediction, presenting results at different orders in QCD, and showing its
+perturbative stability. Its definition as a single scalar number and its linear sensitivity to mW al-
+low a clean extraction of the latter and a straightforward discussion of the associated theoretical
+systematics: a determination of mW with a perturbative QCD uncertainty at the ±5 MeV level is
+viable, with the advantage of solely relying on charged-current Drell-Yan information. The observ-
+able displays desirable properties also from the experimental viewpoint, especially for the unfolding
+of detector effects. We show that, with a conservative estimate of systematic errors, it can lead to
+an experimental determination of mW at the level of ±15 MeV at the LHC.
+Introduction.
+The experimental determination of the
+W-boson mass mW [1–4] plays a central role in the pro-
+gramme of precision tests of the Standard Model (SM)
+at hadron colliders. A potential discrepancy between the
+measured value and precise mW predictions [5, 6] within
+the SM may immediately hint at the presence of New-
+Physics effects, as comprehensively discussed in the con-
+text of global fits [7, 8] of electroweak (EW) precision
+observables.
+At hadron colliders, the value of mW is primarily in-
+ferred from the analysis of the charged-current Drell-Yan
+(CCDY) process. Of particular relevance are the proper-
+ties of final-state kinematical distributions defined in the
+transverse plane with respect to the collision axis, such as
+the charged-lepton transverse momentum pℓ
+⊥, the lepton-
+pair transverse mass M ℓν
+⊥ and transverse momentum pℓν
+⊥ ,
+and the missing transverse energy ET [1–4].
+Experimental analyses aiming at the measurement of
+mW typically employ a QCD modelling of CCDY based
+on parton-shower Monte Carlo (MC) event generators,
+whose parameters are tuned on high-precision neutral-
+current Drell-Yan (NCDY) measurements, chiefly the
+lepton-pair transverse momentum pℓ+ℓ−
+⊥
+. A data-driven
+tuning step is in general necessary, as a standalone pre-
+diction of CCDY with the relatively low accuracy pro-
+vided by MC simulations typically leads to an insufficient
+description of data. Tuned MC predictions are then used
+to prepare templates of the relevant transverse kinemat-
+ical distributions with different mW hypotheses. Theo-
+retical templates are subsequently compared with CCDY
+experimental data, and a χ2 analysis is performed to de-
+termine the preferred value for mW .
+Such a significant dependence of the χ2-based ap-
+proach on the tuning to NCDY experimental data poses
+however some conceptual issues for mW determination.
+On the one hand, it makes it subtle to interpret the
+extracted mW as the SM-lagrangian parameter, as the
+fit procedure heavily relies on phenomenological models
+rather than on first-principle SM predictions. In turn,
+this exposes the procedure to the risk of hiding New-
+Physics effects in the fit of model parameters. On the
+other hand, and even more severely, it hinders the possi-
+bility to perform meaningful studies of the perturbative
+uncertainty associated with the theoretical prediction:
+even an MC tool with arbitrarily low formal accuracy can
+indeed yield an excellent description of data, provided
+it grants sufficient flexibility for tuning. This approach
+essentially makes no use of the high-quality theoretical
+understanding of NCDY and CCDY lepton-pair produc-
+tion [9], which in recent years has witnessed a substantial
+progress in the description of fixed-order [10–24] and all-
+order [25–36] QCD effects, as well as in the evaluation of
+EW [37–46] and mixed QCD-EW [47–65] corrections.
+Theoretical systematics in the data-driven procedure
+are simply assessed by quantifying to what extent the ex-
+perimental input from pℓ+ℓ−
+⊥
+may be applied to pℓν
+⊥ , given
+the theoretical knowledge of the two distributions [66]:
+this might underestimate uncertainties, as it assumes
+that the procedure works equally well for all observables
+used for mW extraction. The impact of modelling uncer-
+tainties on mW determination has been discussed con-
+sidering the role of parton distribution functions (PDFs)
+[67–74], of non-perturbative contributions to transverse
+spectra [75], as well as of EW and of leading QCD-EW
+corrections [52, 64, 76]; all of these studies assumed the
+existence of an underlying data-driven QCD model as
+the backbone of the description, but did not quote any
+uncertainty on the model itself.
+In this letter we present an alternative strategy to de-
+termine the value of mW which fully exploits the the-
+oretical progress in the description of Drell-Yan lepton-
+pair production. We introduce a new observable based
+on the charged-lepton transverse-momentum distribution
+in CCDY, defined as an asymmetry around its jacobian
+peak at mW /2. On the one hand, its clean definition in
+terms of calculable fiducial rates allows to directly inter-
+arXiv:2301.04059v1  [hep-ph]  10 Jan 2023
+
+2
+pret the extracted mW as the fundamental SM parame-
+ter; on the other hand, the observable displays excellent
+perturbative convergence, which enables a robust study
+of the associated perturbative-QCD (pQCD) uncertain-
+ties, and its theoretical description is systematically im-
+provable by adding subleading QCD and EW effects. The
+simple dependence of the observable upon mW in turn al-
+lows a plain study of the impact of non-perturbative QCD
+(npQCD) effects, as well as a consistent propagation of
+their uncertainties in the prediction.
+Lepton transverse momentum and sensitivity to mW .
+The modelling of pℓ
+⊥ in CCDY requires a precise descrip-
+tion of the QCD contributions to the transverse and lon-
+gitudinal degrees of freedom of the final state [77]. At
+leading order (LO) the charged lepton and the neutrino
+are back-to-back, pℓν
+⊥ = 0, thus, neglecting lepton masses
+and the W-boson decay width ΓW , the pℓ
+⊥ distribution
+has a sharp kinematical endpoint at pℓ
+⊥ = mW /2, which
+is the origin of its sensitivity to the W-boson mass (see
+also [78, 79]). Beyond LO in QCD, the region around the
+endpoint develops a sensitivity to soft radiation, which in
+turn generates an integrable singularity [80] in the fixed-
+order differential pℓ
+⊥ spectrum. The all-order treatment
+of soft and collinear initial-state QCD radiation, achieved
+by a resummation of enhanced logarithms log(pℓν
+⊥ /mW ),
+is therefore a central ingredient for a reliable descrip-
+tion of pℓ
+⊥. Such a resummation nowadays reaches next-
+to-next-to-next-to-leading-logarithmic (N3LL) accuracy,
+matched with the next-to-next-to-leading-order (NNLO)
+predictions for the transverse-momentum spectrum [27].
+In the following, we consider the pℓ
+⊥ distribution at the
+Large Hadron Collider (LHC) with centre-of-mass energy
+√
+S = 13 TeV and acceptance cuts pℓ
+⊥ > 20 GeV, M ℓν
+⊥ >
+27 GeV, |ηℓ| < 2.5, 66 GeV < M ℓν < 116 GeV (ηℓ and
+M ℓν being the charged-lepton rapidity and the lepton-
+pair invariant mass, respectively), using the central
+replica of the NNPDF4.0 NNLO proton PDF set [81] with
+strong coupling constant αs(mZ) = 0.118 through the
+LHAPDF interface [82]. We give predictions for three dif-
+ferent QCD approximations, NLO+NLL, NNLO+NNLL
+and NNLO+N3LL [83], using the RadISH [31, 84–86] code
+for pℓν
+⊥ resummation, with a fixed-order prediction pro-
+vided by MCFM [87]. We match the two results using the
+qT -subtraction formalism [88], with a technical slicing
+cutoff qcut
+T
+= 0.81 GeV in the MCFM calculation. Linear
+fiducial power corrections are included in the RadISH pre-
+diction through transverse recoil [28, 89] using the pre-
+scription described in [90, 91].
+We consider 21 values
+of mW between 80.329 GeV and 80.429 GeV, in steps
+of 5 MeV. Renormalisation, factorisation and resumma-
+tion scales are chosen as µR,F = ξR,F
+�
+(M ℓν)2 + (pℓν
+⊥ )2,
+and µQ = ξQ M ℓν, respectively. We estimate pQCD un-
+certainties by varying ξR and ξF independently in the
+range (1/2, 1, 2), excluding ξR,F /ξF,R = 4, while keeping
+ξQ = 1/2 (7 variations). In addition, we consider the 2
+34
+36
+38
+40
+42
+44
+pℓ
+⊥ [GeV]
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+dσ/dpℓ
+⊥ [nb /GeV]
+pℓν
+⊥ < 15 GeV
+mW = 80.379 GeV
+NLO+NLL
+NNLO+NNLL
+NNLO+N3LL
+34
+36
+38
+40
+42
+44
+pℓ
+⊥ [GeV]
+0.996
+0.998
+1.000
+1.002
+1.004
+1.006
+1.008
+1.010
+R
+pℓν
+⊥ < 15 GeV
+R = dσ(mW =80.399GeV)
+dpℓ
+⊥
+/ dσ(mW =80.379GeV)
+dpℓ
+⊥
+NLO+NLL
+NNLO+NNLL
+NNLO+N3LL
+Figure 1.
+Upper panel:
+charged-lepton transverse-
+momentum distribution in CCDY, computed with different
+QCD approximations and reference mW = 80.379 GeV. Lower
+panel: ratio of pℓ
+⊥ distributions computed with two mW val-
+ues differing by 20 MeV.
+variations of ξQ in (1/4, 1) at central values ξR = ξF = 1,
+thereby obtaining a total envelope of 9 variations.
+The upper panel of Figure 1 displays the perturbative
+convergence of the pℓ
+⊥ distribution, for a given value of
+mW = 80.379 GeV: one can notice how the inclusion
+of higher-order pQCD effects in resummed predictions
+translates into a significant reduction of theoretical sys-
+tematics. The lower panel of Figure 1 shows the impact
+on the pℓ
+⊥ distribution of a 20-MeV shift of the reference
+mW value. As evinced by the plot, such a shift induces
+a shape distortion at the 0.5%-level around the jacobian
+peak, an effect which is clearly resolvable beyond the the-
+oretical uncertainty.
+We also note that, starting from
+
+3
+a baseline featuring all-order QCD radiation, the effect
+of the mW shift is remarkably independent of the QCD
+perturbative order and scale choice, as a consequence of
+the factorisation of initial-state QCD radiation from W-
+boson production and decay.
+The sensitivity to mW of the N bins σi of the pℓ
+⊥
+distribution can be quantified by means of the covari-
+ance matrix with respect to mW variations, (CmW )ij ≡
+⟨σi σj⟩−⟨σi⟩ ⟨σj⟩, where the ⟨ ⟩ symbol indicates an arith-
+metic average over the different available mW options
+(21 in our case). The N eigenvectors of CmW represent
+the linear combinations of pℓ
+⊥ bins that transform inde-
+pendently under mW variations, and the corresponding
+eigenvalues in turn express the sensitivity of such combi-
+nations to mW .
+For pℓ
+⊥ bins around the jacobian peak, such as those
+contributing to Figure 1, there is a strong hierarchy
+among the CmW eigenvalues, with the first one being
+more than an order of magnitude larger than all others.
+Such a feature, robust against variations of the consid-
+ered pℓ
+⊥ range, suggests the first linear combination to
+be representative of the behaviour of the whole pℓ
+⊥ dis-
+tribution under mW variations. In our simulation setup,
+the coefficients of this linear combination are all positive
+(negative) for bins at pℓ
+⊥ < 37 GeV (pℓ
+⊥ > 37 GeV), irre-
+spectively of the employed QCD approximation or of the
+pℓ
+⊥ range. The pattern of signs is in turn indicative of
+mW sensitivity: the value of 37 GeV is directly related
+to the position of the jacobian peak at mW /2, after con-
+sidering the smearing due to all-order QCD radiation as
+well as to the W-boson decay width (we set ΓW = 2.084
+GeV). Inspection of the lower panel of Figure 1 confirms
+the value pℓ
+⊥ = 37 GeV as separating the spectrum into
+two regions, respectively with (pℓ
+⊥ > 37 GeV) and with-
+out (pℓ
+⊥ < 37 GeV) sensitivity to mW .
+Jacobian asymmetry and mW determination.
+Based
+on the previous considerations, we introduce a pℓ
+⊥ range
+[pℓ,min
+⊥
+, pℓ,max
+⊥
+] which includes the jacobian peak, as well
+as an intermediate value pℓ,min
+⊥
+< pℓ,mid
+⊥
+< pℓ,max
+⊥
+, and
+define two fiducial cross sections,
+Lpℓ
+⊥ ≡
+� pℓ,mid
+⊥
+pℓ,min
+⊥
+dpℓ
+⊥
+dσ
+dpℓ
+⊥
+,
+Upℓ
+⊥ ≡
+� pℓ,max
+⊥
+pℓ,mid
+⊥
+dpℓ
+⊥
+dσ
+dpℓ
+⊥
+,
+(1)
+together with their asymmetry
+Apℓ
+⊥(pℓ,min
+⊥
+, pℓ,mid
+⊥
+, pℓ,max
+⊥
+) ≡
+Lpℓ
+⊥ − Upℓ
+⊥
+Lpℓ
+⊥ + Upℓ
+⊥
+.
+(2)
+In Figure 2 we plot Apℓ
+⊥(32 GeV, 37 GeV, 47 GeV) as a
+function of mW , with different QCD approximations.
+The uncertainty bands computed (with the same scale
+choice for Lpℓ
+⊥ and Upℓ
+⊥) at the various perturbative or-
+ders exhibit an excellent convergence pattern, and in all
+cases encompass predictions at the next orders. Given
+this behaviour, we consider the size of the NNLO+N3LL
+uncertainty band as a good estimator of the uncertainty
+due to missing pQCD higher-order effects. We have stud-
+ied the dependence of this pattern on pℓ,mid
+⊥
+and found
+that for pℓ,mid
+⊥
+≳ 38 GeV, approaching the effective end-
+point of the fixed-order distribution, the perturbative
+convergence slightly deteriorates; on the contrary, choices
+with pℓ,mid
+⊥
+< 37 GeV exhibit a better stability, at the
+price of a reduced sensitivity to mW . We then choose
+pℓ,mid
+⊥
+= 37 GeV as our default, as an excellent compro-
+mise between stability and sensitivity. The convergence
+behaviour is instead fairly stable against variations of
+pℓ,min
+⊥
+and pℓ,max
+⊥
+.
+We remark in Figure 2 that Apℓ
+⊥ has a clear linear
+sensitivity to mW , directly stemming from the linear
+mW -dependence of the jacobian-peak position.
+More-
+over, its slope is extremely stable irrespectively of the
+QCD approximation and the scale choice, and just de-
+pends on the defining pℓ
+⊥ range.
+These features make
+Apℓ
+⊥ an excellent observable to determine mW and to ro-
+bustly quantify the associated uncertainties. For a given
+choice of [pℓ,min
+⊥
+, pℓ,mid
+⊥
+, pℓ,max
+⊥
+], the experimental value
+of Apℓ
+⊥ can be obtained by simply measuring the fidu-
+cial cross sections Lpℓ
+⊥, Upℓ
+⊥ (i.e. a counting experiment),
+eventually resulting in a single scalar number in which
+systematic uncertainties can be straightforwardly propa-
+gated. The relatively large size of the [pℓ,min
+⊥
+, pℓ,mid
+⊥
+] and
+[pℓ,mid
+⊥
+, pℓ,max
+⊥
+] intervals helps taming the statistical error,
+and is beneficial for the unfolding of detector effects; the
+latter is welcome in view of a combination of the results
+obtained by different experiments [92]. For illustrative
+purposes, in Figure 2 we plot a hypothetical experimen-
+tal measurement for Apℓ
+⊥, with statistical and systematic
+errors realistically propagated.
+From Figure 2 we compare the experimental error band
+with a single theoretical curve (arbitrarily chosen, as all
+have the same slope): the intercepts of the curve with
+the edges of the band identify an mW interval that we
+treat as the experimental uncertainty. With a luminosity
+L = 140 fb−1, a relative systematic error of 0.001 in the
+measurement of Lpℓ
+⊥, Upℓ
+⊥, and neglecting experimental
+correlations, we find ∆Apℓ
+⊥ = ± 0.00007 ± 0.0007, which
+translates into an mW uncertainty ∆mstat
+W
++ ∆msyst
+W
+∼
+± 1.3 ± 12.5 MeV. We then take the two edges of the
+scale-variation band at a given perturbative accuracy,
+and use them to estimate the uncertainty on mW due
+to missing pQCD higher orders, by comparison with the
+central experimental result. At NNLO+N3LL we find a
+very competitive ∆mpQCD
+W
+∼ ±6 MeV.
+In Figure 3 we quantify the pQCD uncertainty on mW
+as just outlined, considering different perturbative or-
+ders and choices of [pℓ,min
+⊥
+, pℓ,mid
+⊥
+, pℓ,max
+⊥
+]. For the sake
+of definiteness and consistency, in each setup we em-
+ploy the central-scale NNLO+N3LL Apℓ
+⊥ value com-
+puted with mW
+= 80.379 GeV as our experimental
+
+4
+80.34
+80.36
+80.38
+80.40
+80.42
+mW [GeV]
+−0.1750
+−0.1725
+−0.1700
+−0.1675
+−0.1650
+−0.1625
+−0.1600
+−0.1575
+−0.1550
+Apℓ
+⊥
+pℓν
+⊥ < 15 GeV
+pℓ,min
+⊥
+=32 GeV, pℓ,mid
+⊥
+=37 GeV, pℓ,max
+⊥
+=47 GeV
+NLO+NLL
+NNLO+NNLL
+NNLO+N3LL
+pseudo-experiment syst+stat
+Figure 2.
+The asymmetry Apℓ
+⊥ as a function of mW , in
+different QCD approximations.
+250
+300
+350
+400
+450
+500
+550
+mW − 80000 [MeV]
+[pℓ,min
+⊥
+, pℓ,mid
+⊥
+, pℓ,max
+⊥
+]
+[32, 38, 50]
+[32, 37, 50]
+[32, 36, 50]
+[32, 38, 47]
+[32, 37, 47]
+[32, 36, 47]
+[30, 38, 47]
+[30, 37, 47]
+[30, 36, 47]
+NLO+NLL
+NNLO+NNLL
+NNLO+N3LL
+Figure 3.
+The range of mW values obtained comparing the
+band of theoretical predictions at different orders in pQCD,
+with the central experimental value of Apℓ
+⊥. Different choices
+of [pℓ,min
+⊥
+, pℓ,mid
+⊥
+, pℓ,max
+⊥
+] are considered.
+proxy.
+The pattern of convergence against variations
+of [pℓ,min
+⊥
+, pℓ,mid
+⊥
+, pℓ,max
+⊥
+] largely reflects our considerations
+below Eq. (2). We also remark the need of N3LL resum-
+mation for a sizeable reduction of theoretical uncertainty,
+and a precise mW determination.
+Discussion.
+The asymmetry Apℓ
+⊥ defined in Eq. (2)
+offers some interesting features, compared to a template
+fit of the whole pℓ
+⊥ distribution. First, it is defined in
+terms of inclusive rates integrated over relatively wide
+phase-space regions: this allows to obtain a fairly stable
+QCD prediction on the theoretical side, and an excellent
+statistical precision and the possibility to unfold detec-
+tor effects on the experimental side. Second, the asym-
+metry enables a determination of mW based on CCDY
+data which, upon including state-of-the-art pQCD pre-
+dictions, is not dominated by the tuning of model param-
+eters on NCDY measurements. Third, through its linear
+dependence on mW , the asymmetry offers the possibil-
+ity to cleanly disentangle the impact on mW determina-
+tion of all effects contributing to the pℓ
+⊥ spectrum. On
+top of the pQCD predictions scrutinised in this paper,
+which constitute a robust starting point, it is conceptu-
+ally straightforward to include final-state QED radiation,
+as well as EW and mixed QCD-EW perturbative correc-
+tions. All of these additional effects induce modifications
+to Apℓ
+⊥ that can be separately assessed and systemati-
+cally refined. Effects of npQCD origin, relevant for a fully
+realistic description, can also be included as a separate
+component to the prediction of Apℓ
+⊥, but as opposed to
+template-fitting, their inclusion is not instrumental for
+the whole mW -extraction procedure.
+As they involve
+initial-state QCD radiation, their inclusion is expected
+to simply induce a vertical offset to Apℓ
+⊥ without altering
+its slope, i.e. its sensitivity to mW . This offset in turn
+yields a shift of the preferred mW value, which can be
+easily estimated thanks to the linear mW -dependence of
+Apℓ
+⊥. The underlying npQCD model can be constrained
+via the simultaneous analysis of more observables, other
+than Apℓ
+⊥: the improvement in the accuracy of this model
+is thus a problem fully decoupled from mW determina-
+tion.
+To illustrate how npQCD contributions can be consis-
+tently studied through the asymmetry Apℓ
+⊥, we consider
+effects on mW coming from collinear proton PDFs and
+from the modelling of an intrinsic transverse momentum
+k⊥ of partons in the proton (further details on the results
+of this study can be found in the Appendix). The un-
+certainty on collinear PDFs enters transverse kinematics
+indirectly, through the finite lepton-rapidity acceptance,
+while intrinsic k⊥ directly shifts leptonic momenta.
+As for the effect of collinear PDFs, predictions for
+Apℓ
+⊥(32 GeV, 37 GeV, 47 GeV) obtained using all 100
+replicas of the NNPDF4.0 set yield a PDF uncer-
+tainty of ±11.5 MeV. More conservatively, we also
+consider the central replicas of the CT18NNLO [93],
+MSHT20nnlo [94], and NNPDF3.1 [95] PDF sets. The
+corresponding spread of mW values is of ∼ 30 MeV. A
+reduction of PDF uncertainty can be achieved by profil-
+ing PDF replicas through the simultaneous inclusion of
+additional information, such as data in different rapid-
+ity regions [68, 69], all bins of the pℓ
+⊥ distribution [73],
+different W charges at the LHC [2].
+Turning to the intrinsic k⊥ of partons in the proton,
+it can be precisely modelled studying the pℓ+ℓ−
+⊥
+distribu-
+
+5
+tion in NCDY. Assuming its universality [96], it can be
+applied to the CCDY simulation, inducing a shift in mW .
+We have investigated the interplay between the scale un-
+certainty of the perturbative NCDY SM description and
+the size of the npQCD component extracted from NCDY
+data (using the central NNLO+N3LL NCDY prediction
+as pseudo-data, hence actually extracting a “pseudo-
+npQCD” contribution).
+To this goal, we have deter-
+mined one pseudo-npQCD contribution per scale choice,
+included it in the CCDY simulation, and assessed its im-
+pact on mW determination. The point which emerges
+from this analysis is that, even if the NCDY pseudo-data
+are a unique set of numbers, the propagation of their in-
+formation to CCDY depends on the underlying pQCD
+approximation, and the outcome is not unique.
+The
+CCDY results, improved with the pseudo-npQCD con-
+tribution, are spread in a range compatible with, or even
+larger than the scale uncertainty of the NNLO+NNLL
+calculation.
+This result stresses the importance of us-
+ing state-of-the-art pQCD results in these high-precision
+studies.
+Conclusions.
+We have presented a new observable,
+Apℓ
+⊥,
+sensitive to the value of the W-boson mass
+mW , with promising experimental properties and robust
+pQCD convergence. Its linear dependence on mW allows
+to systematically disentangle the impact of each contribu-
+tion, perturbative or not, affecting the determination of
+mW and to estimate the associated uncertainty, a crucial
+feature for the comparison of data with SM predictions.
+The study of Apℓ
+⊥ highlights the importance of state-of-
+the-art predictions to reduce the pQCD uncertainty on
+mW down to the ±5 MeV level at the LHC. We argue
+that, using Apℓ
+⊥, an experimental error on mW at the
+±15 MeV level is achievable already with Run-2 data;
+moreover, the possibility is given to unfold the data to
+particle level, easing the combination of results from dif-
+ferent experiments. Given these properties, we hope that
+this observable will be considered for an independent de-
+termination of mW from available CCDY data.
+Acknowledgements.
+We
+thank
+Paolo
+Azzurri,
+Emanuele Bagnaschi, Luca Buonocore, Massimiliano
+Grazzini,
+Michelangelo
+Mangano,
+Pier
+Monni
+and
+Stefano Pozzorini for useful discussions.
+We thank
+Tobias Neumann for support in the use of MCFM. LR
+is supported by the Swiss National Science Founda-
+tion contract PZ00P2 201878.
+PT has been partially
+supported by the Italian Ministry of University and
+Research (MUR) through grant PRIN 20172LNEEZ
+and
+by
+Compagnia
+di
+San
+Paolo
+through
+grant
+TORP S1921 EX-POST 21 01. AV is supported by the
+Italian MUR through grant PRIN201719AVICI 01. LR
+and AV thank MIAPbP for hospitality and acknowledge
+the Deutsche Forschungsgemeinschaft under Germany’s
+Excellence Strategy – EXC-2094 – 390783311.
+250
+300
+350
+400
+450
+500
+550
+mW − 80000 [MeV]
+[pℓ,min
+⊥
+, pℓ,mid
+⊥
+, pℓ,max
+⊥
+]
+[32, 38, 50]
+[32, 37, 50]
+[32, 36, 50]
+[32, 38, 47]
+[32, 37, 47]
+[32, 36, 47]
+[30, 38, 47]
+[30, 37, 47]
+[30, 36, 47]
+NNPDF4.0
+CT18NNLO
+MSHTnnlo
+NNPDF3.1
+Figure 4.
+Same as Fig. 3, now comparing the range of mW
+values obtained with different PDF sets.
+APPENDIX
+In this Appendix we detail the study described in the
+main text about the impact of non-perturbative effects
+on mW determination.
+The discussion focuses on the
+uncertainty due to collinear PDFs, and on the modelling
+of an intrinsic k⊥ of partons in the proton.
+Proton-PDF
+uncertainties.
+Concerning
+the
+ef-
+fect
+of
+different
+collinear
+PDFs,
+predictions
+for
+Apℓ
+⊥(32 GeV, 37 GeV, 47 GeV) obtained with the 100
+replicas of the NNPDF4.0 set yield a bundle of parallel
+straight lines, as expected due to the factorisation of
+QCD effects from W-boson production and decay. The
+intercepts with the experimental Apℓ
+⊥ value yield a
+distribution of 100 mW values. We compute mean value
+and standard deviation of this distribution, obtaining
+at NLO+NLL with central scales a spread in mW of
+±11.5 MeV. We also consider the central replicas of the
+CT18NNLO [93], MSHT20nnlo [94], and NNPDF3.1 [95]
+PDF sets.
+The spread induced on mW , using the
+central-scale NNLO+N3LL prediction, is of ∼ 30 MeV.
+We present in Figure 4 the results for different setups.
+Modelling of the parton intrinsic k⊥.
+With the fol-
+lowing exercise, we schematically describe the encod-
+ing of information present in NCDY data and absent
+from a purely perturbative description of the process.
+We then consider the usage of such an information in
+the simulation of CCDY, and eventually its impact on
+mW determination.
+In particular, the pQCD stability
+of Apℓ
+⊥ allows to study the role of scale variations in
+porting these effects from NCDY to CCDY. We simu-
+
+6
+late both processes at NNLO+N3LL QCD, with ξR =
+ξF = 2 ξQ = 1 and take the results as a proxy for ex-
+perimental data (we dub them “pseudo-data”, see also
+[36]). We assume to have available an event generator
+with NNLO+NNLL pQCD accuracy only, and compute
+the NCDY pℓ+ℓ−
+⊥
+distribution with different scale choices.
+The ratio of NNLO+NNLL predictions with pseudo-data
+defines a reweighing factor, as a function of pℓ+ℓ−
+⊥
+, en-
+coding the missing pQCD higher orders (with real data
+as opposed to pseudo-data it would encode npQCD ef-
+fects as well).
+We compute one such reweighing fac-
+tor per pQCD scale choice in NCDY. We then use the
+NNLO+NNLL generator to simulate the CCDY process
+with scale variations, and reweigh all events in each vari-
+ation according to their pℓν
+⊥ value, using the correspond-
+ing factor determined in NCDY. Since by construction
+the reweighed NCDY NNLO+NNLL curves would ex-
+actly match NCDY pseudo-data, one expects to a large
+extent the same to happen with CCDY pseudo-data and
+reweighed NNLO+NNLL CCDY distributions. We ob-
+serve instead that the reweighed distributions do not ex-
+actly reproduce CCDY pseudo-data, the discrepancy be-
+ing comparable with, or larger than the NNLO+NNLL
+scale-uncertainty band, i.e. ∆mW
+∼ ±27 MeV from
+the study of Apℓ
+⊥(32 GeV, 37 GeV, 47 GeV). We conclude
+that the procedure to model npQCD effects due to an
+intrinsic k⊥ is intertwined with the underlying pQCD
+formulation.
+We thus expect that the same approach,
+using a NNLO+N3LL-accurate event generator and the
+real data as a target, would lead to a smaller final spread
+in Apℓ
+⊥, providing a handle for a robust assessment of the
+impact of npQCD effects on the determination of mW .
+We present in Figure 5 the results for different setups.
+∗ luca.rottoli@physik.uzh.ch
+† paolo.torrielli@to.infn.it
+‡ alessandro.vicini@mi.infn.it
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diff --git a/RtE2T4oBgHgl3EQfsggH/content/tmp_files/load_file.txt b/RtE2T4oBgHgl3EQfsggH/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e8c44741acb41cea2f637e292a9363244755acc1
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@@ -0,0 +1,1044 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf,len=1043
+page_content='ZU-TH 01/23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' TIF-UNIMI-2023-1  Determination of the W-boson mass at hadron colliders  Luca Rottoli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' ∗ Paolo Torrielli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' † and Alessandro Vicini3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' ‡  1Physik Institut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Universit¨at Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' CH-8057 Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Switzerland  2Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Universit`a di Torino and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Sezione di Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' I-10125 Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Italy  3Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Universit`a di Milano and INFN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Sezione di Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' I-20133 Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Italy  We introduce an observable relevant for the determination of the W-boson mass mW at hadron  colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' This observable is defined as an asymmetry around the jacobian peak of the charged-  lepton transverse-momentum distribution in the charged-current Drell-Yan process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We discuss the  observable’s theoretical prediction, presenting results at different orders in QCD, and showing its  perturbative stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Its definition as a single scalar number and its linear sensitivity to mW al-  low a clean extraction of the latter and a straightforward discussion of the associated theoretical  systematics: a determination of mW with a perturbative QCD uncertainty at the ±5 MeV level is  viable, with the advantage of solely relying on charged-current Drell-Yan information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The observ-  able displays desirable properties also from the experimental viewpoint, especially for the unfolding  of detector effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We show that, with a conservative estimate of systematic errors, it can lead to  an experimental determination of mW at the level of ±15 MeV at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The experimental determination of the  W-boson mass mW [1–4] plays a central role in the pro-  gramme of precision tests of the Standard Model (SM)  at hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' A potential discrepancy between the  measured value and precise mW predictions [5, 6] within  the SM may immediately hint at the presence of New-  Physics effects, as comprehensively discussed in the con-  text of global fits [7, 8] of electroweak (EW) precision  observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  At hadron colliders, the value of mW is primarily in-  ferred from the analysis of the charged-current Drell-Yan  (CCDY) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Of particular relevance are the proper-  ties of final-state kinematical distributions defined in the  transverse plane with respect to the collision axis, such as  the charged-lepton transverse momentum pℓ  ⊥, the lepton-  pair transverse mass M ℓν  ⊥ and transverse momentum pℓν  ⊥ ,  and the missing transverse energy ET [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Experimental analyses aiming at the measurement of  mW typically employ a QCD modelling of CCDY based  on parton-shower Monte Carlo (MC) event generators,  whose parameters are tuned on high-precision neutral-  current Drell-Yan (NCDY) measurements, chiefly the  lepton-pair transverse momentum pℓ+ℓ−  ⊥  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' A data-driven  tuning step is in general necessary, as a standalone pre-  diction of CCDY with the relatively low accuracy pro-  vided by MC simulations typically leads to an insufficient  description of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Tuned MC predictions are then used  to prepare templates of the relevant transverse kinemat-  ical distributions with different mW hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Theo-  retical templates are subsequently compared with CCDY  experimental data, and a χ2 analysis is performed to de-  termine the preferred value for mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Such a significant dependence of the χ2-based ap-  proach on the tuning to NCDY experimental data poses  however some conceptual issues for mW determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  On the one hand, it makes it subtle to interpret the  extracted mW as the SM-lagrangian parameter, as the  fit procedure heavily relies on phenomenological models  rather than on first-principle SM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' In turn,  this exposes the procedure to the risk of hiding New-  Physics effects in the fit of model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' On the  other hand, and even more severely, it hinders the possi-  bility to perform meaningful studies of the perturbative  uncertainty associated with the theoretical prediction:  even an MC tool with arbitrarily low formal accuracy can  indeed yield an excellent description of data, provided  it grants sufficient flexibility for tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' This approach  essentially makes no use of the high-quality theoretical  understanding of NCDY and CCDY lepton-pair produc-  tion [9], which in recent years has witnessed a substantial  progress in the description of fixed-order [10–24] and all-  order [25–36] QCD effects, as well as in the evaluation of  EW [37–46] and mixed QCD-EW [47–65] corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Theoretical systematics in the data-driven procedure  are simply assessed by quantifying to what extent the ex-  perimental input from pℓ+ℓ−  ⊥  may be applied to pℓν  ⊥ , given  the theoretical knowledge of the two distributions [66]:  this might underestimate uncertainties, as it assumes  that the procedure works equally well for all observables  used for mW extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The impact of modelling uncer-  tainties on mW determination has been discussed con-  sidering the role of parton distribution functions (PDFs)  [67–74], of non-perturbative contributions to transverse  spectra [75], as well as of EW and of leading QCD-EW  corrections [52, 64, 76];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' all of these studies assumed the  existence of an underlying data-driven QCD model as  the backbone of the description, but did not quote any  uncertainty on the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  In this letter we present an alternative strategy to de-  termine the value of mW which fully exploits the the-  oretical progress in the description of Drell-Yan lepton-  pair production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We introduce a new observable based  on the charged-lepton transverse-momentum distribution  in CCDY, defined as an asymmetry around its jacobian  peak at mW /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' On the one hand, its clean definition in  terms of calculable fiducial rates allows to directly inter-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='04059v1  [hep-ph]  10 Jan 2023  2  pret the extracted mW as the fundamental SM parame-  ter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' on the other hand, the observable displays excellent  perturbative convergence, which enables a robust study  of the associated perturbative-QCD (pQCD) uncertain-  ties, and its theoretical description is systematically im-  provable by adding subleading QCD and EW effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The  simple dependence of the observable upon mW in turn al-  lows a plain study of the impact of non-perturbative QCD  (npQCD) effects, as well as a consistent propagation of  their uncertainties in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Lepton transverse momentum and sensitivity to mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The modelling of pℓ  ⊥ in CCDY requires a precise descrip-  tion of the QCD contributions to the transverse and lon-  gitudinal degrees of freedom of the final state [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' At  leading order (LO) the charged lepton and the neutrino  are back-to-back, pℓν  ⊥ = 0, thus, neglecting lepton masses  and the W-boson decay width ΓW , the pℓ  ⊥ distribution  has a sharp kinematical endpoint at pℓ  ⊥ = mW /2, which  is the origin of its sensitivity to the W-boson mass (see  also [78, 79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Beyond LO in QCD, the region around the  endpoint develops a sensitivity to soft radiation, which in  turn generates an integrable singularity [80] in the fixed-  order differential pℓ  ⊥ spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The all-order treatment  of soft and collinear initial-state QCD radiation, achieved  by a resummation of enhanced logarithms log(pℓν  ⊥ /mW ),  is therefore a central ingredient for a reliable descrip-  tion of pℓ  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Such a resummation nowadays reaches next-  to-next-to-next-to-leading-logarithmic (N3LL) accuracy,  matched with the next-to-next-to-leading-order (NNLO)  predictions for the transverse-momentum spectrum [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  In the following, we consider the pℓ  ⊥ distribution at the  Large Hadron Collider (LHC) with centre-of-mass energy  √  S = 13 TeV and acceptance cuts pℓ  ⊥ > 20 GeV, M ℓν  ⊥ >  27 GeV, |ηℓ| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='5, 66 GeV < M ℓν < 116 GeV (ηℓ and  M ℓν being the charged-lepton rapidity and the lepton-  pair invariant mass, respectively), using the central  replica of the NNPDF4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='0 NNLO proton PDF set [81] with  strong coupling constant αs(mZ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='118 through the  LHAPDF interface [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We give predictions for three dif-  ferent QCD approximations, NLO+NLL, NNLO+NNLL  and NNLO+N3LL [83], using the RadISH [31, 84–86] code  for pℓν  ⊥ resummation, with a fixed-order prediction pro-  vided by MCFM [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We match the two results using the  qT -subtraction formalism [88], with a technical slicing  cutoff qcut  T  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='81 GeV in the MCFM calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Linear  fiducial power corrections are included in the RadISH pre-  diction through transverse recoil [28, 89] using the pre-  scription described in [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We consider 21 values  of mW between 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='329 GeV and 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='429 GeV, in steps  of 5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Renormalisation, factorisation and resumma-  tion scales are chosen as µR,F = ξR,F  �  (M ℓν)2 + (pℓν  ⊥ )2,  and µQ = ξQ M ℓν, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We estimate pQCD un-  certainties by varying ξR and ξF independently in the  range (1/2, 1, 2), excluding ξR,F /ξF,R = 4, while keeping  ξQ = 1/2 (7 variations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' In addition, we consider the 2  34  36  38  40  42  44  pℓ  ⊥ [GeV]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='25  dσ/dpℓ  ⊥ [nb /GeV]  pℓν  ⊥ < 15 GeV  mW = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='379 GeV  NLO+NLL  NNLO+NNLL  NNLO+N3LL  34  36  38  40  42  44  pℓ  ⊥ [GeV]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='998  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='002  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='004  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='006  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='008  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='010  R  pℓν  ⊥ < 15 GeV  R = dσ(mW =80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='399GeV)  dpℓ  ⊥  / dσ(mW =80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='379GeV)  dpℓ  ⊥  NLO+NLL  NNLO+NNLL  NNLO+N3LL  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Upper panel:  charged-lepton transverse-  momentum distribution in CCDY, computed with different  QCD approximations and reference mW = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='379 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Lower  panel: ratio of pℓ  ⊥ distributions computed with two mW val-  ues differing by 20 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  variations of ξQ in (1/4, 1) at central values ξR = ξF = 1,  thereby obtaining a total envelope of 9 variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The upper panel of Figure 1 displays the perturbative  convergence of the pℓ  ⊥ distribution, for a given value of  mW = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='379 GeV: one can notice how the inclusion  of higher-order pQCD effects in resummed predictions  translates into a significant reduction of theoretical sys-  tematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The lower panel of Figure 1 shows the impact  on the pℓ  ⊥ distribution of a 20-MeV shift of the reference  mW value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' As evinced by the plot, such a shift induces  a shape distortion at the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='5%-level around the jacobian  peak, an effect which is clearly resolvable beyond the the-  oretical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We also note that, starting from  3  a baseline featuring all-order QCD radiation, the effect  of the mW shift is remarkably independent of the QCD  perturbative order and scale choice, as a consequence of  the factorisation of initial-state QCD radiation from W-  boson production and decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The sensitivity to mW of the N bins σi of the pℓ  ⊥  distribution can be quantified by means of the covari-  ance matrix with respect to mW variations, (CmW )ij ≡  ⟨σi σj⟩−⟨σi⟩ ⟨σj⟩, where the ⟨ ⟩ symbol indicates an arith-  metic average over the different available mW options  (21 in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The N eigenvectors of CmW represent  the linear combinations of pℓ  ⊥ bins that transform inde-  pendently under mW variations, and the corresponding  eigenvalues in turn express the sensitivity of such combi-  nations to mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  For pℓ  ⊥ bins around the jacobian peak, such as those  contributing to Figure 1, there is a strong hierarchy  among the CmW eigenvalues, with the first one being  more than an order of magnitude larger than all others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Such a feature, robust against variations of the consid-  ered pℓ  ⊥ range, suggests the first linear combination to  be representative of the behaviour of the whole pℓ  ⊥ dis-  tribution under mW variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' In our simulation setup,  the coefficients of this linear combination are all positive  (negative) for bins at pℓ  ⊥ < 37 GeV (pℓ  ⊥ > 37 GeV), irre-  spectively of the employed QCD approximation or of the  pℓ  ⊥ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The pattern of signs is in turn indicative of  mW sensitivity: the value of 37 GeV is directly related  to the position of the jacobian peak at mW /2, after con-  sidering the smearing due to all-order QCD radiation as  well as to the W-boson decay width (we set ΓW = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='084  GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Inspection of the lower panel of Figure 1 confirms  the value pℓ  ⊥ = 37 GeV as separating the spectrum into  two regions, respectively with (pℓ  ⊥ > 37 GeV) and with-  out (pℓ  ⊥ < 37 GeV) sensitivity to mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Jacobian asymmetry and mW determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Based  on the previous considerations, we introduce a pℓ  ⊥ range  [pℓ,min  ⊥  , pℓ,max  ⊥  ] which includes the jacobian peak, as well  as an intermediate value pℓ,min  ⊥  < pℓ,mid  ⊥  < pℓ,max  ⊥  , and  define two fiducial cross sections,  Lpℓ  ⊥ ≡  � pℓ,mid  ⊥  pℓ,min  ⊥  dpℓ  ⊥  dσ  dpℓ  ⊥  ,  Upℓ  ⊥ ≡  � pℓ,max  ⊥  pℓ,mid  ⊥  dpℓ  ⊥  dσ  dpℓ  ⊥  ,  (1)  together with their asymmetry  Apℓ  ⊥(pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ) ≡  Lpℓ  ⊥ − Upℓ  ⊥  Lpℓ  ⊥ + Upℓ  ⊥  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  (2)  In Figure 2 we plot Apℓ  ⊥(32 GeV, 37 GeV, 47 GeV) as a  function of mW , with different QCD approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The uncertainty bands computed (with the same scale  choice for Lpℓ  ⊥ and Upℓ  ⊥) at the various perturbative or-  ders exhibit an excellent convergence pattern, and in all  cases encompass predictions at the next orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Given  this behaviour, we consider the size of the NNLO+N3LL  uncertainty band as a good estimator of the uncertainty  due to missing pQCD higher-order effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We have stud-  ied the dependence of this pattern on pℓ,mid  ⊥  and found  that for pℓ,mid  ⊥  ≳ 38 GeV, approaching the effective end-  point of the fixed-order distribution, the perturbative  convergence slightly deteriorates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' on the contrary, choices  with pℓ,mid  ⊥  < 37 GeV exhibit a better stability, at the  price of a reduced sensitivity to mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We then choose  pℓ,mid  ⊥  = 37 GeV as our default, as an excellent compro-  mise between stability and sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The convergence  behaviour is instead fairly stable against variations of  pℓ,min  ⊥  and pℓ,max  ⊥  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We remark in Figure 2 that Apℓ  ⊥ has a clear linear  sensitivity to mW , directly stemming from the linear  mW -dependence of the jacobian-peak position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  More-  over, its slope is extremely stable irrespectively of the  QCD approximation and the scale choice, and just de-  pends on the defining pℓ  ⊥ range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  These features make  Apℓ  ⊥ an excellent observable to determine mW and to ro-  bustly quantify the associated uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' For a given  choice of [pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ], the experimental value  of Apℓ  ⊥ can be obtained by simply measuring the fidu-  cial cross sections Lpℓ  ⊥, Upℓ  ⊥ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' a counting experiment),  eventually resulting in a single scalar number in which  systematic uncertainties can be straightforwardly propa-  gated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The relatively large size of the [pℓ,min  ⊥  , pℓ,mid  ⊥  ] and  [pℓ,mid  ⊥  , pℓ,max  ⊥  ] intervals helps taming the statistical error,  and is beneficial for the unfolding of detector effects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' the  latter is welcome in view of a combination of the results  obtained by different experiments [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' For illustrative  purposes, in Figure 2 we plot a hypothetical experimen-  tal measurement for Apℓ  ⊥, with statistical and systematic  errors realistically propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  From Figure 2 we compare the experimental error band  with a single theoretical curve (arbitrarily chosen, as all  have the same slope): the intercepts of the curve with  the edges of the band identify an mW interval that we  treat as the experimental uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' With a luminosity  L = 140 fb−1, a relative systematic error of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='001 in the  measurement of Lpℓ  ⊥, Upℓ  ⊥, and neglecting experimental  correlations, we find ∆Apℓ  ⊥ = ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='00007 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='0007, which  translates into an mW uncertainty ∆mstat  W  + ∆msyst  W  ∼  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='3 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We then take the two edges of the  scale-variation band at a given perturbative accuracy,  and use them to estimate the uncertainty on mW due  to missing pQCD higher orders, by comparison with the  central experimental result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' At NNLO+N3LL we find a  very competitive ∆mpQCD  W  ∼ ±6 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  In Figure 3 we quantify the pQCD uncertainty on mW  as just outlined, considering different perturbative or-  ders and choices of [pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' For the sake  of definiteness and consistency, in each setup we em-  ploy the central-scale NNLO+N3LL Apℓ  ⊥ value com-  puted with mW  = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='379 GeV as our experimental  4  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='34  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='36  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='38  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='40  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='42  mW [GeV]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1750  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1725  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1700  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1675  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1650  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1625  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1600  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1575  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1550  Apℓ  ⊥  pℓν  ⊥ < 15 GeV  pℓ,min  ⊥  =32 GeV, pℓ,mid  ⊥  =37 GeV, pℓ,max  ⊥  =47 GeV  NLO+NLL  NNLO+NNLL  NNLO+N3LL  pseudo-experiment syst+stat  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The asymmetry Apℓ  ⊥ as a function of mW , in  different QCD approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  250  300  350  400  450  500  550  mW − 80000 [MeV]  [pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ]  [32, 38, 50]  [32, 37, 50]  [32, 36, 50]  [32, 38, 47]  [32, 37, 47]  [32, 36, 47]  [30, 38, 47]  [30, 37, 47]  [30, 36, 47]  NLO+NLL  NNLO+NNLL  NNLO+N3LL  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The range of mW values obtained comparing the  band of theoretical predictions at different orders in pQCD,  with the central experimental value of Apℓ  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Different choices  of [pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ] are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The pattern of convergence against variations  of [pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ] largely reflects our considerations  below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We also remark the need of N3LL resum-  mation for a sizeable reduction of theoretical uncertainty,  and a precise mW determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The asymmetry Apℓ  ⊥ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' (2)  offers some interesting features, compared to a template  fit of the whole pℓ  ⊥ distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' First, it is defined in  terms of inclusive rates integrated over relatively wide  phase-space regions: this allows to obtain a fairly stable  QCD prediction on the theoretical side, and an excellent  statistical precision and the possibility to unfold detec-  tor effects on the experimental side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Second, the asym-  metry enables a determination of mW based on CCDY  data which, upon including state-of-the-art pQCD pre-  dictions, is not dominated by the tuning of model param-  eters on NCDY measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Third, through its linear  dependence on mW , the asymmetry offers the possibil-  ity to cleanly disentangle the impact on mW determina-  tion of all effects contributing to the pℓ  ⊥ spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' On  top of the pQCD predictions scrutinised in this paper,  which constitute a robust starting point, it is conceptu-  ally straightforward to include final-state QED radiation,  as well as EW and mixed QCD-EW perturbative correc-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' All of these additional effects induce modifications  to Apℓ  ⊥ that can be separately assessed and systemati-  cally refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Effects of npQCD origin, relevant for a fully  realistic description, can also be included as a separate  component to the prediction of Apℓ  ⊥, but as opposed to  template-fitting, their inclusion is not instrumental for  the whole mW -extraction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  As they involve  initial-state QCD radiation, their inclusion is expected  to simply induce a vertical offset to Apℓ  ⊥ without altering  its slope, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' its sensitivity to mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' This offset in turn  yields a shift of the preferred mW value, which can be  easily estimated thanks to the linear mW -dependence of  Apℓ  ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The underlying npQCD model can be constrained  via the simultaneous analysis of more observables, other  than Apℓ  ⊥: the improvement in the accuracy of this model  is thus a problem fully decoupled from mW determina-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  To illustrate how npQCD contributions can be consis-  tently studied through the asymmetry Apℓ  ⊥, we consider  effects on mW coming from collinear proton PDFs and  from the modelling of an intrinsic transverse momentum  k⊥ of partons in the proton (further details on the results  of this study can be found in the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The un-  certainty on collinear PDFs enters transverse kinematics  indirectly, through the finite lepton-rapidity acceptance,  while intrinsic k⊥ directly shifts leptonic momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  As for the effect of collinear PDFs, predictions for  Apℓ  ⊥(32 GeV, 37 GeV, 47 GeV) obtained using all 100  replicas of the NNPDF4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='0 set yield a PDF uncer-  tainty of ±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' More conservatively, we also  consider the central replicas of the CT18NNLO [93],  MSHT20nnlo [94], and NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1 [95] PDF sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The  corresponding spread of mW values is of ∼ 30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' A  reduction of PDF uncertainty can be achieved by profil-  ing PDF replicas through the simultaneous inclusion of  additional information, such as data in different rapid-  ity regions [68, 69], all bins of the pℓ  ⊥ distribution [73],  different W charges at the LHC [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Turning to the intrinsic k⊥ of partons in the proton,  it can be precisely modelled studying the pℓ+ℓ−  ⊥  distribu-  5  tion in NCDY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Assuming its universality [96], it can be  applied to the CCDY simulation, inducing a shift in mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We have investigated the interplay between the scale un-  certainty of the perturbative NCDY SM description and  the size of the npQCD component extracted from NCDY  data (using the central NNLO+N3LL NCDY prediction  as pseudo-data, hence actually extracting a “pseudo-  npQCD” contribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  To this goal, we have deter-  mined one pseudo-npQCD contribution per scale choice,  included it in the CCDY simulation, and assessed its im-  pact on mW determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The point which emerges  from this analysis is that, even if the NCDY pseudo-data  are a unique set of numbers, the propagation of their in-  formation to CCDY depends on the underlying pQCD  approximation, and the outcome is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The  CCDY results, improved with the pseudo-npQCD con-  tribution, are spread in a range compatible with, or even  larger than the scale uncertainty of the NNLO+NNLL  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  This result stresses the importance of us-  ing state-of-the-art pQCD results in these high-precision  studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We have presented a new observable,  Apℓ  ⊥,  sensitive to the value of the W-boson mass  mW , with promising experimental properties and robust  pQCD convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Its linear dependence on mW allows  to systematically disentangle the impact of each contribu-  tion, perturbative or not, affecting the determination of  mW and to estimate the associated uncertainty, a crucial  feature for the comparison of data with SM predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The study of Apℓ  ⊥ highlights the importance of state-of-  the-art predictions to reduce the pQCD uncertainty on  mW down to the ±5 MeV level at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We argue  that, using Apℓ  ⊥, an experimental error on mW at the  ±15 MeV level is achievable already with Run-2 data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  moreover, the possibility is given to unfold the data to  particle level, easing the combination of results from dif-  ferent experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Given these properties, we hope that  this observable will be considered for an independent de-  termination of mW from available CCDY data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We  thank  Paolo  Azzurri,  Emanuele Bagnaschi, Luca Buonocore, Massimiliano  Grazzini,  Michelangelo  Mangano,  Pier  Monni  and  Stefano Pozzorini for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We thank  Tobias Neumann for support in the use of MCFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' LR  is supported by the Swiss National Science Founda-  tion contract PZ00P2 201878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  PT has been partially  supported by the Italian Ministry of University and  Research (MUR) through grant PRIN 20172LNEEZ  and  by  Compagnia  di  San  Paolo  through  grant  TORP S1921 EX-POST 21 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' AV is supported by the  Italian MUR through grant PRIN201719AVICI 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' LR  and AV thank MIAPbP for hospitality and acknowledge  the Deutsche Forschungsgemeinschaft under Germany’s  Excellence Strategy – EXC-2094 – 390783311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  250  300  350  400  450  500  550  mW − 80000 [MeV]  [pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ]  [32, 38, 50]  [32, 37, 50]  [32, 36, 50]  [32, 38, 47]  [32, 37, 47]  [32, 36, 47]  [30, 38, 47]  [30, 37, 47]  [30, 36, 47]  NNPDF4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='0  CT18NNLO  MSHTnnlo  NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' 3, now comparing the range of mW  values obtained with different PDF sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  APPENDIX  In this Appendix we detail the study described in the  main text about the impact of non-perturbative effects  on mW determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The discussion focuses on the  uncertainty due to collinear PDFs, and on the modelling  of an intrinsic k⊥ of partons in the proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Proton-PDF  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Concerning  the  ef-  fect  of  different  collinear  PDFs,  predictions  for  Apℓ  ⊥(32 GeV, 37 GeV, 47 GeV) obtained with the 100  replicas of the NNPDF4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='0 set yield a bundle of parallel  straight lines, as expected due to the factorisation of  QCD effects from W-boson production and decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' The  intercepts with the experimental Apℓ  ⊥ value yield a  distribution of 100 mW values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We compute mean value  and standard deviation of this distribution, obtaining  at NLO+NLL with central scales a spread in mW of  ±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='5 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We also consider the central replicas of the  CT18NNLO [93], MSHT20nnlo [94], and NNPDF3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='1 [95]  PDF sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The spread induced on mW , using the  central-scale NNLO+N3LL prediction, is of ∼ 30 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We present in Figure 4 the results for different setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Modelling of the parton intrinsic k⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  With the fol-  lowing exercise, we schematically describe the encod-  ing of information present in NCDY data and absent  from a purely perturbative description of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We then consider the usage of such an information in  the simulation of CCDY, and eventually its impact on  mW determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  In particular, the pQCD stability  of Apℓ  ⊥ allows to study the role of scale variations in  porting these effects from NCDY to CCDY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We simu-  6  late both processes at NNLO+N3LL QCD, with ξR =  ξF = 2 ξQ = 1 and take the results as a proxy for ex-  perimental data (we dub them “pseudo-data”, see also  [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We assume to have available an event generator  with NNLO+NNLL pQCD accuracy only, and compute  the NCDY pℓ+ℓ−  ⊥  distribution with different scale choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  The ratio of NNLO+NNLL predictions with pseudo-data  defines a reweighing factor, as a function of pℓ+ℓ−  ⊥  , en-  coding the missing pQCD higher orders (with real data  as opposed to pseudo-data it would encode npQCD ef-  fects as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We compute one such reweighing fac-  tor per pQCD scale choice in NCDY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We then use the  NNLO+NNLL generator to simulate the CCDY process  with scale variations, and reweigh all events in each vari-  ation according to their pℓν  ⊥ value, using the correspond-  ing factor determined in NCDY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Since by construction  the reweighed NCDY NNLO+NNLL curves would ex-  actly match NCDY pseudo-data, one expects to a large  extent the same to happen with CCDY pseudo-data and  reweighed NNLO+NNLL CCDY distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We ob-  serve instead that the reweighed distributions do not ex-  actly reproduce CCDY pseudo-data, the discrepancy be-  ing comparable with, or larger than the NNLO+NNLL  scale-uncertainty band, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' ∆mW  ∼ ±27 MeV from  the study of Apℓ  ⊥(32 GeV, 37 GeV, 47 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' We conclude  that the procedure to model npQCD effects due to an  intrinsic k⊥ is intertwined with the underlying pQCD  formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We thus expect that the same approach,  using a NNLO+N3LL-accurate event generator and the  real data as a target, would lead to a smaller final spread  in Apℓ  ⊥, providing a handle for a robust assessment of the  impact of npQCD effects on the determination of mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  We present in Figure 5 the results for different setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  ∗ luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='rottoli@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='ch  † paolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='torrielli@to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='it  ‡ alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='vicini@mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='it  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Aaltonen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' (CDF, D0), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' D 88,  052018 (2013), arXiv:1307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='7627 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Aaboud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' (ATLAS), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' C 78,  110 (2018), [Erratum:  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='C 78, 898 (2018)],  arXiv:1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='07240 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Aaij  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  (LHCb),  JHEP  01,  036  (2022),  arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='01113 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Gambino,  and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Haller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Hoecker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Kogler,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' M¨onig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Stelzer (Gfitter Group),  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content='3792 [hep-  ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' de Blas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Ciuchini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Franco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Goncalves,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Mishima, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Pierini, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Reina,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Silvestrini,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' D 106, 033003 (2022), arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='07274  250  300  350  400  450  500  550  mW − 80000 [MeV]  [pℓ,min  ⊥  , pℓ,mid  ⊥  , pℓ,max  ⊥  ]  [32, 38, 50]  [32, 37, 50]  [32, 36, 50]  [32, 38, 47]  [32, 37, 47]  [32, 36, 47]  [30, 38, 47]  [30, 37, 47]  [30, 36, 47]  NNLO+NNLL w/ rescaling  NNLO+NNLL  NNLO+N3LL  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' 3, now including the reweighed  NCDY NNLO+NNLL predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Alioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=', Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content='02330 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Duhr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Dulat, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Mistlberger, JHEP 11, 143  (2020), arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='13313 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Dulat, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Mistlberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Mistlberger, JHEP 03, 116 (2022),  arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Gehrmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Glover, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Huss, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' 128, 052001 (2022),  arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='09085 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Liu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' 115, 062002 (2015), arXiv:1504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Gehrmann-De Ridder, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' Gehrmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content='  Glover, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Focke,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Liu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+page_content=' Liu,  and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE2T4oBgHgl3EQfsggH/content/2301.04059v1.pdf'}
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+Safe Policy Improvement for POMDPs via Finite-State Controllers
+Thiago D. Sim˜ao,*
+Marnix Suilen,*
+Nils Jansen
+Department of Software Science
+Radboud University
+Nijmegen, The Netherlands
+{thiago.simao, marnix.suilen, nils.jansen}@ru.nl
+Abstract
+We study safe policy improvement (SPI) for partially observ-
+able Markov decision processes (POMDPs). SPI is an offline
+reinforcement learning (RL) problem that assumes access to
+(1) historical data about an environment, and (2) the so-called
+behavior policy that previously generated this data by inter-
+acting with the environment. SPI methods neither require ac-
+cess to a model nor the environment itself, and aim to reliably
+improve the behavior policy in an offline manner. Existing
+methods make the strong assumption that the environment is
+fully observable. In our novel approach to the SPI problem
+for POMDPs, we assume that a finite-state controller (FSC)
+represents the behavior policy and that finite memory is suf-
+ficient to derive optimal policies. This assumption allows us
+to map the POMDP to a finite-state fully observable MDP,
+the history MDP. We estimate this MDP by combining the
+historical data and the memory of the FSC, and compute an
+improved policy using an off-the-shelf SPI algorithm. The
+underlying SPI method constrains the policy-space accord-
+ing to the available data, such that the newly computed policy
+only differs from the behavior policy when sufficient data was
+available. We show that this new policy, converted into a new
+FSC for the (unknown) POMDP, outperforms the behavior
+policy with high probability. Experimental results on several
+well-established benchmarks show the applicability of the ap-
+proach, even in cases where finite memory is not sufficient.
+1
+Introduction
+Reinforcement learning (RL) is a standard approach to solve
+sequential decision-making problems when the environment
+dynamics are unknown (Sutton and Barto 1998). Typically,
+an RL agent interacts with the environment and optimizes
+its behavior according to environment’s feedback. How-
+ever, in offline RL (Levine et al. 2020), the RL agent re-
+ceives a fixed dataset of past interactions between a behav-
+ior policy and the environment and derives a new policy
+with no direct interaction with the environment. One of the
+challenges in offline RL is to ensure that the new policy
+outperforms the behavior policy (Cheng et al. 2022). This
+problem is called safe policy improvement (SPI; Thomas,
+Theocharous, and Ghavamzadeh 2015b). Most of the ap-
+proaches to SPI assume fully observable environments,
+*Equal contribution.
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+z, r
+learning
+deployment
+πβ
+a
+data collection
+[(zi, ai, ri, z′
+i)]
+buffer
+z, r
+a
+πI
+πI
+πI
+D
+Figure 1: Illustration of the offline reinforcement learning
+problem in partially observable environments (adapted from
+Levine et al. 2020). The dashed arrow indicates the setting
+where the behavior policy is available during learning.
+see for instance (Petrik, Ghavamzadeh, and Chow 2016;
+Laroche, Trichelair, and des Combes 2019).
+The restriction to full observability poses a serious lim-
+itation on the applicability of SPI, as most real-world
+problems are partially observable, due to, for instance,
+noisy sensors (Kochenderfer 2015). Partially observable
+Markov decision processes (POMDPs) are the standard
+model for decision-making problems under partial observ-
+ability (Kaelbling, Littman, and Cassandra 1998). So far,
+SPI for POMDPs was only studied for memoryless poli-
+cies (Thomas, Theocharous, and Ghavamzadeh 2015a; Yea-
+ger et al. 2022). However, POMDP policies often require a
+notion of memory. In general, optimal policies for POMDPs
+with infinite horizons require infinite memory, rendering this
+problem undecidable (Madani, Hanks, and Condon 2003).
+Nevertheless, finite memory can make good approximations
+of the optimal policy (Bonet 2002) and are often used in
+practice for being more explainable (Dujardin, Dietterich,
+and Chades 2017). Policies with finite memory may take the
+form of finite-state controllers (FSCs; Meuleau et al. 1999a).
+Our approach.
+We contribute a novel SPI approach for
+POMDPs. First, to account for the inherent memory re-
+quirement in partially observable domains, we present the
+behavior policy as an FSC. To create a tractable method,
+we assume that there exists a finite-memory policy for the
+POMDP that is optimal, also known as the finite-history-
+window approach (Kaelbling, Littman, and Moore 1996,
+Section 7.3). This assumption allows us to cast the POMDP
+arXiv:2301.04939v1  [cs.AI]  12 Jan 2023
+
+as an equivalent, fully observable, history MDP that is finite,
+instead of the standard infinite-history MDP (Silver and Ve-
+ness 2010). We are then able to reliably estimate the transi-
+tion and reward models of this finite-history MDP from the
+available data. We employ a specific SPI method for MDPs,
+called safe policy improvement with baseline bootstrap-
+ping (SPIBB; Laroche, Trichelair, and des Combes 2019).
+In particular, we compute an improved policy that outper-
+forms the behavior policy up to an admissible performance
+loss with high probability. In comparison to the approach for
+mere MDPs (Laroche, Trichelair, and des Combes 2019), we
+derive an improved bound on this admissible performance
+loss by exploiting the specific structure of the history MDP.
+The general flow of our approach is illustrated in Figure 1.
+Real-world applications.
+This setting captures multiple
+real-world applications, such as predictive maintenance,
+conservation of endangered species, and management of in-
+vasive species. We may, for instance, have data from the
+degradation process of a certain asset, which includes logs of
+inspections and maintenance that were performed according
+to a fixed schedule (represented, for instance, as a finite-state
+controller). Once we acquire a new asset, we can formalize
+the optimization problem with offline RL to compute a new
+schedule, using the original schedule as a behavior policy.
+We demonstrate the applicability of our method on three
+standard POMDP problems. The evaluation confirms the
+theoretical findings of our SPIBB approach, in comparison
+to standard offline RL. We highlight results for varying sizes
+of memory, and show that we can achieve reliable perfor-
+mance improvement even for problems where finite memory
+is not sufficient in general.
+2
+Background
+For a countable set X we write |X| for the number of ele-
+ments in X, and ∆(X) for the set of probability distributions
+over X with finite support. Given two probability distribu-
+tions P, Q ∈ ∆(X), the L1-distance between P and Q is
+∥P − Q∥1 =
+�
+x∈X
+|P(x) − Q(x)|.
+The L1-error of an estimated probability distribution ˜P is
+given by the L1-distance between ˜P and the true distribution
+P: ∥ ˜P − P∥1. Finally, we write I(x = y) for the indicator
+function returning 1 when x = y and 0 otherwise, and [l : m]
+for the set of natural numbers {l, . . . , m} ⊂ N.
+2.1
+MDPs, POMDPs, and FSCs
+Definition
+1
+(POMDP).
+A
+partially
+observable
+Markov
+decision
+process
+(POMDP)
+is
+a
+tuple
+M = ⟨S, A, T, R, γ, Z, O⟩, where S, A and Z are finite
+sets of states, actions, and observations, T : S ×A → ∆(S)
+is the transition function, R: S × A → [Rmin, Rmax] ⊂ R
+is the reward function with known bounds, γ ∈ [0, 1) ⊂ R
+is the discount factor, and O: S × A → ∆(Z) is the
+observation function.
+As a special case we have the (fully observable) Markov
+decision process (MDP; Puterman 1994), where Z = S and
+O(z|s, a) = I(z = s), so it can be defined as a POMDP
+without observations: M = ⟨S, A, T, R, γ⟩.
+A history is a sequence of observations and actions: h ∈
+(Z×A)∗×Z. We denote the set of all histories by H, and Hk
+denotes all histories of maximal length k, where the length
+|h| is the number of observations in the history h.
+A belief b ∈ ∆(S) is a distribution over the states
+of a POMDP. Beliefs are sufficient statistics for histories
+in POMDPs ( ˚Astr¨om 1965; Smallwood and Sondik 1973).
+That is, they provide just as much information as the his-
+tories themselves. A belief b can be updated into a new be-
+lief b′ upon taking an action a and receiving an observation z
+by performing a Bayesian belief update (Kaelbling, Littman,
+and Cassandra 1998). The belief b′ of being in state s given
+some history h′, denoted b(s | h′), can be recursively com-
+puted by repeated applications of the belief update on ac-
+tions a and observations z in the history h′ = haz:
+b′(s′ | haz) = b′(s′ | b(· | h), a, z).
+until the history h is empty, denoted ∅, and where b(· | ∅) is
+the initial belief.
+A POMDP is equivalent to an infinite-state fully observ-
+able MDP called the history MDP (Silver and Veness 2010).
+Definition 2 (History MDP). The fully observable history
+MDP of a POMDP is ⟨H, A, TH, RH, γ⟩, where H is the
+set of all histories, A and γ are the actions and discount
+factor from the POMDP, and TH : H × A → ∆(H) and
+RH : H × A → R are the transition and reward functions:
+TH(haz | h, a) =
+�
+s∈S
+b(s | h)
+�
+s′∈S
+T(s′ | s, a)O(z | s′, a),
+RH(h, a) =
+�
+s∈S
+b(s | h)R(s, a).
+Since belief states are sufficient statistics for histories, the
+so-called belief MDP of a POMDP serves as a common
+alternative for the history MDP, we refer to to Kaelbling,
+Littman, and Cassandra (1998) for more details.
+A policy for a POMDP is a function π: H → ∆(A),
+mapping histories to distributions over actions. The set of all
+policies is Π. The goal is to compute a policy that maximizes
+the expected discounted reward in the infinite horizon:
+max
+π∈Π E
+� ∞
+�
+t=1
+γtrt
+�
+,
+where rt is the reward the agent receives at time step t when
+following policy π. In general, a policy that maximizes the
+expected discounted reward requires infinite memory, that
+is, it needs to account for all possible histories. As such,
+computing optimal policies in POMDPs is undecidable in
+general (Madani, Hanks, and Condon 2003).
+We may instead use policies that only use a finite amount
+of memory. In general, such policies are not optimal, that is,
+they do not maximize the expected discounted reward, yet
+they are computationally more tractable. A finite-memory
+policy maps finite histories to actions, π: Hk → ∆(A), and
+can be represented by a finite-state controller of size κ =
+(|Z| + 1)k (Meuleau et al. 1999a; Junges et al. 2018), as
+
+we need to account for all possible combinations of obser-
+vations of size k, with a possible empty observation.
+Definition 3 (Finite-state controller). A finite-state con-
+troller (FSC) is a tuple ⟨N, n0, ψ, η⟩ where N is a finite set
+of memory nodes, n0 ∈ N is an initial node, ψ: N × Z →
+∆(A) is an action mapping, and η: N × Z × A → N is a
+memory update function. A κ-FSC is an FSC with |N| = κ.
+In any time step t, given the current memory node nt and
+observation zt, the action at is randomly drawn from the
+distribution ψ(· | nt, zt), and the memory node is updated
+to nt+1 = η(nt, zt, at). A policy represented by an FSC can
+get arbitrarily close to the optimal policy as κ grows (Bonet
+2002). Computing finite-state controllers that aim to maxi-
+mize the expected (discounted) reward can be done in sev-
+eral ways, among those gradient descent (Meuleau et al.
+1999b) and convex optimization (Amato, Bernstein, and Zil-
+berstein 2010; Junges et al. 2018; Cubuktepe et al. 2021).
+We denote the set of finite-memory policies of size k
+by Πk, and the set of finite-memory policies of size k rep-
+resented by FSCs with some fixed memory update η by Πη
+k.
+Furthermore, FSCs of size k can be represented by FSCs
+with more memory. These sets are related by the following
+inclusions, where k′ > k: Πη
+k ⊂ Πk ⊂ Πk′ ⊂ Π.
+Finally, we define the state-based and state-action-based
+value functions on an (PO)MDP M with policy π as V M
+π (s)
+and QM
+π (s, a) respectively. Whenever M or π are clear from
+the context we will omit them. The performance of a policy
+π in M is denoted by ρ(π, M) and is defined as the expected
+value in some initial state s0, that is, ρ(π, M) = V M
+π (s0).
+Furthermore, we write Vmax for a known upper bound on the
+absolute value of the performance: Vmax ≤ Rmax/1−γ.
+2.2
+Safe Policy Improvement on MDPs
+Here, we review safe policy improvement (SPI) for MDPs.
+A dataset is a sequence D of trajectories collected under a
+behavior policy πβ in an MDP M ∗. For MDPs, the datasets
+we consider for SPI are of the form D = ⟨st, at, rt⟩t∈[1:m],
+and we write #D(x) for the number of times x occurs in D.
+The goal of SPI is to compute a new policy πI based on D
+that outperforms πβ with an allowed performance loss ζ ∈
+R with high probability 1 − δ.
+SPI operates on a set of admissible MDPs Ξ, that is,
+MDPs M = (S, A, T, R, γ) which are ‘close’ to an MDP
+˜
+M = (S, A, ˜T, ˜R, γ) estimated from a dataset D by maxi-
+mum likelihood estimation (MLE).
+Definition 4 (MLE-MDP). The MLE-MDP of an unknown
+true MDP M ∗ = (S, A, T, R, γ) and a dataset D is a tuple
+˜
+M = (S, A, ˜T, ˜R, γ) with transition and reward functions ˜T
+and ˜R derived from D via maximum likelihood estimation:
+˜T(s′ | s, a) = #D(s, a, s′)
+#D(s, a) , and ˜R(s, a) = Rtotal(s, a)
+#D(s, a) ,
+where Rtotal(s, a) is the sum of all rewards in (s, a):
+Rtotal(s, a) = �
+(st,at,rt)∈D I(st = s ∧ at = a) · rt.
+For an MLE-MDP ˜
+M and error function e: S × A → R,
+we define the set of admissible MDPs Ξ ˜
+M
+e with a transition
+function T that has L1 distance to the estimated transition
+function ˜T bounded by the error function e:
+Ξ
+˜
+M
+e = {M | ∀(s, a).∥T(· | s, a) − ˜T(· | s, a)∥1 ≤ e(s, a)}.
+The general idea behind SPI methods is to define this er-
+ror function e such that Ξ ˜
+M
+e includes the true MDP M ∗ with
+high probability 1 − δ (Petrik, Ghavamzadeh, and Chow
+2016, Proposition 9). Then one can compute a new policy
+which is an improvement for all MDPs within Ξ ˜
+M
+e . An al-
+ternative is to simply solve the MLE-MDP, but this could
+lead to arbitrarily poor policies when the amount of data
+is insufficient. If, however, the amount of data is sufficient
+for all state-action pairs, then we can guarantee with high
+probability that the improved policy computed on the MLE-
+MDP has a higher performance. Specifically, as pointed by
+Laroche, Trichelair, and des Combes (2019), the amount of
+data is sufficient when for all state-action pairs
+#D(s, a) ≥
+8V 2
+max
+ζ2(1 − γ)2 log 2|S||A|2|S|
+δ
+.
+(1)
+Then with probability 1 − δ an optimal policy πI for ˜
+M is
+ζ-approximately safe with respect to the true MDP M ∗ for
+some admissible performance loss ζ ∈ R. That is,
+ρ(πI, M ∗) ≥ ρ(π∗, M ∗) − ζ ≥ ρ(πβ, M ∗) − ζ,
+where π∗ is an optimal policy in the true MDP M ∗. Intu-
+itively this ensures that the estimated transition function is
+close enough to the true MDP to guarantee that the policy
+computed in the MLE-MDP approximately outperforms the
+behavior policy in the underlying MDP.
+2.3
+SPI with Baseline Bootstrapping on MDPs
+The bound in Equation (1) needs to hold for every state-
+action pair, which limits the practical use of optimizing the
+MLE-MDP. The SPI with baseline bootstrapping (SPIBB;
+Laroche, Trichelair, and des Combes 2019) algorithm over-
+comes this limitation, allowing the constraint in (1) to be
+violated on some state-action pairs. These state-action pairs
+are collected in U, the set of unknown state-action pairs with
+counts smaller than a given hyperparameter N∧:
+U = {(s, a) ∈ S × A | #D(s, a) ≤ N∧}.
+SPIBB computes an improved policy πI on ˜
+M as above, ex-
+cept that πI is constrained to follow πβ for unknown state-
+action pairs: ∀(s, a) ∈ U. πI(a | s) = πβ(a | s). Then, πI is
+a ζ-approximately safe improvement of πβ with high proba-
+bility 1 − δ, where ζ is computed via (Theorem 2; Laroche,
+Trichelair, and des Combes 2019):
+ζ = 4Vmax
+1 − γ
+�
+2
+N∧
+log 2|S||A|2|S|
+δ
+− ρ(πI, ˜
+M) + ρ(πβ, ˜
+M).
+(2)
+3
+SPIBB for POMDPs
+Now we detail our approach to apply SPIBB to POMDPs.
+
+Formal problem statement.
+Given a POMDP M =
+⟨S, A, T, R, γ, Z, O⟩ of which the transition and observa-
+tion functions are unknown, some initial belief b ∈ ∆(S),
+and a finite-memory behavior policy represented as a κ-FSC
+πβ = (N, n0, ψ, η), the goal is to apply SPIBB to construct
+a new κ-FSC πI = (N, n0, ψ′, η) with the same nodes and
+memory structure η, i.e. πβ, πI ∈ Πη
+k, such that with high
+probability 1−δ, πI is a ζ-approximately safe improvement
+over πβ with respect to M. That is, with a probability of at
+least 1 − δ we have
+ρ(πI, M) ≥ ρ(πβ, M) − ζ.
+3.1
+From POMDP to Finite-History MDP
+While a POMDP can be mapped to a fully observable history
+MDP (Definition 2), this MDP has infinitely many states,
+making a direct application of SPI(BB) methods infeasible.
+To mitigate this issue, we make an assumption on the struc-
+ture of the history MDP (and inherently on the POMDP)
+that implies that the history MDP is equivalent to a smaller,
+finite, MDP. We formalize this assumption via stochastic
+bisimulation (Givan, Dean, and Greig 2003). Intuitively, this
+bisimulation is an equivalence relation that relates (history)
+states that behave similarly according to reward signals.
+Definition 5 (Bisimilarity of history states). A stochastic
+bisimulation relation E ⊆ H × H on history states h1, h2 ∈
+H is an equivalence relation satisfying
+E(h1, h2) ⇐⇒ ∀a ∈ A.
+RH(h1, a) = RH(h2, a) and
+∀h′
+1, h′
+2 ∈ H with E(h′
+1, h′
+2) we have
+TH(h′
+1 | h1, a) = TH(h′
+2 | h2, a).
+The largest stochastic bisimulation relation is called
+(stochastic) bisimulation, denoted by ∼. We write [h]∼ for
+the equivalence class of history h under ∼, and H/∼ for the
+set of equivalence classes.
+Assumption 1 (Sufficiency of finite histories). Every his-
+tory state h of size |h| > k in the history MDP is bisimilar
+to a history state h′ of size |h′| ≤ k. That is, h ∼ h′.
+As a consequence, the history MDP satisfying Assumption 1
+has a finite bisimulation quotient MDP (Givan, Dean, and
+Greig 2003), and we call it a finite-history MDP instead.
+This finite-history MDP consists of states that are the equiv-
+alence classes of histories under ∼.
+Definition 6 (Finite-history MDP). A POMDP satisfying
+Assumption 1 is a fully observable finite-state MDP M =
+(H/∼, A, TH, RH, γ) where the states are given by the set
+of equivalence classes, the actions and discount factor from
+the POMDP, and transition and reward functions defined as
+TH([haz]∼ | [h]∼, a) =
+�
+s∈S
+b(s | [h]∼)
+�
+s′∈S
+T(s′ | s, a)O(z | s′, a),
+RH([h]∼, a) =
+�
+s∈S
+b(s | [h]∼)R(s, a).
+Under bisimulation equivalence, the finite-history MDP and
+the POMDP are related in the following fundamental way.
+Theorem 1 (Optimal finite-memory policies under bisimi-
+larity). An optimal policy π∗ in the finite-history MDP is an
+optimal finite-memory policy for the POMDP.
+Theorem 1 is a direct result of bisimilarity (Givan, Dean,
+and Greig 2003). We may number the equivalence classes in
+the finite-history MDP in such a way that they correspond
+to memory nodes of an FSC. As a result, the finite-history
+MDP can be defined on a state-space consisting of memory
+nodes and observations, rather than histories.
+Definition 7 (Finite-history MDP via FSC). A POMDP sat-
+isfying Assumption 1 is a fully observable finite-state MDP
+M = (N × Z, A, TH, RH, γ) where the states are given by
+pairs of memory nodes from an FSC and observations, the
+actions from the POMDP, and transition and reward func-
+tions defined as
+TH(⟨n′, z′⟩ | ⟨n, z⟩, a) =
+�
+s∈S
+b(s | ⟨n, z⟩)
+�
+s′∈S
+T(s′ | s, a)O(z′ | s′, a)η(n′ | n, z′, a),
+RH(⟨n, z⟩, a) =
+�
+s∈S
+b(s | ⟨n, z⟩)R(s, a).
+Where b(s | ⟨n, z⟩) is the belief of being in state s of the
+POMDP given memory node n and observation z.
+This finite-history MDP will serve as the (unknown) true
+MDP M ∗ in our application of SPIBB, and η is the memory
+update function of the FSC.
+3.2
+Estimating the Finite-History MDP
+Next, we describe how to estimate the true finite-history
+MDP M ∗ by an MLE-MDP ˜
+M. The approach is similar to
+that of SPI for MDPs described in Section 2, except that the
+dataset D is different. Here, D is collected from simulating
+the POMDP M under (FSC) policy πβ. This yields a dataset
+of the form
+D = ⟨⟨nt, zt⟩, at, rt⟩t∈[1:m]
+(3)
+where the observations zt come from the observation func-
+tion, and the memory nodes nt are observed from the FSC.
+Definition 8 (Finite-history MLE-MDP). The MDP from
+Definition 7 can be estimated from a dataset D of the
+form (3), following the same approach for estimating a stan-
+dard MLE-MDP as in Definition 4:
+˜TH(⟨n′, z′⟩ | ⟨n, z⟩, a) = #D(⟨n, z⟩, a, ⟨n′, z′⟩)
+#D(⟨n, z⟩, a)
+, and
+˜RH(⟨n, z⟩, a) = Rtotal(⟨n, z⟩, a)
+#D(⟨n, z⟩, a) , where
+Rtotal(⟨n, z⟩, a) =
+�
+(⟨nt,zt⟩,at,rt)∈D
+I(⟨nt, zt⟩ = ⟨n, z⟩ ∧ at = a) · rt.
+3.3
+Applying SPIBB to the Finite-History MDP
+In this section we apply the theory of SPIBB, as introduced
+in Section 2, to our setting. In particular, we have just defined
+a true MDP M ∗ (the finite-history MDP, Definition 6) and
+an MLE-MDP ˜
+M estimating M ∗ (Definition 8). Let
+U = {(⟨n, z⟩, a) ∈ N × Z × A | #D(⟨n, z⟩, a) ≤ N∧}
+
+9
+11
+10
+6
+8
+7
+1
+5
+4
+3
+2
+Figure 2: The Maze environment. The locations are colored
+according to the agent’s perception.
+be the set of tuples (⟨n, z⟩, a) which occur less than N∧
+times in the dataset D for some hyperparameter N∧. Just
+as in SPIBB for MDPs, we compute a new policy πI ∈ Πη
+k
+for the MLE-MDP ˜
+M that estimates the finite-history MDP,
+constrained to follow the behavior policy πβ used to collect
+D for all (⟨n, z⟩, a) ∈ U.
+Theorem 2 (ζ-bound on history MDP). Let Πβ be the
+set of policies under the constraint of following πβ when
+(⟨n, z⟩, a) ∈ U. Then, the policy πI computed by the
+SPIBB algorithm on the history MDP (Definition 2) is a ζ-
+approximate safe policy improvement over the behavior pol-
+icy πβ with high probability 1 − δ, where:
+ζ = 4Vmax
+1 − γ
+�
+2
+N∧
+log 2|H||A|2|Z|
+δ
+−ρ(πI, ˜
+M)+ρ(πβ, ˜
+M).
+The proof replaces the regular MDP from the SPIBB al-
+gorithm by the (infinite) history MDP. We can reduce the
+exponent from |H|, which would be the result of naively ap-
+plying the SPIBB algorithm, to |Z| because of the structure
+of the transition function of the history MDP. In particular,
+the transition function of the history MDP is defined for his-
+tories h which are appended by an action a and an observa-
+tion z to haz, see Definition 2. As such, the successor states
+of h in the history MDP are fully determined by the obser-
+vation z instead of the full state-space, and thus we may re-
+place 2|S| from Equation (1) by 2|Z|. The full proof can be
+found in Appendix A.
+While Theorem 2 and its proof reason over the full history
+MDP, these results extend to the finite-history MDP when
+Assumption 1 is satisfied. We have the following corollary.
+Corollary 1 (ζ-bound on finite-history MDP). Let Πβ be
+the set of policies under the constraint of following πβ when
+(⟨n, z⟩, a) ∈ U. Then, the policy πI computed by the SPIBB
+algorithm in the finite-history MDP M ∗ of a POMDP satis-
+fying Assumption 1 is a ζ-approximate safe policy improve-
+ment over the behavior policy πβ with high probability 1−δ,
+where the admissible performance loss ζ is given by
+ζ = 4Vmax
+1 − γ
+�
+2
+N∧
+log 2|N×Z||A|2|Z|
+δ
+− ρ(πI, ˜
+M) + ρ(πβ, ˜
+M).
+Since bisimilarity is an equivalence relation, the finite-
+history MDP is equivalent to the full history MDP, and thus
+also the POMDP, see Theorem 1. As a consequence, the
+proof of Corollary 1 follows immediately from Theorem 2
+and the fact that bisimulation is an equivalence relation.
+4
+Empirical Analysis
+This section contains the empirical evaluation of our ap-
+proach to SPI on POMDPs. We first describe the setup
+of the experiments, and then present and analyze the re-
+sults. We provide further details in Appendix B and code
+at https://github.com/LAVA-LAB/spi pomdp.
+4.1
+Setup
+Environments.
+We consider three POMDP problems:
+i) CheeseMaze (McCallum 1993): An agent navigates a
+maze, moving in the four cardinal directions, but in each
+state it only perceives whether or not there is a barrier in
+each direction (see Figure 2). The agent is placed at a ran-
+dom location at the beginning of an episode, and receives
+a positive reward (+1) if it reaches the goal (�), and
+a small negative reward (−0.01) otherwise. The episode
+ends when the agent reaches the goal.
+ii) Tiger (Kaelbling, Littman, and Cassandra 1998): An
+agent is in front of two doors, and a tiger is randomly
+positioned behind one of them at the beginning of each
+episode. The agent has three actions: Listening, or open-
+ing one of the doors. Listening gives a noisy observa-
+tion of the position of the tiger, and a small negative re-
+ward (−1). Opening the door with the tiger gives a large
+negative reward (−100), while opening the other door
+gives a positive reward (+10).
+iii) Voicemail (Williams and Young 2007): An agent con-
+trols a voicemail machine, at the beginning of the episode
+the user listens to a message and decides if they want
+to keep it. This information is hidden from the agent,
+which has three actions: ask, save, and delete. Asking
+the user if they want to keep the message gives the agent
+a small negative reward (−1) and a noisy observation of
+the user’s intention. Correctly saving the message gives
+a positive reward (+5), and a negative reward (−10) oth-
+erwise. Correctly deleting the message gives a positive
+reward (+5), and a negative reward (−20) otherwise.
+Satisfaction of Assumption 1.
+Note that the Maze envi-
+ronment is close to satisfying Assumption 1 for memory that
+looks back two steps, i.e., k = 2, with the exceptions of his-
+tories with equal observations. Tiger and Voicemail do not
+satisfy the assumption for any k.
+Behavior policies.
+The behavior policies are generated via
+Q-learning using the memory of an FSC that keeps track of
+the last k ∈ {1, 2} observations as the state. After conver-
+gence, we extract a softmax policy, to ensure the behavior
+policy explores different actions during data collection.
+Data collection.
+We consider datasets of different sizes,
+namely: 1, 2, 5, 10, 20, 50, · · · , 5 000, and 10 000 trajecto-
+ries. For each environment, number of trajectories, and be-
+havior policy, we generate 500 datasets.
+Learning.
+We consider two algorithms to compute a new
+policy: SPIBB, and Basic RL. Both algorithms operate on
+the finite-history MLE-MDP (Definition 8) related to the
+finite-history MDP of the POMDP. We implement Basic
+RL as an unconstrained SPIBB where N∧ = 0, that is,
+
+101
+103
+105
+|D|
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Expected Return
+CheeseMaze(N∧ = 5, k = 2, k′ = 2)
+101
+103
+105
+|D|
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Expected Return
+CheeseMaze(N∧ = 20, k = 2, k′ = 2)
+101
+103
+105
+|D|
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Expected Return
+CheeseMaze(N∧ = 5, k = 2, k′ = 3)
+101
+103
+105
+|D|
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Expected Return
+CheeseMaze(N∧ = 20, k = 2, k′ = 3)
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 5, k = 2, k′ = 2)
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 20, k = 2, k′ = 2)
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 5, k = 2, k′ = 3)
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 20, k = 2, k′ = 3)
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 5, k = 2, k′ = 2)
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 20, k = 2, k′ = 2)
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 5, k = 2, k′ = 3)
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 20, k = 2, k′ = 3)
+Algorithm
+Basic RL
+SPIBB
+Metric
+Mean
+10%-CVaR
+1%-CVaR
+Behavior Policy
+PO-UCT
+Figure 3: Policy improvement on the environments Maze, Tiger and Voicemail (first, second and third row, respectively) for
+datasets collected by a behavior policy with history size k = 2, varying the hyperparameters pairs column-wise: (N∧ = 5, k′ =
+2), (N∧ = 20, k′ = 2), (N∧ = 5, k′ = 3), and (N∧ = 20, k′ = 3). The plots show the mean (solid line), 10%-CVaR (dashed
+line) and 1%-CVaR (dotted line). The performance of the behavior policy is shown in green (dash-dotted line).
+it solves the MLE-MDP using value iteration. For each
+dataset, we compute new policies πI using each offline
+RL algorithm, considering different hyperparameters: N∧ ∈
+{5, 7, 10, 15, 20, 30, 50, 70, 100} and k′ ∈ {k, k+1}, where
+k′ is the history size encoded in the FSC of πI.
+Evaluation metrics.
+Each policy is evaluated over 10 000
+episodes to obtain an estimate of the performance of the im-
+proved policy, ρ(πI, M ∗). For an evaluation across environ-
+ments and behavior policies, we also consider the normal-
+ized policy improvement:
+¯ρ(πI) =
+ρ(πI, M ∗) − ρ(πβ, M ∗)
+ρ(πmax, M ∗) − ρ(πβ, M ∗),
+where πmax is the policy with the highest expected return
+in each environment. To aggregate the results across the 500
+repetitions, we compute the mean and Conditional Value at
+Risk (CVaR; Rockafellar and Uryasev 2000). We use x%-
+CVaR to indicate the mean of the x% lowest performances.
+As an approximation of the optimal value, we show the per-
+formance of PO-UCT (Silver and Veness 2010), which uses
+the environment as a simulator to compute a policy.
+4.2
+Results
+Figure 3 shows results on the three environments (ordered
+by row). The data was collected using a behavior policy with
+k = 2. The first column shows the results where SPIBB uses
+a low threshold to consider a history-action pair known and
+the same memory size as the behavior policy (N∧ = 5 and
+k′ = 2). The second column shows the results with a higher
+threshold (N∧ = 20 and k′ = 2). The third column shows
+the results for increased memory (N∧ = 5 and k′ = 3).
+Finally, the fourth column shows the results with a higher
+threshold and increased memory (N∧ = 20 and k′ = 3).
+Basic RL is included everywhere to give a perspective on
+the influence of different hyperparameters.
+Figures 4 and 5 extend the empirical analysis on the
+Voicemail and Tiger environments for memoryless behavior
+policy (k = 1), since they demonstrated to be more chal-
+lenging for the safe policy improvement problem. Figure 4
+considers the Voicemail environment, while Figure 5 shows
+the normalized results for a range of thresholds in the Tiger
+environment. We provide further results in the Appendix C.
+4.3
+Analysis
+Basic RL is unreliable.
+Across all environments, the Ba-
+sic RL algorithm shows a considerable performance drop
+compared to the behavior policy, even in terms of the mean
+performance for smaller datasets. Notice that for Tiger and
+Voicemail, the CVaR metrics are often outside the graph.
+SPIBB outperforms Basic RL.
+In the environments Tiger
+and Voicemail (Figure 3, second and third row), the SPIBB
+algorithm shows better performance than the Basic RL
+across all dataset sizes. This is likely due to the SPIBB al-
+gorithm retaining the randomization of the behavior policy
+when insufficient data is available.
+
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 5, k = 1, k′ = 1)
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 50, k = 1, k′ = 1)
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 5, k = 1, k′ = 2)
+101
+103
+105
+|D|
+−4
+−2
+0
+2
+Expected Return
+Voicemail(N∧ = 50, k = 1, k′ = 2)
+Algorithm
+Basic RL
+SPIBB
+Metric
+Mean
+10%-CVaR
+1%-CVaR
+Behavior Policy
+PO-UCT
+Figure 4: Policy improvement on the Voicemail environment for datasets collected with a memoryless policy (k = 1), varying
+the hyperparameters pairs column-wise: (N∧ = 5, k′ = 1), (N∧ = 50, k′ = 1), (N∧ = 5, k′ = 2), and (N∧ = 50, k′ = 2).
+The plots show the mean (solid line), 10%-CVaR (dashed line) and 1%-CVaR (dotted line). The performance of the behavior
+policy is shown in green (dash-dotted line).
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 1, k′′ = 1) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 1, k′′ = 1) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 1, k′′ = 1) 1%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 1, k′′ = 2) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 1, k′′ = 2) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 1, k′′ = 2) 1%-CVaR
+−3
+−2
+−1
+0
+1
+¯ρ(πI)
+Figure 5: Normalized performance ¯ρ(πI) on the Tiger envi-
+ronment (k = 1). The left, middle and right columns show
+the mean, 10%-CVaR and 1%-CVaR, respectively. The first
+row shows the results where the improved policy uses the
+same memory as the behavior policy (k′ = k), while the
+second row shows the results for an improved policy with
+more memory (k′ = k + 1).
+SPIBB is reliable when Assumption 1 is satisfied.
+Ana-
+lyzing the results for the Maze environment (Figure 3, first
+row), we observe that SPIBB shows reliably outperforms
+the behavior policy even for a small N∧ (first column), for
+which only the 1%-CVaR shows a performance drop.
+More memory improves the reliability.
+SPIBB shows
+slightly unreliable behavior for small values of N∧ in the
+Tiger and Voicemail environments (Figure 3), as evidenced
+by both the CVaR curves, which can be alleviated by increas-
+ing the N∧ or the memory of the new policy (second, third
+and fourth column). When Assumption 1 is violated, the per-
+formance drop may be significant, as seen in the first two
+columns of Figure 4. In this case, merely increasing the N∧
+threshold is not enough to guarantee a policy improvement.
+Increasing the memory size, however, allows the SPIBB al-
+gorithm to reliably improve the behavior policy, as Figure 4
+(last column) and Figure 5 (second row) show.
+Deterministic policies may require more memory.
+Fig-
+ure 4 shows an interesting phenomenon. In partially observ-
+able settings, the stochastic behavior policy might perform
+better than the new deterministic policy, since randomiza-
+tion can trade-off some amount of memory. We observe that
+when k = 1, SPIBB and Basic RL converge to determinis-
+tic policies with an expected return lower than the behavior
+policy. When SPIBB has sufficient data, it is not constrained
+to follow the behavior policy, and thus does not inherit any
+randomization from that policy. As stated in the previous
+paragraph, more memory can then yield a new deterministic
+policy with a higher return than the behavior policy.
+5
+Related Work
+Offline RL, also known as batch RL, learns or evaluates a
+policy from a fixed batch of historical data (Levine et al.
+2020). Overall, these algorithms rely on pessimism to miti-
+gate the lack of feedback from the environment and can be
+divided in two categories (Jin, Yang, and Wang 2021): those
+that constrain the final policy to stay close to the behav-
+ior policy (Laroche, Trichelair, and des Combes 2019), and
+those that penalize rare experiences (Petrik, Ghavamzadeh,
+and Chow 2016). Our method belongs to the first category.
+Various extensions of SPIBB could be adapted for
+POMDPs, such as soft-SPIBB (Nadjahi, Laroche, and des
+Combes 2019; Scholl et al. 2022), deep-SPIBB (Brand-
+fonbrener, des Combes, and Laroche 2022), and factored-
+SPIBB (Sim˜ao and Spaan 2019a,b). SPI has also been stud-
+ied without the behavior policy (Sim˜ao, Laroche, and des
+Combes 2020) and for multi-objective (Satija et al. 2021)
+and non-stationary settings (Chandak et al. 2020).
+When the behavior policy is influenced by unobserved
+variables, we may come across confounding variables. The
+problem of evaluating a policy offline was studied in this set-
+ting, for instance, assuming that observed and unobserved
+variables are decoupled (Tennenholtz, Shalit, and Mannor
+2020), or that the influence of the confounding variable on
+the behavior policy is limited (Namkoong et al. 2020). Since
+we assume that the behavior policy only depends on the ob-
+served history, we have no confounding variables.
+6
+Conclusions
+We presented a new approach to safe policy improvement
+in POMDPs. Our experiments show the applicability of the
+approach, even in cases where finite-history is not sufficient
+to obtain optimal results. In the future, it would be interest-
+ing to relax Assumption 1 to distance metrics (Ferns, Panan-
+gaden, and Precup 2004, 2005) instead of exact bisimilarity.
+
+Acknowledgments
+We would like to thank Alberto Castellini, Alessandro
+Farinelli, Matthijs Spaan, and Edoardo Zorzi for discus-
+sions on related topics. We also thank Maris Galesloot for
+help with the implementation of the PO-UCT algorithm.
+This research has been partially funded by NWO grants
+OCENW.KLEIN.187 (Provably Correct Policies for Uncer-
+tain Partially Observable Markov Decision Processes) and
+NWA.1160.18.238 (PrimaVera).
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+able Markov decision processes for spoken dialog systems.
+Comput. Speech Lang., 21(2): 393–422.
+Yeager, J.; Moss, J. E. B.; Norrish, M.; and Thomas, P. S.
+2022. Mechanizing Soundness of Off-Policy Evaluation. In
+ITP, volume 237, 32:1–32:20.
+
+A
+Proof of Theorem 2
+Theorem 2
+(ζ-bound on history MDP).
+Let Πβ be the set of policies under the constraint of following πβ when (⟨n, z⟩, a) ∈
+U. Then, the policy πI computed by the SPIBB algorithm on the history MDP (Definition 2) is a ζ-approximate safe policy
+improvement over the behavior policy πβ with high probability 1 − δ, where:
+ζ = 4Vmax
+1 − γ
+�
+2
+N∧
+log 2|H||A|2|Z|
+δ
+− ρ(πI, ˜
+M) + ρ(πβ, ˜
+M).
+(4)
+Proof of theorem 2. First, we restate Theorem 2.1 by Weissman et al. (2003) that bounds the L1-error of an empirical probability
+distribution ˜P given a finite number of samples.
+Proposition (Theorem 2.1 by Weissman et al. (2003)). Given m i.i.d. random variables distributed according to P and given
+an estimated probability distribution ˜P computed from those m variables, we have
+Pr(∥P − ˜P∥1 ≥ ϵ) ≤ (2n − 2)e−mϵ2/2.
+To bound the L1-error of the estimated transition function of given a (h, a) pair of the history MDP, given #D(h, a) samples
+for (h, a) ∈ D we can apply the Theorem 2.1 (Weissman et al. 2003) to obtain
+Pr
+�
+∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ
+�
+≤ (2|Z| − 2)e−#D(h,a)ϵ2/2.
+Choosing an appropriate N∧, we can do a union bound over all history-action pairs, we get that the probability of having an
+arbitrary history-action pair (h, a) of which the L1-error is larger than ϵ is bounded by δ:
+Pr
+�
+∃(h, a) ∈ H × A : ∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ
+�
+≤
+�
+h,a∈H×A
+Pr
+�
+∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ
+�
+≤ |H| |A|(2|Z| − 2)e−N∧ϵ2/2
+= δ.
+Reordering the equation above, we get
+ϵ =
+�
+2
+N∧
+log 2|H||A|2|Z|
+δ
+,
+(5)
+and then
+Pr
+�
+∃h, a ∈ H × A.∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ
+�
+≤ δ.
+The last step of the proof is to replace the ϵ from Equation 36 of (Laroche, Trichelair, and des Combes 2019) by the value
+from Equation (5), resulting in
+ζ = 4Vmax
+1 − γ
+�
+2
+N∧
+log 2|H||A|2|Z|
+δ
+− ρ(πI, ˜
+M) + ρ(πβ, ˜
+M),
+as stated in the Theorem.
+B
+Experiments Details
+Further details on the problems.
+All experiments use a discount factor γ = 0.95, with at most 300 steps in an episode.
+Training the behavior policies.
+We train a Q-learning agent over 5 000 episodes with learning rate(α) and exploration rate (ϵ)
+decaying exponentially after each episode.
+αi ← α0 exp(−λ ∗ i),
+ϵi ← ϵ0 exp(−λ ∗ i),
+where i is the episode index, λ is the decay rate, α0 and ϵ0 are the initial learning rate and initial exploration rate, respectively.
+After training, we extract the behavior policy πβ using a softmax function
+πβ(a | s) =
+e−τQ(s,a)
+�
+a′∈A e−τQ(s,a′) ,
+where Q(s, a) is the final value of the state-action pair s, a and τ a hyperparameter to control the randomness of the behavior
+policy. As mentioned in the main paper, the state in this case is the history of the last k observations.
+
+CheeseMaze
+Tiger
+Voicemail
+Initial exploration rate
+ϵ0
+0.500
+0.500
+0.500
+Initial learning rate
+α0
+1.000
+1.000
+1.000
+Decay rate
+λ
+0.002
+0.002
+0.002
+Softmax temperature
+τ
+15.000
+0.050
+0.300
+Table 1: Hyperparameters to generate the behavior policies.
+Computing specifications.
+All experiments were performed on a machine with a 4GHz Intel Core i9 CPU and 64Gb of
+memory, using a single core for each experiment.
+C
+Supplementary Experimental Results
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 5, k = 1, k′ = 1)
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 50, k = 1, k′ = 1)
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 5, k = 1, k′ = 2)
+101
+103
+105
+|D|
+−30
+−20
+−10
+0
+Expected Return
+Tiger(N∧ = 50, k = 1, k′ = 2)
+Algorithm
+Basic RL
+SPIBB
+Metric
+Mean
+10%-CVaR
+1%-CVaR
+Behavior Policy
+PO-UCT
+Figure 6: Policy improvement on the Tiger environment for datasets collected with a memoryless policy (k = 1), varying the
+hyperparameters pairs column-wise: (N∧ = 5, k′ = 1), (N∧ = 50, k′ = 1), (N∧ = 5, k′ = 2), and (N∧ = 50, k′ = 2). The
+plots show the mean (solid line), 10%-CVaR (dashed line) and 1%-CVaR (dotted line). The performance of the behavior policy
+is shown in green (dash-dotted line).
+
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+CheeseMaze (k = 2, k′′ = 2) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+CheeseMaze (k = 2, k′′ = 2) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+CheeseMaze (k = 2, k′′ = 2) 1%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+CheeseMaze (k = 2, k′′ = 3) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+CheeseMaze (k = 2, k′′ = 3) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+CheeseMaze (k = 2, k′′ = 3) 1%-CVaR
+−3
+−2
+−1
+0
+1
+¯ρ(πI)
+Figure 7: Normalized results (¯ρ(πI)) on the Maze environment (k = 2). The left, middle and right columns show the mean,
+10%-CVaR and 1%-CVaR, respectively. The first row shows the results where the improved policy uses the same memory as
+the behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k + 1).
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 2, k′′ = 2) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 2, k′′ = 2) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 2, k′′ = 2) 1%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 2, k′′ = 3) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 2, k′′ = 3) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 2, k′′ = 3) 1%-CVaR
+−3
+−2
+−1
+0
+1
+¯ρ(πI)
+Figure 8: Normalized result on the Voicemail environment (k = 2). The left, middle and right columns show the mean, 10%-
+CVaR and 1%-CVaR, respectively. The first row shows the results where the improved policy uses the same memory as the
+behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k + 1).
+
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 2, k′′ = 2) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 2, k′′ = 2) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 2, k′′ = 2) 1%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 2, k′′ = 3) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 2, k′′ = 3) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Tiger (k = 2, k′′ = 3) 1%-CVaR
+−3
+−2
+−1
+0
+1
+¯ρ(πI)
+Figure 9: Normalized results (¯ρ(πI)) on the Tiger environment (k = 2). The left, middle and right columns show the mean,
+10%-CVaR and 1%-CVaR, respectively. The first row shows the results where the improved policy uses the same memory as
+the behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k + 1).
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 1, k′′ = 1) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 1, k′′ = 1) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 1, k′′ = 1) 1%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 1, k′′ = 2) Mean
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 1, k′′ = 2) 10%-CVaR
+1
+2
+5
+10
+20
+50
+100
+200
+500
+1000
+2000
+5000
+10000
+20000
+50000
+100000
+|D|
+100
+70
+50
+30
+20
+15
+10
+7
+5
+0
+N∧
+Voicemail (k = 1, k′′ = 2) 1%-CVaR
+−3
+−2
+−1
+0
+1
+¯ρ(πI)
+Figure 10: Normalized results (¯ρ(πI)) on the Voicemail environment (k = 1). The left, middle and right columns show the
+mean, 10%-CVaR and 1%-CVaR, respectively. The first row shows the results where the improved policy uses the same memory
+as the behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k+1).
+
diff --git a/TNE4T4oBgHgl3EQfLgyY/content/tmp_files/load_file.txt b/TNE4T4oBgHgl3EQfLgyY/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4e00b97a8e37cdfe028df989845c98ed507f1bb8
--- /dev/null
+++ b/TNE4T4oBgHgl3EQfLgyY/content/tmp_files/load_file.txt
@@ -0,0 +1,909 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf,len=908
+page_content='Safe Policy Improvement for POMDPs via Finite-State Controllers  Thiago D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Sim˜ao,*  Marnix Suilen,*  Nils Jansen  Department of Software Science  Radboud University  Nijmegen, The Netherlands  {thiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='simao, marnix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='suilen, nils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='jansen}@ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='nl  Abstract  We study safe policy improvement (SPI) for partially observ-  able Markov decision processes (POMDPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' SPI is an offline  reinforcement learning (RL) problem that assumes access to  (1) historical data about an environment, and (2) the so-called  behavior policy that previously generated this data by inter-  acting with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' SPI methods neither require ac-  cess to a model nor the environment itself, and aim to reliably  improve the behavior policy in an offline manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Existing  methods make the strong assumption that the environment is  fully observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In our novel approach to the SPI problem  for POMDPs, we assume that a finite-state controller (FSC)  represents the behavior policy and that finite memory is suf-  ficient to derive optimal policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' This assumption allows us  to map the POMDP to a finite-state fully observable MDP,  the history MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We estimate this MDP by combining the  historical data and the memory of the FSC, and compute an  improved policy using an off-the-shelf SPI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The  underlying SPI method constrains the policy-space accord-  ing to the available data, such that the newly computed policy  only differs from the behavior policy when sufficient data was  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We show that this new policy, converted into a new  FSC for the (unknown) POMDP, outperforms the behavior  policy with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Experimental results on several  well-established benchmarks show the applicability of the ap-  proach, even in cases where finite memory is not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  1  Introduction  Reinforcement learning (RL) is a standard approach to solve  sequential decision-making problems when the environment  dynamics are unknown (Sutton and Barto 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Typically,  an RL agent interacts with the environment and optimizes  its behavior according to environment’s feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' How-  ever, in offline RL (Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2020), the RL agent re-  ceives a fixed dataset of past interactions between a behav-  ior policy and the environment and derives a new policy  with no direct interaction with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' One of the  challenges in offline RL is to ensure that the new policy  outperforms the behavior policy (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' This  problem is called safe policy improvement (SPI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Thomas,  Theocharous, and Ghavamzadeh 2015b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Most of the ap-  proaches to SPI assume fully observable environments,  Equal contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  z, r  learning  deployment  πβ  a  data collection  [(zi, ai, ri, z′  i)]  buffer  z, r  a  πI  πI  πI  D  Figure 1: Illustration of the offline reinforcement learning  problem in partially observable environments (adapted from  Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The dashed arrow indicates the setting  where the behavior policy is available during learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  see for instance (Petrik, Ghavamzadeh, and Chow 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Laroche, Trichelair, and des Combes 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The restriction to full observability poses a serious lim-  itation on the applicability of SPI, as most real-world  problems are partially observable, due to, for instance,  noisy sensors (Kochenderfer 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Partially observable  Markov decision processes (POMDPs) are the standard  model for decision-making problems under partial observ-  ability (Kaelbling, Littman, and Cassandra 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' So far,  SPI for POMDPs was only studied for memoryless poli-  cies (Thomas, Theocharous, and Ghavamzadeh 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Yea-  ger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' However, POMDP policies often require a  notion of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In general, optimal policies for POMDPs  with infinite horizons require infinite memory, rendering this  problem undecidable (Madani, Hanks, and Condon 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Nevertheless, finite memory can make good approximations  of the optimal policy (Bonet 2002) and are often used in  practice for being more explainable (Dujardin, Dietterich,  and Chades 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Policies with finite memory may take the  form of finite-state controllers (FSCs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Meuleau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 1999a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We contribute a novel SPI approach for  POMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' First, to account for the inherent memory re-  quirement in partially observable domains, we present the  behavior policy as an FSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' To create a tractable method,  we assume that there exists a finite-memory policy for the  POMDP that is optimal, also known as the finite-history-  window approach (Kaelbling, Littman, and Moore 1996,  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' This assumption allows us to cast the POMDP  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='04939v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='AI]  12 Jan 2023  as an equivalent, fully observable, history MDP that is finite,  instead of the standard infinite-history MDP (Silver and Ve-  ness 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We are then able to reliably estimate the transi-  tion and reward models of this finite-history MDP from the  available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We employ a specific SPI method for MDPs,  called safe policy improvement with baseline bootstrap-  ping (SPIBB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Laroche, Trichelair, and des Combes 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  In particular, we compute an improved policy that outper-  forms the behavior policy up to an admissible performance  loss with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In comparison to the approach for  mere MDPs (Laroche, Trichelair, and des Combes 2019), we  derive an improved bound on this admissible performance  loss by exploiting the specific structure of the history MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The general flow of our approach is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  This setting captures multiple  real-world applications, such as predictive maintenance,  conservation of endangered species, and management of in-  vasive species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We may, for instance, have data from the  degradation process of a certain asset, which includes logs of  inspections and maintenance that were performed according  to a fixed schedule (represented, for instance, as a finite-state  controller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Once we acquire a new asset, we can formalize  the optimization problem with offline RL to compute a new  schedule, using the original schedule as a behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We demonstrate the applicability of our method on three  standard POMDP problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The evaluation confirms the  theoretical findings of our SPIBB approach, in comparison  to standard offline RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We highlight results for varying sizes  of memory, and show that we can achieve reliable perfor-  mance improvement even for problems where finite memory  is not sufficient in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2  Background  For a countable set X we write |X| for the number of ele-  ments in X, and ∆(X) for the set of probability distributions  over X with finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Given two probability distribu-  tions P, Q ∈ ∆(X), the L1-distance between P and Q is  ∥P − Q∥1 =  �  x∈X  |P(x) − Q(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The L1-error of an estimated probability distribution ˜P is  given by the L1-distance between ˜P and the true distribution  P: ∥ ˜P − P∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Finally, we write I(x = y) for the indicator  function returning 1 when x = y and 0 otherwise, and [l : m]  for the set of natural numbers {l, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' , m} ⊂ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='1  MDPs, POMDPs, and FSCs  Definition  1  (POMDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  A  partially  observable  Markov  decision  process  (POMDP)  is  a  tuple  M = ⟨S, A, T, R, γ, Z, O⟩, where S, A and Z are finite  sets of states, actions, and observations, T : S ×A → ∆(S)  is the transition function, R: S × A → [Rmin, Rmax] ⊂ R  is the reward function with known bounds, γ ∈ [0, 1) ⊂ R  is the discount factor, and O: S × A → ∆(Z) is the  observation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  As a special case we have the (fully observable) Markov  decision process (MDP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Puterman 1994), where Z = S and  O(z|s, a) = I(z = s), so it can be defined as a POMDP  without observations: M = ⟨S, A, T, R, γ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  A history is a sequence of observations and actions: h ∈  (Z×A)∗×Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We denote the set of all histories by H, and Hk  denotes all histories of maximal length k, where the length  |h| is the number of observations in the history h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  A belief b ∈ ∆(S) is a distribution over the states  of a POMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Beliefs are sufficient statistics for histories  in POMDPs ( ˚Astr¨om 1965;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Smallwood and Sondik 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  That is, they provide just as much information as the his-  tories themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A belief b can be updated into a new be-  lief b′ upon taking an action a and receiving an observation z  by performing a Bayesian belief update (Kaelbling, Littman,  and Cassandra 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The belief b′ of being in state s given  some history h′, denoted b(s | h′), can be recursively com-  puted by repeated applications of the belief update on ac-  tions a and observations z in the history h′ = haz:  b′(s′ | haz) = b′(s′ | b(· | h), a, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  until the history h is empty, denoted ∅, and where b(· | ∅) is  the initial belief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  A POMDP is equivalent to an infinite-state fully observ-  able MDP called the history MDP (Silver and Veness 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Definition 2 (History MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The fully observable history  MDP of a POMDP is ⟨H, A, TH, RH, γ⟩, where H is the  set of all histories, A and γ are the actions and discount  factor from the POMDP, and TH : H × A → ∆(H) and  RH : H × A → R are the transition and reward functions:  TH(haz | h, a) =  �  s∈S  b(s | h)  �  s′∈S  T(s′ | s, a)O(z | s′, a),  RH(h, a) =  �  s∈S  b(s | h)R(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Since belief states are sufficient statistics for histories, the  so-called belief MDP of a POMDP serves as a common  alternative for the history MDP, we refer to to Kaelbling,  Littman, and Cassandra (1998) for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  A policy for a POMDP is a function π: H → ∆(A),  mapping histories to distributions over actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The set of all  policies is Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The goal is to compute a policy that maximizes  the expected discounted reward in the infinite horizon:  max  π∈Π E  � ∞  �  t=1  γtrt  �  ,  where rt is the reward the agent receives at time step t when  following policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In general, a policy that maximizes the  expected discounted reward requires infinite memory, that  is, it needs to account for all possible histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' As such,  computing optimal policies in POMDPs is undecidable in  general (Madani, Hanks, and Condon 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We may instead use policies that only use a finite amount  of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In general, such policies are not optimal, that is,  they do not maximize the expected discounted reward, yet  they are computationally more tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A finite-memory  policy maps finite histories to actions, π: Hk → ∆(A), and  can be represented by a finite-state controller of size κ =  (|Z| + 1)k (Meuleau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 1999a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Junges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2018), as  we need to account for all possible combinations of obser-  vations of size k, with a possible empty observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Definition 3 (Finite-state controller).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A finite-state con-  troller (FSC) is a tuple ⟨N, n0, ψ, η⟩ where N is a finite set  of memory nodes, n0 ∈ N is an initial node, ψ: N × Z →  ∆(A) is an action mapping, and η: N × Z × A → N is a  memory update function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A κ-FSC is an FSC with |N| = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  In any time step t, given the current memory node nt and  observation zt, the action at is randomly drawn from the  distribution ψ(· | nt, zt), and the memory node is updated  to nt+1 = η(nt, zt, at).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A policy represented by an FSC can  get arbitrarily close to the optimal policy as κ grows (Bonet  2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Computing finite-state controllers that aim to maxi-  mize the expected (discounted) reward can be done in sev-  eral ways, among those gradient descent (Meuleau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  1999b) and convex optimization (Amato, Bernstein, and Zil-  berstein 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Junges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Cubuktepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We denote the set of finite-memory policies of size k  by Πk, and the set of finite-memory policies of size k rep-  resented by FSCs with some fixed memory update η by Πη  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Furthermore, FSCs of size k can be represented by FSCs  with more memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' These sets are related by the following  inclusions, where k′ > k: Πη  k ⊂ Πk ⊂ Πk′ ⊂ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Finally, we define the state-based and state-action-based  value functions on an (PO)MDP M with policy π as V M  π (s)  and QM  π (s, a) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Whenever M or π are clear from  the context we will omit them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The performance of a policy  π in M is denoted by ρ(π, M) and is defined as the expected  value in some initial state s0, that is, ρ(π, M) = V M  π (s0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Furthermore, we write Vmax for a known upper bound on the  absolute value of the performance: Vmax ≤ Rmax/1−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='2  Safe Policy Improvement on MDPs  Here, we review safe policy improvement (SPI) for MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  A dataset is a sequence D of trajectories collected under a  behavior policy πβ in an MDP M ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' For MDPs, the datasets  we consider for SPI are of the form D = ⟨st, at, rt⟩t∈[1:m],  and we write #D(x) for the number of times x occurs in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The goal of SPI is to compute a new policy πI based on D  that outperforms πβ with an allowed performance loss ζ ∈  R with high probability 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  SPI operates on a set of admissible MDPs Ξ, that is,  MDPs M = (S, A, T, R, γ) which are ‘close’ to an MDP  ˜  M = (S, A, ˜T, ˜R, γ) estimated from a dataset D by maxi-  mum likelihood estimation (MLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Definition 4 (MLE-MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The MLE-MDP of an unknown  true MDP M ∗ = (S, A, T, R, γ) and a dataset D is a tuple  ˜  M = (S, A, ˜T, ˜R, γ) with transition and reward functions ˜T  and ˜R derived from D via maximum likelihood estimation:  ˜T(s′ | s, a) = #D(s, a, s′)  #D(s, a) , and ˜R(s, a) = Rtotal(s, a)  #D(s, a) ,  where Rtotal(s, a) is the sum of all rewards in (s, a):  Rtotal(s, a) = �  (st,at,rt)∈D I(st = s ∧ at = a) · rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  For an MLE-MDP ˜  M and error function e: S × A → R,  we define the set of admissible MDPs Ξ ˜  M  e with a transition  function T that has L1 distance to the estimated transition  function ˜T bounded by the error function e:  Ξ  ˜  M  e = {M | ∀(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='∥T(· | s, a) − ˜T(· | s, a)∥1 ≤ e(s, a)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The general idea behind SPI methods is to define this er-  ror function e such that Ξ ˜  M  e includes the true MDP M ∗ with  high probability 1 − δ (Petrik, Ghavamzadeh, and Chow  2016, Proposition 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Then one can compute a new policy  which is an improvement for all MDPs within Ξ ˜  M  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' An al-  ternative is to simply solve the MLE-MDP, but this could  lead to arbitrarily poor policies when the amount of data  is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' If, however, the amount of data is sufficient  for all state-action pairs, then we can guarantee with high  probability that the improved policy computed on the MLE-  MDP has a higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Specifically, as pointed by  Laroche, Trichelair, and des Combes (2019), the amount of  data is sufficient when for all state-action pairs  #D(s, a) ≥  8V 2  max  ζ2(1 − γ)2 log 2|S||A|2|S|  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  (1)  Then with probability 1 − δ an optimal policy πI for ˜  M is  ζ-approximately safe with respect to the true MDP M ∗ for  some admissible performance loss ζ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' That is,  ρ(πI, M ∗) ≥ ρ(π∗, M ∗) − ζ ≥ ρ(πβ, M ∗) − ζ,  where π∗ is an optimal policy in the true MDP M ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Intu-  itively this ensures that the estimated transition function is  close enough to the true MDP to guarantee that the policy  computed in the MLE-MDP approximately outperforms the  behavior policy in the underlying MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='3  SPI with Baseline Bootstrapping on MDPs  The bound in Equation (1) needs to hold for every state-  action pair, which limits the practical use of optimizing the  MLE-MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The SPI with baseline bootstrapping (SPIBB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Laroche, Trichelair, and des Combes 2019) algorithm over-  comes this limitation, allowing the constraint in (1) to be  violated on some state-action pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' These state-action pairs  are collected in U, the set of unknown state-action pairs with  counts smaller than a given hyperparameter N∧:  U = {(s, a) ∈ S × A | #D(s, a) ≤ N∧}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  SPIBB computes an improved policy πI on ˜  M as above, ex-  cept that πI is constrained to follow πβ for unknown state-  action pairs: ∀(s, a) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' πI(a | s) = πβ(a | s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Then, πI is  a ζ-approximately safe improvement of πβ with high proba-  bility 1 − δ, where ζ is computed via (Theorem 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Laroche,  Trichelair, and des Combes 2019):  ζ = 4Vmax  1 − γ  �  2  N∧  log 2|S||A|2|S|  δ  − ρ(πI, ˜  M) + ρ(πβ, ˜  M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  (2)  3  SPIBB for POMDPs  Now we detail our approach to apply SPIBB to POMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Formal problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Given a POMDP M =  ⟨S, A, T, R, γ, Z, O⟩ of which the transition and observa-  tion functions are unknown, some initial belief b ∈ ∆(S),  and a finite-memory behavior policy represented as a κ-FSC  πβ = (N, n0, ψ, η), the goal is to apply SPIBB to construct  a new κ-FSC πI = (N, n0, ψ′, η) with the same nodes and  memory structure η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' πβ, πI ∈ Πη  k, such that with high  probability 1−δ, πI is a ζ-approximately safe improvement  over πβ with respect to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' That is, with a probability of at  least 1 − δ we have  ρ(πI, M) ≥ ρ(πβ, M) − ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='1  From POMDP to Finite-History MDP  While a POMDP can be mapped to a fully observable history  MDP (Definition 2), this MDP has infinitely many states,  making a direct application of SPI(BB) methods infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  To mitigate this issue, we make an assumption on the struc-  ture of the history MDP (and inherently on the POMDP)  that implies that the history MDP is equivalent to a smaller,  finite, MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We formalize this assumption via stochastic  bisimulation (Givan, Dean, and Greig 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Intuitively, this  bisimulation is an equivalence relation that relates (history)  states that behave similarly according to reward signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Definition 5 (Bisimilarity of history states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A stochastic  bisimulation relation E ⊆ H × H on history states h1, h2 ∈  H is an equivalence relation satisfying  E(h1, h2) ⇐⇒ ∀a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  RH(h1, a) = RH(h2, a) and  ∀h′  1, h′  2 ∈ H with E(h′  1, h′  2) we have  TH(h′  1 | h1, a) = TH(h′  2 | h2, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The largest stochastic bisimulation relation is called  (stochastic) bisimulation, denoted by ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We write [h]∼ for  the equivalence class of history h under ∼, and H/∼ for the  set of equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Assumption 1 (Sufficiency of finite histories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Every his-  tory state h of size |h| > k in the history MDP is bisimilar  to a history state h′ of size |h′| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' That is, h ∼ h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  As a consequence, the history MDP satisfying Assumption 1  has a finite bisimulation quotient MDP (Givan, Dean, and  Greig 2003), and we call it a finite-history MDP instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  This finite-history MDP consists of states that are the equiv-  alence classes of histories under ∼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Definition 6 (Finite-history MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A POMDP satisfying  Assumption 1 is a fully observable finite-state MDP M =  (H/∼, A, TH, RH, γ) where the states are given by the set  of equivalence classes, the actions and discount factor from  the POMDP, and transition and reward functions defined as  TH([haz]∼ | [h]∼, a) =  �  s∈S  b(s | [h]∼)  �  s′∈S  T(s′ | s, a)O(z | s′, a),  RH([h]∼, a) =  �  s∈S  b(s | [h]∼)R(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Under bisimulation equivalence, the finite-history MDP and  the POMDP are related in the following fundamental way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Theorem 1 (Optimal finite-memory policies under bisimi-  larity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' An optimal policy π∗ in the finite-history MDP is an  optimal finite-memory policy for the POMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Theorem 1 is a direct result of bisimilarity (Givan, Dean,  and Greig 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We may number the equivalence classes in  the finite-history MDP in such a way that they correspond  to memory nodes of an FSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' As a result, the finite-history  MDP can be defined on a state-space consisting of memory  nodes and observations, rather than histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Definition 7 (Finite-history MDP via FSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' A POMDP sat-  isfying Assumption 1 is a fully observable finite-state MDP  M = (N × Z, A, TH, RH, γ) where the states are given by  pairs of memory nodes from an FSC and observations, the  actions from the POMDP, and transition and reward func-  tions defined as  TH(⟨n′, z′⟩ | ⟨n, z⟩, a) =  �  s∈S  b(s | ⟨n, z⟩)  �  s′∈S  T(s′ | s, a)O(z′ | s′, a)η(n′ | n, z′, a),  RH(⟨n, z⟩, a) =  �  s∈S  b(s | ⟨n, z⟩)R(s, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Where b(s | ⟨n, z⟩) is the belief of being in state s of the  POMDP given memory node n and observation z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  This finite-history MDP will serve as the (unknown) true  MDP M ∗ in our application of SPIBB, and η is the memory  update function of the FSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='2  Estimating the Finite-History MDP  Next, we describe how to estimate the true finite-history  MDP M ∗ by an MLE-MDP ˜  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The approach is similar to  that of SPI for MDPs described in Section 2, except that the  dataset D is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Here, D is collected from simulating  the POMDP M under (FSC) policy πβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' This yields a dataset  of the form  D = ⟨⟨nt, zt⟩, at, rt⟩t∈[1:m]  (3)  where the observations zt come from the observation func-  tion, and the memory nodes nt are observed from the FSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Definition 8 (Finite-history MLE-MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The MDP from  Definition 7 can be estimated from a dataset D of the  form (3), following the same approach for estimating a stan-  dard MLE-MDP as in Definition 4:  ˜TH(⟨n′, z′⟩ | ⟨n, z⟩, a) = #D(⟨n, z⟩, a, ⟨n′, z′⟩)  #D(⟨n, z⟩, a)  , and  ˜RH(⟨n, z⟩, a) = Rtotal(⟨n, z⟩, a)  #D(⟨n, z⟩, a) , where  Rtotal(⟨n, z⟩, a) =  �  (⟨nt,zt⟩,at,rt)∈D  I(⟨nt, zt⟩ = ⟨n, z⟩ ∧ at = a) · rt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='3  Applying SPIBB to the Finite-History MDP  In this section we apply the theory of SPIBB, as introduced  in Section 2, to our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In particular, we have just defined  a true MDP M ∗ (the finite-history MDP, Definition 6) and  an MLE-MDP ˜  M estimating M ∗ (Definition 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Let  U = {(⟨n, z⟩, a) ∈ N × Z × A | #D(⟨n, z⟩, a) ≤ N∧}  9  11  10  6  8  7  1  5  4  3  2  Figure 2: The Maze environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The locations are colored  according to the agent’s perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  be the set of tuples (⟨n, z⟩, a) which occur less than N∧  times in the dataset D for some hyperparameter N∧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Just  as in SPIBB for MDPs, we compute a new policy πI ∈ Πη  k  for the MLE-MDP ˜  M that estimates the finite-history MDP,  constrained to follow the behavior policy πβ used to collect  D for all (⟨n, z⟩, a) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Theorem 2 (ζ-bound on history MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Let Πβ be the  set of policies under the constraint of following πβ when  (⟨n, z⟩, a) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Then, the policy πI computed by the  SPIBB algorithm on the history MDP (Definition 2) is a ζ-  approximate safe policy improvement over the behavior pol-  icy πβ with high probability 1 − δ, where:  ζ = 4Vmax  1 − γ  �  2  N∧  log 2|H||A|2|Z|  δ  −ρ(πI, ˜  M)+ρ(πβ, ˜  M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The proof replaces the regular MDP from the SPIBB al-  gorithm by the (infinite) history MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We can reduce the  exponent from |H|, which would be the result of naively ap-  plying the SPIBB algorithm, to |Z| because of the structure  of the transition function of the history MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In particular,  the transition function of the history MDP is defined for his-  tories h which are appended by an action a and an observa-  tion z to haz, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' As such, the successor states  of h in the history MDP are fully determined by the obser-  vation z instead of the full state-space, and thus we may re-  place 2|S| from Equation (1) by 2|Z|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The full proof can be  found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  While Theorem 2 and its proof reason over the full history  MDP, these results extend to the finite-history MDP when  Assumption 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We have the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Corollary 1 (ζ-bound on finite-history MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Let Πβ be  the set of policies under the constraint of following πβ when  (⟨n, z⟩, a) ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Then, the policy πI computed by the SPIBB  algorithm in the finite-history MDP M ∗ of a POMDP satis-  fying Assumption 1 is a ζ-approximate safe policy improve-  ment over the behavior policy πβ with high probability 1−δ,  where the admissible performance loss ζ is given by  ζ = 4Vmax  1 − γ  �  2  N∧  log 2|N×Z||A|2|Z|  δ  − ρ(πI, ˜  M) + ρ(πβ, ˜  M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Since bisimilarity is an equivalence relation, the finite-  history MDP is equivalent to the full history MDP, and thus  also the POMDP, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' As a consequence, the  proof of Corollary 1 follows immediately from Theorem 2  and the fact that bisimulation is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  4  Empirical Analysis  This section contains the empirical evaluation of our ap-  proach to SPI on POMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We first describe the setup  of the experiments, and then present and analyze the re-  sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We provide further details in Appendix B and code  at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='com/LAVA-LAB/spi pomdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='1  Setup  Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We consider three POMDP problems:  i) CheeseMaze (McCallum 1993): An agent navigates a  maze, moving in the four cardinal directions, but in each  state it only perceives whether or not there is a barrier in  each direction (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The agent is placed at a ran-  dom location at the beginning of an episode, and receives  a positive reward (+1) if it reaches the goal (�), and  a small negative reward (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='01) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The episode  ends when the agent reaches the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  ii) Tiger (Kaelbling, Littman, and Cassandra 1998): An  agent is in front of two doors, and a tiger is randomly  positioned behind one of them at the beginning of each  episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The agent has three actions: Listening, or open-  ing one of the doors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Listening gives a noisy observa-  tion of the position of the tiger, and a small negative re-  ward (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Opening the door with the tiger gives a large  negative reward (−100), while opening the other door  gives a positive reward (+10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  iii) Voicemail (Williams and Young 2007): An agent con-  trols a voicemail machine, at the beginning of the episode  the user listens to a message and decides if they want  to keep it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' This information is hidden from the agent,  which has three actions: ask, save, and delete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Asking  the user if they want to keep the message gives the agent  a small negative reward (−1) and a noisy observation of  the user’s intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Correctly saving the message gives  a positive reward (+5), and a negative reward (−10) oth-  erwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Correctly deleting the message gives a positive  reward (+5), and a negative reward (−20) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Satisfaction of Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Note that the Maze envi-  ronment is close to satisfying Assumption 1 for memory that  looks back two steps, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=', k = 2, with the exceptions of his-  tories with equal observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Tiger and Voicemail do not  satisfy the assumption for any k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Behavior policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The behavior policies are generated via  Q-learning using the memory of an FSC that keeps track of  the last k ∈ {1, 2} observations as the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' After conver-  gence, we extract a softmax policy, to ensure the behavior  policy explores different actions during data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We consider datasets of different sizes,  namely: 1, 2, 5, 10, 20, 50, · · · , 5 000, and 10 000 trajecto-  ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' For each environment, number of trajectories, and be-  havior policy, we generate 500 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We consider two algorithms to compute a new  policy: SPIBB, and Basic RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Both algorithms operate on  the finite-history MLE-MDP (Definition 8) related to the  finite-history MDP of the POMDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We implement Basic  RL as an unconstrained SPIBB where N∧ = 0, that is,  101  103  105  |D|  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  Expected Return  CheeseMaze(N∧ = 5, k = 2, k′ = 2)  101  103  105  |D|  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  Expected Return  CheeseMaze(N∧ = 20, k = 2, k′ = 2)  101  103  105  |D|  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  Expected Return  CheeseMaze(N∧ = 5, k = 2, k′ = 3)  101  103  105  |D|  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='00  Expected Return  CheeseMaze(N∧ = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 3)  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 3)  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 3)  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 3)  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 3)  Algorithm  Basic RL  SPIBB  Metric  Mean  10%-CVaR  1%-CVaR  Behavior Policy  PO-UCT  Figure 3: Policy improvement on the environments Maze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Tiger and Voicemail (first,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' second and third row,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' respectively) for  datasets collected by a behavior policy with history size k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' varying the hyperparameters pairs column-wise: (N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ =  2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (N∧ = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' and (N∧ = 20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The plots show the mean (solid line), 10%-CVaR (dashed  line) and 1%-CVaR (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The performance of the behavior policy is shown in green (dash-dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  it solves the MLE-MDP using value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' For each  dataset, we compute new policies πI using each offline  RL algorithm, considering different hyperparameters: N∧ ∈  {5, 7, 10, 15, 20, 30, 50, 70, 100} and k′ ∈ {k, k+1}, where  k′ is the history size encoded in the FSC of πI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Each policy is evaluated over 10 000  episodes to obtain an estimate of the performance of the im-  proved policy, ρ(πI, M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' For an evaluation across environ-  ments and behavior policies, we also consider the normal-  ized policy improvement:  ¯ρ(πI) =  ρ(πI, M ∗) − ρ(πβ, M ∗)  ρ(πmax, M ∗) − ρ(πβ, M ∗),  where πmax is the policy with the highest expected return  in each environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' To aggregate the results across the 500  repetitions, we compute the mean and Conditional Value at  Risk (CVaR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Rockafellar and Uryasev 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We use x%-  CVaR to indicate the mean of the x% lowest performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  As an approximation of the optimal value, we show the per-  formance of PO-UCT (Silver and Veness 2010), which uses  the environment as a simulator to compute a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='2  Results  Figure 3 shows results on the three environments (ordered  by row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The data was collected using a behavior policy with  k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The first column shows the results where SPIBB uses  a low threshold to consider a history-action pair known and  the same memory size as the behavior policy (N∧ = 5 and  k′ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The second column shows the results with a higher  threshold (N∧ = 20 and k′ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The third column shows  the results for increased memory (N∧ = 5 and k′ = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Finally, the fourth column shows the results with a higher  threshold and increased memory (N∧ = 20 and k′ = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Basic RL is included everywhere to give a perspective on  the influence of different hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Figures 4 and 5 extend the empirical analysis on the  Voicemail and Tiger environments for memoryless behavior  policy (k = 1), since they demonstrated to be more chal-  lenging for the safe policy improvement problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Figure 4  considers the Voicemail environment, while Figure 5 shows  the normalized results for a range of thresholds in the Tiger  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We provide further results in the Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='3  Analysis  Basic RL is unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Across all environments, the Ba-  sic RL algorithm shows a considerable performance drop  compared to the behavior policy, even in terms of the mean  performance for smaller datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Notice that for Tiger and  Voicemail, the CVaR metrics are often outside the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  SPIBB outperforms Basic RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  In the environments Tiger  and Voicemail (Figure 3, second and third row), the SPIBB  algorithm shows better performance than the Basic RL  across all dataset sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' This is likely due to the SPIBB al-  gorithm retaining the randomization of the behavior policy  when insufficient data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1)  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1)  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  101  103  105  |D|  −4  −2  0  2  Expected Return  Voicemail(N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  Algorithm  Basic RL  SPIBB  Metric  Mean  10%-CVaR  1%-CVaR  Behavior Policy  PO-UCT  Figure 4: Policy improvement on the Voicemail environment for datasets collected with a memoryless policy (k = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' varying  the hyperparameters pairs column-wise: (N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' and (N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The plots show the mean (solid line), 10%-CVaR (dashed line) and 1%-CVaR (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The performance of the behavior  policy is shown in green (dash-dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 1) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 1) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 1) 1%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 1%-CVaR  −3  −2  −1  0  1  ¯ρ(πI)  Figure 5: Normalized performance ¯ρ(πI) on the Tiger envi-  ronment (k = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The left, middle and right columns show  the mean, 10%-CVaR and 1%-CVaR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The first  row shows the results where the improved policy uses the  same memory as the behavior policy (k′ = k), while the  second row shows the results for an improved policy with  more memory (k′ = k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  SPIBB is reliable when Assumption 1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Ana-  lyzing the results for the Maze environment (Figure 3, first  row), we observe that SPIBB shows reliably outperforms  the behavior policy even for a small N∧ (first column), for  which only the 1%-CVaR shows a performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  More memory improves the reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  SPIBB shows  slightly unreliable behavior for small values of N∧ in the  Tiger and Voicemail environments (Figure 3), as evidenced  by both the CVaR curves, which can be alleviated by increas-  ing the N∧ or the memory of the new policy (second, third  and fourth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' When Assumption 1 is violated, the per-  formance drop may be significant, as seen in the first two  columns of Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In this case, merely increasing the N∧  threshold is not enough to guarantee a policy improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Increasing the memory size, however, allows the SPIBB al-  gorithm to reliably improve the behavior policy, as Figure 4  (last column) and Figure 5 (second row) show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Deterministic policies may require more memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Fig-  ure 4 shows an interesting phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In partially observ-  able settings, the stochastic behavior policy might perform  better than the new deterministic policy, since randomiza-  tion can trade-off some amount of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We observe that  when k = 1, SPIBB and Basic RL converge to determinis-  tic policies with an expected return lower than the behavior  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' When SPIBB has sufficient data, it is not constrained  to follow the behavior policy, and thus does not inherit any  randomization from that policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' As stated in the previous  paragraph, more memory can then yield a new deterministic  policy with a higher return than the behavior policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  5  Related Work  Offline RL, also known as batch RL, learns or evaluates a  policy from a fixed batch of historical data (Levine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Overall, these algorithms rely on pessimism to miti-  gate the lack of feedback from the environment and can be  divided in two categories (Jin, Yang, and Wang 2021): those  that constrain the final policy to stay close to the behav-  ior policy (Laroche, Trichelair, and des Combes 2019), and  those that penalize rare experiences (Petrik, Ghavamzadeh,  and Chow 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Our method belongs to the first category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Various extensions of SPIBB could be adapted for  POMDPs, such as soft-SPIBB (Nadjahi, Laroche, and des  Combes 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Scholl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2022), deep-SPIBB (Brand-  fonbrener, des Combes, and Laroche 2022), and factored-  SPIBB (Sim˜ao and Spaan 2019a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' SPI has also been stud-  ied without the behavior policy (Sim˜ao, Laroche, and des  Combes 2020) and for multi-objective (Satija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2021)  and non-stationary settings (Chandak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  When the behavior policy is influenced by unobserved  variables, we may come across confounding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The  problem of evaluating a policy offline was studied in this set-  ting, for instance, assuming that observed and unobserved  variables are decoupled (Tennenholtz, Shalit, and Mannor  2020), or that the influence of the confounding variable on  the behavior policy is limited (Namkoong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Since  we assume that the behavior policy only depends on the ob-  served history, we have no confounding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  6  Conclusions  We presented a new approach to safe policy improvement  in POMDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Our experiments show the applicability of the  approach, even in cases where finite-history is not sufficient  to obtain optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In the future, it would be interest-  ing to relax Assumption 1 to distance metrics (Ferns, Panan-  gaden, and Precup 2004, 2005) instead of exact bisimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Acknowledgments  We would like to thank Alberto Castellini, Alessandro  Farinelli, Matthijs Spaan, and Edoardo Zorzi for discus-  sions on related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' We also thank Maris Galesloot for  help with the implementation of the PO-UCT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  This research has been partially funded by NWO grants  OCENW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='KLEIN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='187 (Provably Correct Policies for Uncer-  tain Partially Observable Markov Decision Processes) and  NWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='1160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Ad-  versarially Trained Actor Critic for Offline Reinforcement  Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Reinforcement learn-  ing - an introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Adaptive computation and machine  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Tennenholtz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Shalit, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' and Mannor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Off-Policy  Evaluation in Partially Observable Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  AAAI, 10276–10283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Thomas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Theocharous, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' and Ghavamzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2015a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' High-Confidence Off-Policy Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In AAAI,  3000–3006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Thomas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Theocharous, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' and Ghavamzadeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2015b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' High Confidence Policy Improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In ICML,  volume 37, 2380–2388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Ordentlich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Seroussi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Verdu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' and  Weinberger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+page_content=' Inequalities for the L1 Deviation  of the Empirical Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Technical report, Hewlett-  Packard Labs, Palo Alto, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Williams, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' and Young, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Partially observ-  able Markov decision processes for spoken dialog systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Speech Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=', 21(2): 393–422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Yeager, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Moss, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Norrish, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' and Thomas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Mechanizing Soundness of Off-Policy Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' In  ITP, volume 237, 32:1–32:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  A  Proof of Theorem 2  Theorem 2  (ζ-bound on history MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Let Πβ be the set of policies under the constraint of following πβ when (⟨n, z⟩, a) ∈  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Then, the policy πI computed by the SPIBB algorithm on the history MDP (Definition 2) is a ζ-approximate safe policy  improvement over the behavior policy πβ with high probability 1 − δ, where:  ζ = 4Vmax  1 − γ  �  2  N∧  log 2|H||A|2|Z|  δ  − ρ(πI, ˜  M) + ρ(πβ, ˜  M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  (4)  Proof of theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' First, we restate Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='1 by Weissman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (2003) that bounds the L1-error of an empirical probability  distribution ˜P given a finite number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Proposition (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='1 by Weissman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (2003)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' Given m i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' random variables distributed according to P and given  an estimated probability distribution ˜P computed from those m variables, we have  Pr(∥P − ˜P∥1 ≥ ϵ) ≤ (2n − 2)e−mϵ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  To bound the L1-error of the estimated transition function of given a (h, a) pair of the history MDP, given #D(h, a) samples  for (h, a) ∈ D we can apply the Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='1 (Weissman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' 2003) to obtain  Pr  �  ∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ  �  ≤ (2|Z| − 2)e−#D(h,a)ϵ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Choosing an appropriate N∧, we can do a union bound over all history-action pairs, we get that the probability of having an  arbitrary history-action pair (h, a) of which the L1-error is larger than ϵ is bounded by δ:  Pr  �  ∃(h, a) ∈ H × A : ∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ  �  ≤  �  h,a∈H×A  Pr  �  ∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ  �  ≤ |H| |A|(2|Z| − 2)e−N∧ϵ2/2  = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Reordering the equation above, we get  ϵ =  �  2  N∧  log 2|H||A|2|Z|  δ  ,  (5)  and then  Pr  �  ∃h, a ∈ H × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='∥T(· | h, a) − ˜T(· | h, a)∥1 ≥ ϵ  �  ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  The last step of the proof is to replace the ϵ from Equation 36 of (Laroche, Trichelair, and des Combes 2019) by the value  from Equation (5), resulting in  ζ = 4Vmax  1 − γ  �  2  N∧  log 2|H||A|2|Z|  δ  − ρ(πI, ˜  M) + ρ(πβ, ˜  M),  as stated in the Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  B  Experiments Details  Further details on the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  All experiments use a discount factor γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='95, with at most 300 steps in an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Training the behavior policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  We train a Q-learning agent over 5 000 episodes with learning rate(α) and exploration rate (ϵ)  decaying exponentially after each episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  αi ← α0 exp(−λ ∗ i),  ϵi ← ϵ0 exp(−λ ∗ i),  where i is the episode index, λ is the decay rate, α0 and ϵ0 are the initial learning rate and initial exploration rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  After training, we extract the behavior policy πβ using a softmax function  πβ(a | s) =  e−τQ(s,a)  �  a′∈A e−τQ(s,a′) ,  where Q(s, a) is the final value of the state-action pair s, a and τ a hyperparameter to control the randomness of the behavior  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' As mentioned in the main paper, the state in this case is the history of the last k observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  CheeseMaze  Tiger  Voicemail  Initial exploration rate  ϵ0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='500  Initial learning rate  α0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='000  Decay rate  λ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='002  Softmax temperature  τ  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='300  Table 1: Hyperparameters to generate the behavior policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  Computing specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  All experiments were performed on a machine with a 4GHz Intel Core i9 CPU and 64Gb of  memory, using a single core for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  C  Supplementary Experimental Results  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1)  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1)  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  101  103  105  |D|  −30  −20  −10  0  Expected Return  Tiger(N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2)  Algorithm  Basic RL  SPIBB  Metric  Mean  10%-CVaR  1%-CVaR  Behavior Policy  PO-UCT  Figure 6: Policy improvement on the Tiger environment for datasets collected with a memoryless policy (k = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' varying the  hyperparameters pairs column-wise: (N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' (N∧ = 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' and (N∧ = 50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The  plots show the mean (solid line), 10%-CVaR (dashed line) and 1%-CVaR (dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The performance of the behavior policy  is shown in green (dash-dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  CheeseMaze (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  CheeseMaze (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  CheeseMaze (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 1%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  CheeseMaze (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  CheeseMaze (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  CheeseMaze (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) 1%-CVaR  −3  −2  −1  0  1  ¯ρ(πI)  Figure 7: Normalized results (¯ρ(πI)) on the Maze environment (k = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The left, middle and right columns show the mean,  10%-CVaR and 1%-CVaR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The first row shows the results where the improved policy uses the same memory as  the behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 1%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) 1%-CVaR  −3  −2  −1  0  1  ¯ρ(πI)  Figure 8: Normalized result on the Voicemail environment (k = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The left, middle and right columns show the mean, 10%-  CVaR and 1%-CVaR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The first row shows the results where the improved policy uses the same memory as the  behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 1%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Tiger (k = 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 3) 1%-CVaR  −3  −2  −1  0  1  ¯ρ(πI)  Figure 9: Normalized results (¯ρ(πI)) on the Tiger environment (k = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The left, middle and right columns show the mean,  10%-CVaR and 1%-CVaR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The first row shows the results where the improved policy uses the same memory as  the behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content='  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 1) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 1) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 1) 1%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) Mean  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 10%-CVaR  1  2  5  10  20  50  100  200  500  1000  2000  5000  10000  20000  50000  100000  |D|  100  70  50  30  20  15  10  7  5  0  N∧  Voicemail (k = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' k′′ = 2) 1%-CVaR  −3  −2  −1  0  1  ¯ρ(πI)  Figure 10: Normalized results (¯ρ(πI)) on the Voicemail environment (k = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The left, middle and right columns show the  mean, 10%-CVaR and 1%-CVaR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
+page_content=' The first row shows the results where the improved policy uses the same memory  as the behavior policy (k′ = k), while the second row shows the results for an improved policy with more memory (k′ = k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE4T4oBgHgl3EQfLgyY/content/2301.04939v1.pdf'}
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+Polarized image of a rotating black hole surrounded by a cold dark matter halo
+Xin Qin1, Songbai Chen1,2∗, Zelin Zhang1, Jiliang Jing1,2 †
+1 Department of Physics, Key Laboratory of Low Dimensional
+Quantum Structures and Quantum Control of Ministry of Education,
+Synergetic Innovation Center for Quantum Effects and Applications,
+Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
+2Center for Gravitation and Cosmology, College of Physical Science and Technology,
+Yangzhou University, Yangzhou 225009, People’s Republic of China
+Abstract
+We have studied the polarized image of an equatorial emitting ring around a rotating black hole
+surrounded by a cold dark matter (CDM) halo. Results show that the CDM halo density has the
+similar effects of the halo’s characteristic radius on the polarized image for the black hole. The
+effects of the CDM halo on the polarized image depend on the magnetic field configuration, the
+fluid velocity and the observed inclination. With the increase of the CDM halo parameters, the
+observed polarization intensity decreases when the magnetic field lies in equatorial plane, but in the
+case where the magnetic field is perpendicular to the equatorial plane, the change of the observed
+polarization intensity with CDM halo also depends on the position of the emitting point in the
+ring. The change of the electric vector position angle (EVPA) with the CDM halo becomes more
+complicated. Our results also show that the influence of the CDM halo on the polarized image is
+generally small, which are consistent with the effects of dark matter halo on black hole shadows.
+These results could help to further understand dark matter from black hole images.
+PACS numbers:
+04.70.Cs, 98.62.Mw, 97.60.Lf
+∗ Corresponding author: csb3752@hunnu.edu.cn
+† jljing@hunnu.edu.cn
+
+2
+I.
+INTRODUCTION
+The releasing of black hole images of M87* [1–6] and Sgr A*[7] by the Event Horizon Telescope (EHT)
+collaboration, together with the corresponding polarized patterns [8, 9], means that the observational black
+hole astronomy has been entered an new era of rapid progress. It is helpful to confirm the existence of black
+holes, to test general relativity and to explore matter distribution around black holes. The polarized image of
+M87* [8, 9] revealed that there is a strong magnetic field around the black hole. It is beneficial to understand
+the process of accretion and the generation of jet in the vicinity of black holes in universe. Therefore, a lot of
+polarized images of black holes have been studied in both theoretical and experimental aspects [10–30].
+In general, numerical simulations are applied to obtain an accurate description of polarization image of a
+black hole. However, they are computationally expensive because of the broad parameter surveys and the
+complicated couplings among astrophysical and relativistic effects. Recently, a simple model of an equatorial
+ring of magnetized fluid has been proposed to study the polarized images of synchrotron emission around
+Schwarzschild [10] and Kerr black hole [11]. Although only the emission from a single radius is considered in
+this model, it can clearly reveal the dependence of the polarization signatures on the magnetic field config-
+uration, the black hole spin and the observer inclination. In addition, the studies [10, 11] also indicate that
+the ring model image is broadly consistent with the polarization morphology of the EHT image, even though
+there is a high fractional polarization after blurring. Thus, the simple ring model has been recently applied to
+study the polarized image of an equatorial emitting ring in various spacetimes, such as, a 4D Gauss-Bonnet
+black hole spacetime [12], a rotating black hole spacetime in the STVG-MOG theory [13], regular black hole
+spacetimes [16], a Schwarzschilld-Melvin black hole spacetime [17], and so on.
+Dark matter is a theoretical model to explain the discrepancy between the observed dynamics and the
+amount of luminous matter. However, the nature of dark matter is still open. Dark matter is assumed to
+be an invisible matter and has feeble couplings with the common visible matter at most. Despite extensive
+observational data supporting its presence on a large scale [31–33], dark matter has not been directly detected
+at present.
+The CDM model is one of the most popular dark matter models [34, 35], which has a good
+consistency with the observation of large scale structure of cosmology. The spacetime metric of Schwarzschild
+black hole and Kerr black hole surrounded by a CDM halo were studied in [36] and the effects of the CDM
+halo on the quasinormal modes have also been studied in the spherical symmetric black hole spacetime [37].
+The studies of black hole shadows [38, 39] show that the effects of the CDM halo on shadows are generally
+
+3
+small and only become significant when the CDM parameter k = ρc ˜R3 increases to order of magnitude of 107.
+However, they are very important for understanding dark matter from black hole shadows. Along this line, we
+here will study the polarized image of the equatorial emitting ring around a rotating black hole surrounded
+by a CDM halo and probe the effects of the CDM halo on the polarization image.
+The paper is organized as follows: Section II briefly introduces the rotating black hole surrounded by a CDM
+halo and presents the calculation formulas for the observed polarization vector in the image of an emitting ring
+in this spacetime. Section III presents the polarization images of the synchrotron emitting ring and probes
+the effects of the CDM halo. Finally, this paper ends with a summary.
+II.
+OBSERVED POLARIZATION FIELD IN A ROTATING BLACK HOLE SPACETIME
+SURROUNDED BY DARK MATTER HALO
+The metric analytical form of a rotating black hole surrounded by a CDM halo was obtained in [36]. In the
+Boyer-Lindquist coordinates, it has the form
+ds2 = −
+�
+1 − r2 − f(r)r2
+ρ2
+�
+dt2 + ρ2
+∆ dr2 + 2
+�
+r2 − f(r)r2�
+a sin2 θ
+ρ2
+dφdt+
+ρ2dθ2 + sin θ2
+ρ2
+��
+r2 + a2�2 − a2∆ sin θ2�
+dφ2,
+(1)
+with
+f(r) =
+�
+1 + r
+˜R
+�− 8πGρc ˜
+R3
+c2r
+− 2GM
+rc2 ,
+∆ = r2f(r) + a2,
+ρ2 = r2 + a2 cos θ2,
+(2)
+where M is the mass of the black hole, G and c are Newtonian gravitational constant and the speed of light,
+respectively. ρc is the density of the halo at the moment when the dark matter halo collapsed and ˜R is the
+corresponding characteristic radius. According to the observations on the Sgr A∗ [38], the best fit values of
+the rotation curve of the Galaxy are ρc = 1.936×107 M⊙kpc−3 and ˜R = 17.46 kpc. The metric can reduce to
+the Kerr metric in the limit ρc = 0. Here, we set G = 1 and c = 1.
+In the spacetime of a rotating black hole surrounded by a CDM halo (1), the geodesic equation for photons
+can be expressed as
+ρ2
+E pt = r2 + a2
+∆
+�
+r2 + a2 − aλ
+�
++ a
+�
+λ − a sin θ2�
+,
+(3)
+ρ2
+E pφ = a
+∆
+�
+r2 + a2 − aλ
+�
++
+λ
+sin θ2 − a,
+ρ2
+E pr = ±r
+�
+R(r),
+ρ2
+E pθ = ±θ
+�
+Θ(θ).
+
+4
+The conserved quantities λ and η are the energy-rescaled angular momentum parallel to the axis of symmetry
+and Carter constant, respectively. The radial potential R(r) and the angular potential Θ(θ) take the form
+R(r) =
+�
+r2 + a2 − aλ
+�2 − ∆
+�
+η + (a − λ)2�
+,
+(4)
+Θ(θ) = η + a2 cos θ2 − λ2 cot θ2.
+Utilizing the null geodesic Eqs.(3), the coordinates (x, y) for the photon’s arrival position on the observer’s
+screen can be obtained as
+x = −
+λ
+sin θo
+,
+y = ±o
+�
+Θ(θ),
+(5)
+where θo is the observed inclination angle from the normal direction of the accretion disk. The radial and
+angle integrals of the photon trajectories from the initial position (rs, θs) to the final initial (ro, θo) can be
+expressed as [11, 47]
+Ir ≡
+ ro
+rs
+dr
+±r
+�
+R(r)
+=
+ θo
+θs
+dθ
+±r
+�
+Θ(θ)
+≡ Gθ.
+(6)
+The slash in the path integral denotes that the sign of ±r or ±θ changes when the photon passes through
+at the radial or angular turning point. For a photon’s trajectory with m turning points, the angular path
+integral can be written as
+Gm
+θ =
+1
+�
+−u−a2
+�
+2mK
+�u+
+u−
+�
+− sign(β)Fo
+�
+,
+(7)
+where
+Fo = F
+�
+arcsin cos θo
+√u+
+���u+
+u−
+�
+,
+u± = ∆θ ±
+�
+∆2
+θ + η
+a2 ,
+∆θ = 1
+2
+�
+1 − η + λ2
+a2
+�
+.
+(8)
+Combining Eqs.(5), Eqs.(6) and Eqs.(7), one can numerically get the set of celestial coordinates (x, y) for
+a photon emitted on an equatorial ring with a radius rs. The four-momentum of the photon at the source
+(rs, θs = π
+2 ) is given by
+pt = −1,
+pr = ±r
+�
+R(rs)
+∆s
+,
+pθ = ±s
+√η,
+pφ = λ.
+(9)
+Here, the sign of pθ at the source ±s = (−1)m±o and the sign of ±r can be computed by a semi-analytic
+method [11, 47]. The four-momentum pµ of photon at the source can be obtained as
+pt = 1
+r2s
+�r2
+s + a2
+∆s
+�
+r2
+s + a2 − aλ
+�
++ a(λ − a)
+�
+,
+pr = ±r
+1
+r2s
+�
+R(rs),
+(10)
+pφ = 1
+r2s
+� a
+∆s
+�
+r2
+s + a2 − aλ
+�
++ λ − a
+�
+,
+pθ = ±s
+√η
+r2s
+.
+
+5
+The emitted ring around a rotating black hole surrounded by a CDM halo (1) is assumed to lie in the equatorial
+plane (θs = π
+2 ). In the local orthonormal zero-angular-momentum-observer (ZAMO) frame of the point P,
+the boosting velocity of the boosted emitter is assumed to be in the r-φ plane and has a form
+⃗β = βν
+�
+cos χ (ˆr) + sin χ(ˆφ)
+�
+.
+(11)
+In the boosted orthonormal frame, the four-momentum p(a) of point emitter can be obtained by the four-
+momentum pµ in a rotating black hole spacetime surrounded by a CDM halo, i.e.,
+p(a) = Λ(a)
+(b)η(b)(c)eµ
+(c)pµ.
+(12)
+where η(b)(c) is the flat Minkowski metric. The zero-angular-momentum-observer (ZAMO) tetrad eµ
+(c) and
+the Lorentz transformation Λ(a)
+(b) are respectively given by
+eµ
+(c) =
+�
+�����
+1
+rs
+�
+Ξs
+∆s
+0
+ωs
+rs
+�
+Ξs
+∆s
+0
+0
+√∆s
+rs
+0
+0
+0
+0
+rs
+√Ξs
+0
+0
+0
+0
+− 1
+rs
+�
+�����
+,
+(13)
+and
+Λ(a)
+(b) =
+�
+��
+γ
+−βνγ cos χ
+−βνγ sin χ
+0
+−βνγ cos χ (γ − 1) cos χ2 + 1 (γ − 1) sin χ cos χ 0
+−βνγ sin χ (γ − 1) sin χ cos χ (γ − 1) sin χ2 + 1 0
+0
+0
+0
+1
+�
+�� ,
+(14)
+where γ =
+1
+√
+1−β2 is the Lorentz factor. In the local frame, the temporal components of polarization vector
+f (t) = 0. Since the three-dimensional electric vector ⃗E of photon is along the direction of vector ⃗p × ⃗B, the
+spatial components of polarization vector ⃗f can be expressed as
+f (r) = p(φ) × B(θ)
+|⃗p|
+,
+f (φ) = p(θ) × B(r)
+|⃗p|
+,
+f (θ) = p(r) × B(φ)
+|⃗p|
+.
+(15)
+Thus, the photon polarization vector in the black hole spacetime can be obtained from the local polarization
+vector by using an inverse transformation as
+f µ = eµ
+(c)Λ
+(c)
+(a) f (a),
+(16)
+where Λ
+(c)
+(a)
+is the inverse matrix of Λ(a)
+(b). And the photon polarization vector satisfies the relationship as
+f µfµ = sin ζ2| ⃗B|2.
+(17)
+Here, the ζ is the angle between photon momentum ⃗p and the magnetic field ⃗B and has the form
+sin ζ = |⃗p × ⃗B|
+|⃗p|| ⃗B|
+.
+(18)
+
+6
+FIG. 1: Effects of ρc on the polarized intensity and EVPA in the rotating black hole spacetime (1). ˜R is
+fixed to be 17.46kpc. Here rs = 6, a = 0.3, θo = 20◦, βν = 0.3, and χ = −90◦.
+FIG. 2: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1). ρc is
+fixed to be 1.936×107 M⊙·kpc−3. Here rs = 6, a = 0.3, θo = 20◦, βν = 0.3, and χ = −90◦.
+As the photon propagate along the null geodesic in the rotating black hole spacetime surrounded by a CDM
+halo (1), the polarization vector f µ obeys
+f µpµ = 0,
+pµ∇µf ν = 0.
+(19)
+Since the black hole spacetime (1) belong to a type-D spacetime, the conserved Penrose-Walker constant κ
+
+R=17.46kpc,Br=1,Bμ=0,Bs=0
+R=17.46kpc,Br=1,B=0,Bg=0
+R=17.46kpc,Br=1,Bμ=0,Be=0
+R=17.46kpc,Br=1,Bg=0,Be=0
+0.000 E
+1.5E
+0.50
+-0.002
+1.0E
+0.45
+-0.004
+0.5E
+EVPA
+EVPA
+-0.006
+= 0.40 
+0.0
+-0.008
+- 0.5
+0.35
+-0.010
+-1.0
+0.30
+-0.012
+1.5
+0
+2
+3
+4
+6
+2
+4
+5
+6
+0
+G
+6
+5
+3
+1
+3
+Φ
+Φ
+R=17.46kpc,Br=0,B=1,Bg=0
+R=17.46kpc,Br=0,Bμ=1,Bg=0
+R=17.46kpc,Br=0,B=1,B=0
+R=17.46kpc,Br=0,Bμ=1,Ba=0
+0.6F
+1.5F
+0.010F
+1.0[
+0.005
+0.5
+0.5
+△EVPA
+0.000
+EVPA
+三 0.4
+0.0
+-0.005
+-0.5
+0.3F
+-0.010
+-1.0
+-1.5
+-0.015
+150
+0.2E
+0
+2
+3
+5
+6
+0
+3
+4
+5
+0
+2
+3
+4
+5
+0
+1
+2
+6
+X
+Φ
+Pc =0 Mo'kpc-3
+Pc=1×101° Mokpc-3
+Pc=5×101° Mo·kpc-3
+Pc=1×1011 Mo~kpc-3Pc=1.936x107Mo·kpc-3,Br=1,Bμ=0,Bo=0
+Pc=1.936×107 Mo:kpc-3,Br=1,Bg=0,Be=0
+R=1.936×107Mokpc-3,Br=1,Bg=0,Be=0
+=1.936×107Mo:kpc-3,Br=1,Bg=0,Be=0
+1.5
+0.50
+5
+1.0
+-0.005
+0.45
+0.5
+EVPA
+0.40F
+-0.010
+0.0
+AEV
+0.35[
+-0.5
+-0.015
+-1.0
+0.30E
+1.5
+0.25
+2
+4
+5
+6
+0
+2
+3
+5
+6
+0
+2
+3
+4
+5
+6
+d
+Φ
+,Br=0,Bμ=1,Bg=0
+Pc=1.936×10′Mokpc-3,B,=0,Bμ=1,Be=0
+e=1.936×10′Mo:kpc-3,Br=0,Bg=1,Be=0
+0.6
+1.5
+0.015
+1.0[
+0.010E
+0.5
+0.5E
+0.005E
+EVPA
+VPA
+0.000
+三 0.4
+0.0F
+-0.005
+-0.5
+-0.010
+0.3
+1.0
+-0.015
+79 
+.73 /
+1.5
+=Φ
+Φ=
+-0.020
+150
+50
+0.2E
+0
+2
+4
+5
+0
+2
+3
+5
+6
+0
+2
+3
+4
+5
+6
+0
+Φ
+Φ
+R=0 kpc
+R=500 kpc
+R=1000 kpc
+R=1500 kpcPcPcPcPc7
+FIG. 3: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for
+different fluid direction angle χ. ρc is fixed to be 1.936×107 M⊙·kpc−3. Here rs = 6, a = 0.3, θo = 20◦,
+βν = 0.3, Br = 0.87, Bφ = 0.5 and Bθ = 0.
+can be expressed as
+κ = pif j(linj − ljni − mi ¯mj + ¯mimj)Ψ(− 1
+3)
+2
+,
+(20)
+with
+κ = κ1 + iκ2 = (A − iB)Ψ
+− 1
+3
+2
+,
+(21)
+A = (ptf r − prf t) + a sin θ2(prf φ − pφf r),
+B =
+�
+(r2 + a2)(pφf θ − pθf φ) − a(ptf θ − pθf t)
+�
+sin θ.
+Here Ψ2 is the Weyl scalar with the form
+Ψ2 =
+1
+r2 + a2 cos θ2
+� r2
+12h′′(r) −
+�
+r2 + 2iar cos θ
+�
+6 (r − ia cos θ) h′(r) +
+�
+r2 + 4iar cos θ − a2 cos θ2�
+6(r − ia cos θ)2
+h(r)
+− 6Mr + r2 + a cos θ(6iM + 4ir − a cos θ)
+6(r − ia cos θ)2
+�
+,
+(22)
+where h(r) =
+�
+1 + r
+˜
+R
+�− 8πGρc ˜
+R3
+c2r
+. The Walker-Penrose constant build a bridge between polarization vectors at
+
+-1200
+120
+1.5
+0.006 E
+0.40b
+0.004
+1.0E
+0.35
+0.002
+0.5
+EVPA
+AEVPA
+0.000
+0.30
+0.0
+-0.002
+N
+0.25[
+-0.5
+-0.004
+0.20
+-1.0E
+-0.006
+43元
+269元
+-0.008
+0.15
+1.5
+50 :
+150 :
+0
+2
+3
+4
+5
+6
+0
+2
+3
+5
+0
+2
+3
+4
+6
+Φ
+(=-150°
+X=-150°
+0.35F
+1.5
+0.015
+1.0E
+0.30
+0.5
+△EVPA
+0.010
+0.25
+EVPA
+0.0 E
+0.20
+-0.5
+0.005
+-1.0
+0.15
+-1.5
+0.000
+0
+0
+2
+3
+4
+5
+6
+0
+2
+3
+4
+5
+6
+0
+2
+3
+4
+5
+6
+X
+Φ
+Φ
+Φ
+-180°
+-180°
+X=-180°
+180°
+1.5
+0.020
+0.30
+1.0[
+0.5 
+0.015
+0.25
+EVPA
+VPA
+三
+0.0 E
+0.010
+0.20
+-0.5 E
+0.005
+-1.0F
+0.15
+0.000 
+0
+0
+2
+3
+4
+5
+6
+0
+2
+3
+4
+5
+6
+0
+1
+2
+3
+1
+4
+5
+6
+x
+Φ
+Φ
+R=0 kpc
+R=500 kpc
+R=1000 kpc
+R=1500 kpc8
+FIG. 4: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for
+different observer inclination angle θo. ρc is fixed to be 1.936×107 M⊙·kpc−3. Here rs = 6, a = 0.3, βν = 0.3,
+χ = −90◦, Br = 0.87, Bφ = 0.5 and Bθ = 0.
+the source and at the observer. Using the celestial coordinates (x, y) and the Penrose-Walker constant κ at
+the source, the polarization vector on the observer’s screen can be expressed as [41, 42, 49]
+f x =
+�
+−1 − 5ξ
+12 + ξ
+2 ln
+�R0
+˜R
+�� 1
+3 yκ2 − µκ1
+µ2 + y2 ,
+(23)
+f y =
+�
+−1 − 5ξ
+12 + ξ
+2 ln
+�R0
+˜R
+�� 1
+3 yκ1 + µκ2
+µ2 + y2 ,
+µ = −(x + a sin θo).
+Where the quantity ξ = − 8πGρc ˜
+R3
+c2
+. R0 is the distance between the centre of the black hole and the observer.
+The intensity of the linearly polarized synchrotron radiation from hot gas near the black hole to the observer
+can be expressed as [10, 11]
+|I| = g3+ανlp| ⃗B|1+αν sin ζ1+αν,
+(24)
+Here, g is the redshift factor of the photon travelling from the source to the observer. The quantity lp is the
+geodesic path length which the photon through the emitting material and has a form lp = p(t)
+s
+p(z)
+s H. The height
+of the disk H can be taken be a constant 1. The power αν of the magnetic field ⃗B is related to the properties
+of the accretion disk and can be set to αν = 1 [10, 11]. Then, the observed components of the polarization
+vector can be expressed as
+f x
+obs =
+�
+lpg2|B| sin ζf x,
+f y
+obs =
+�
+lpg2|B| sin ζf y.
+(25)
+
+0,=50°
+=50
+=50°
+1.5
+0.010
+1.2
+1.0
+0.005
+1.0
+0.5
+0.000
+0.8 
+EVPA
+-0.005
+0.0
+山
+0.6
+-0.010
+-0.5
+-0.015
+0.4 
+-1.0
+-0.020
+101 元 
+_ 269 元:
+0.2
+1.5
+-0.025
+100
+150 
+0
+2
+3
+4
+6
+0
+3
+5
+6
+0
+2
+3
+4
+5
+6
+Φ
+Φ
+800
+0=800
+0。=80
+080
+1.5[
+0.06[
+5
+1.0F
+0.04
+4
+0.5
+EVPA
+AEVPA
+0.02
+J=
+0.0
+-0.5
+0.00
+-1.0
+-0.02
+101元
+-1.5
+100
+150
+U
+3
+5
+0
+2
+3
+5
+6
+0
+2
+3
+4
+5
+6
+Φ
+Φ
+Φ
+R=0 kpc
+R=500 kpc
+R=1000 kpc
+R=1500 kpc。=50°o=50°o=80°o=80°A=8C9
+FIG. 5: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for
+different spin parameter a. ρc is fixed to be 1.936×107 M⊙·kpc−3. Here rs = 6, χ = −90◦, θo = 20◦,
+βν = 0.3, Br = 0.87, Bφ = 0.5 and Bθ = 0.
+The total polarization intensity and the EVPA in the observer’s screen can be expressed as
+I = (f x
+obs)2 + (f y
+obs)2 ,
+EVPA = 1
+2 arctan U
+Q,
+(26)
+where Q and U are the Stokes parameters
+Q = (f y
+obs)2 − (f x
+obs)2 ,
+U = −2f x
+obsf y
+obs.
+(27)
+For a rotating black hole surrounded by a CDM halo (1), the polarization intensity and EVPA in the pixel
+related to the point source can be obtained by making use of the set of celestial coordinates (x, y) and Eqs.
+
+a=0,Br=1,Bs=0,Be=0
+a=0,Br=1,Bμ=0,Be=0
+a=0,Br=1,Bμ=0,Be=0
+a=0,Br=1,Bμ=0,B=0
+1.5E
+0.50F
+1.0
+-0.005
+0.45
+0.5
+△EVPA
+EVPA
+三 0.40
+0.0
+-0.010
+0.35
+-0.5
+-1.0
+-0.015
+0.30F
+1.5
+0.25
+3
+4
+5
+0
+2
+3
+4
+5
+6
+6
+0
+2
+3
+4
+5
+6
+=-0.3,B,=1,Bμ=0,Be=0
+a=-0.3,Br=1,Bμ=0,B=0
+-0.3,Br=1,B=0,Bg=0
+a=-0.3,Br=1,B=0,Be=0
+0.55
+1.5
+0.50
+1.0E
+-0.005
+0.45[
+0.5E
+VPA
+三 0.40
+0.0E
+△EV
+-0.010
+0.35
+-0.5
+0.30
+-1.0E
+-0.015
+0.25E
+1.5
+0
+5
+6
+2
+3
+4
+5
+6
+0
+3
+4
+5
+2
+4
+Φ
+Φ
+a=0,Br=0,Bμ=1,Bg=0
+=0,Br=0,Bμ=1,Be=0
+a=0,Br=0,Bμ=1,B=0
+0.6
+1.5
+0.015
+1.0E
+0.010
+0.5/
+0.005
+0.5
+EVPA
+EVPA
+0.000
+三 0.4
+0.0
+-0.005
+-0.5
+-0.010
+0.3
+.1.0
+-0.015
+-1.5
+-0.020
+0.2
+100
+300
+0
+2
+2
+4
+5
+6
+3
+0
+3
+4
+6
+C
+5
+7
+6
+Φ
+Φ
+X
+a=-0.3,Br=0,B=1,Bg=0
+-0.3,B,=0,Bμ=1,Bg=0
+a=-0.3,B,=0,B=1,Bg=0
+-0.3,Br=0,B=1,Be=0
+0.6k
+1.5
+1.0E
+0.01
+0.5
+0.5E
+EVPA
+△EVPA
+0.00
+0.4
+0.0
+-0.5
+-0.01
+0.3
+1.0
+143 /
+229 元
+-1.5
+-0.02
+300
+150
+0.2上
+0
+3
+4
+0
+0
+2
+4
+5
+6
+2
+5
+3
+4
+5
+6
+0
+3
+7
+X
+Φ
+Φ
+R=0 kpc
+R=500kpc
+R=1000 kpc
+R=1500 kpc10
+FIG. 6: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for
+different χ in the case with the magnetic field owned only the vertical component Bθ. ρc is fixed to be
+1.936×107 M⊙·kpc−3. Here rs = 6, a = 0.3, θo = 20◦, βν = 0.3, Br = 0, Bφ = 0 and Bθ = 1.
+(20), (23), (25), (26) and (27). The effects of dark matter halo on the total polarization image can be presented
+through repeating similar operations along the emitting ring.
+III.
+EFFECTS OF THE CDM HALO ON THE POLARIZED IMAGE OF A BLACK HOLE
+Now, we can probe the effects of the CDM halo on the polarization image of the equatorial emitting ring
+with radius rs = 6 around the rotating black hole (1). Figs.(1) and (2) show the effects of the density ρc and
+the characteristic radius ˜R of CDM halo on the polarized image of the emitting ring in the case where the
+magnetic field lies in the equatorial plane. With increasing ρc or ˜R, the polarization intensity decreases, while
+
+X==-90°
+1.5E
+0.00
+0.15
+1.0 E
+-0.05
+0.5
+EVPA
+△EVPA
+0.0
+-0.10
+三 0.10
+0.5
+0.15
+0.05
+1.0
+0.20
+1.5
+19元
+289 元
+150
+150
+6
+0
+3
+4
+5
+6
+0
+2
+3
+4
+5
+3
+4
+5
+2
+6
+Φ
+Φ
+K==120°
+=-120°
+-120°
+1.5//
+0.00
+0.16
+1.0[
+-0.02
+0.14
+0.5
+-0.04 E
+0.12
+EVPA
+AEVPA
+0.0
+-0.06
+0.10
+-0.5
+-0.08
+0.08
+-1.0
+-0.10E
+0.06
+191元
+25
+0.04
+-0.12
+100 
+-5
+0
+0
+1
+2
+5
+6
+0
+2
+3
+4
+6
+0
+2
+3
+5
+6
+x
+Φ
+Φ
+Φ
+X=-150°
+X=-150°
+X==150°
+X=-150°
+1.5E
+0.02日
+0.16E
+1.0
+0.14F
+0.00
+0.5E
+0.12
+0.02
+EVPA
+0.0E
+0.10
+0.5
+0.08F
+1.0
+-0.06
+0.06
+31元
+523元
+1.5
+100
+-0.08
+300
+0
+0
+1
+2
+4
+5
+6
+0
+2
+3
+4
+5
+6
+0
+2
+3
+4
+5
+6
+-5
+5
+X
+Φ
+Φ
+Φ
+X=-180°
+=-180°
+X=-180°
+-180°
+0.16F
+1.5
+0.14
+1.0
+0.02
+0.5
+0.12
+0.00
+EVPA
+AEVPA
+0.0 E
+0.10
+-0.02
+-0.5E
+0.08F
+-1.0 E
+-0.04
+143 /
+151元
+0.06
+-1.5 E
+-0.06
+300
+100
+0
+1
+2
+3
+5
+6
+0
+1
+2
+3
+4
+5
+6
+4
+2
+4
+5
+6
+X
+Φ
+Φ
+Φ
+R=0 kpc
+R=500kpc
+R=1000kpc
+R=1500kpc11
+FIG. 7: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for
+different θo in the case with the magnetic field owned only the vertical component Bθ. ρc is fixed to be
+1.936×107 M⊙·kpc−3. Here rs = 6, a = 0.3, θo = 20◦, βν = 0.3, Br = 0, Bφ = 0 and Bθ = 1.
+FIG. 8: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for
+different a in the case with the magnetic field owned only the vertical component Bθ. ρc is fixed to be
+1.936×107 M⊙·kpc−3. Here rs = 6, a = 0.3, θo = 20◦, βν = 0.3, Br = 0, Bφ = 0 and Bθ = 1.
+the EVPA decreases in the case with pure radial magnetic field and the change of the EVPA depends on the
+coordinate φ in the case of the pure angular magnetic field. In other words, the effects of the density ρc are
+similar to those of the characteristic radius ˜R. It can be explained by a fact that the metric function f(r) in
+Eq. (1) decreases with ρc and ˜R. Thus, in the following parts, we only discuss the effects of the characteristic
+
+。=50°
+6=50°
+0560°
+0=560°
+0.7 
+0.5
+0.10
+0.6
+0.05F
+0.5[
+0.0
+0.4 
+EVPA
+EVPA
+0.00
+0.3
+-0.05
+-0.5
+0.2
+-0.10
+0.1
+523元
+1.0
+0.15
+Φ
+00
+300 :
+0.05
+0
+2
+3
+5
+6
+0
+2
+3
+4
+5
+6
+0
+2
+3
+4
+5
+6
+Φ
+Φ
+Φ
+0。=800
+0。=80°
+@880°
+@,880°
+0.06
+5
+0.2
+0.04
+0.0
+0.02
+0.00
+EVP
+-0.2
+-0.02
+2
+-0.4
+-0.04
+-0.6
+-0.06
+59 元
+131元
+150
+100
+0
+2
+1
+3
+5
+6
+0
+1
+2
+3
+4
+5
+6
+0
+1
+2
+3
+4
+5
+6
+x
+Φ
+d
+Φ
+R=0 kpc
+R=500 kpc
+R=1000 kpc
+R=1500 kpc。=50°=50°0=80°。=80°a=0,Br=0,Bμ=0,B=1
+a=0,Br=0,Bμ=0,Be=1
+a=0,Br=0,Bμ=0,Be=1
+=0.Br=0.B=0.Be=1
+1.5
+0.00
+0.15
+1.0
+-0.05
+0.5
+△EVPA
+EVPA
+0.0
+-0.10
+三 0.10
+-0.5
+-0.15
+0.05
+-1.0
+-5
+-0.20
+19 /
+,191 元
+d=
+150
+100
+0
+2
+3
+4
+5
+6
+3
+9
+0
+2
+3
+5
+6
+Φ
+Φ
+Φ
+-0.3,Br=0,B=0,B=1
+a=-0.3,Br=0,Bμ=0,Be=1
+a=-0.3,Br=0,Bμ=0,Bs=1
+=-0.3,Br=0,Bμ=0,Be=1
+1.5
+0.00
+0.15
+1.0 
+-0.05
+0.5 
+EVPA
+VPA
+0.0
+-0.10
+三 0.10
+AEV
+-0.5E
+0.15
+0.05
+-1.0
+-0.20
+19 元
+_142 元
+150
+75.
+0
+2
+3
+4
+5
+6
+2
+3
+4
+5
+6
+0
+2
+3
+4
+5
+6
+Φ
+Φ
+Φ
+R=0 kpc
+R=500 kpc
+R=1000 kpc
+R=1500 kpc12
+radius ˜R of the CDM halo on the polarized image.
+Figs.(3)-(5) present respectively the effects of the radius ˜R of the CDM halo on the polarized image in the
+case the magnetic field lies in the equatorial plane for different fluid direction angles χ, observer inclination
+angles θo and spin parameters of the black hole. When χ changes from −120◦ to −180◦, the polarization
+intensity still decreases with ˜R, the change of the EVPA becomes more complicated. The region where the
+EVPA increases with ˜R becomes gradually broader with χ, so that the EVPA finally becomes a increasing
+function of ˜R. This is also shown in the change of the quantity ∆EVPA ≡ EVPA − EVPAK, where EVPAK
+denotes the corresponding EVPA in the Kerr black hole spacetime. Similarly, with the increase of the observer
+inclination angle θo, the polarization intensity decreases with ˜R and the region where the EVPA increases with
+˜R becomes gradually broader. However, the EVPA finally does not become a increasing function of ˜R in the
+high spin case. Moreover, one can find that in the rotating black hole case the dependence of the polarization
+intensity and EVPA on the CDM halo radius ˜R is qualitatively similar to that in the non-rotating case.
+Figs.(6)-(8) also present the effects of ˜R on the polarized image in the case where the magnetic field is
+perpendicular to the equatorial plane. For different χ, the changes of the polarization intensity are different
+from those in the case with the pure equatorial magnetic field. As the χ changes from −90◦ to −180◦, the
+region where the polarized intensity decreases with ˜R becomes broad, while the region where EVPA decrease
+with ˜R becomes narrow. However, in the case of χ = −90◦, the decreasing of EVPA with ˜R is dominated.
+With the increase of θo, the dependence of the polarization intensity on the angular coordinate φ is changed,
+but the polarization intensity still decreases with ˜R. The region where EVPA decrease with ˜R becomes narrow,
+which is similar to that in the case where the χ changes from −90◦ to −180◦. Fig.(8) shows the changes of
+the polarization intensity and EVPA with ˜R are also qualitatively similar for the different spin parameter a
+in this case.
+Finally, Figs.(9)-(10) show the effects of ˜R on the Q − U loops patterns in the image of the emitting
+ring around a rotating black hole surrounded by the CDM halo, which depend heavily on the magnetic field
+configuration, the fluid velocity, the observation inclination angle and the spin parameter of the black hole.
+In the case with the lower observed inclination, the size of the two loops decreases with the increase of ˜R as
+the magnetic field lies in the equatorial plane, while in the case of the magnetic field being perpendicular to
+the equatorial plane, the size of the outer loop decreases and the inner loop increases. For the case of the
+magnetic field lies in the equatorial plane, the sizes of two loops decrease with ˜R although the inner loop
+dramatically shrinks in the higher observed inclination case. For the case where the direction of magnetic field
+
+13
+FIG. 9: Effects of ˜R on the Q − U diagram for different χ in the rotating black hole spacetime (1). Here
+rs = 6, θo = 20◦, a = 0.3, and βν = 0.3. As ρc is fixed to be 1.936×107 M⊙·kpc−3. The red solid line and the
+blue dashed line correspond to the cases with the ˜R = 0 kpc and ˜R = 1500 kpc, respectively. Black
+crosshairs indicate the origin of each plot.
+FIG. 10: Effects of ˜R on the Q − U diagram for different θo in the rotating black hole spacetime (1). Here
+rs = 6, χ = −90◦, a = 0.3, and βν = 0.3. As ρc is fixed to be 1.936×107 M⊙·kpc−3. The red solid line and
+the blue dashed line correspond to the cases with the ˜R = 0 kpc and ˜R = 1500 kpc, respectively. Black
+crosshairs indicate the origin of each plot.
+
+X=-120°,Br=0.87,B=0.5,Be=0
+X=-150°Br=0.87,B=0.5,Bg=0
+X=-180°,Br=0.87,Ba=0.5,Be=0
+0.6
+0.6
+0.6
+0.4
+0.4
+0.4
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+0.2
+0.2
+-0.2
+0.4
+0.4
+0.4
+0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.6
+0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+Q
+Q
+Q
+X=-120°Br=0,B=0,Be=1
+-150°B=0,B=0,Bg=1
+=-180°,B,=0,B=0,B=1
+0.2
+0.1
+0.1
+0.1
+0.0
+0.0
+0.0
+-0.1
+-0.1
+0.1
+-0.2
+0.2
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+0.2
+-0.1
+0.1 
+0.2
+-0.2
+0.3
+0.0
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+Q
+Q
+Q800B-087B05BE0
+。=20°,B,=0.87,B=0.5,Bg=0
+=50°B,=0.87,B=0.5,Be=0
+1.5
+0.5
+1.0
+0.5
+0.0
+0.0
+0.5
+0.5
+ 0.5
+0.0
+0.5
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+2
+0
+2
+4
+6
+X
+Q
+Q
+Q
+=20°,B,=0,Bμ=0,Bg=1
+=50°B,=0,B=0,Bg=1
+=80°,Br=0,B=0,B=1
+0.2
+0.4
+0.5
+0.2
+0.1
+0.0
+0.0
+U
+0.0
+U
+-0.5
+0.2
+0.1
+-1.0
+0.4
+0.2
+0.6
+0.2
+.15
+0.3
+0.2
+0.1
+0.0
+0.1
+0.2
+=0.8
+0.6
+0.4
+0.0
+0.2
+-8
+-6
+2
+0
+Q
+Q
+Qo=80°,B,=0.87,B=0.5,Bg=0o6o14
+is perpendicular to the equatorial plane, the size of the outer loops decreases and the inner loops increases
+with ˜R in the lower observed inclination θo = 20◦. However, in the higher inclination angle case, the inner
+loop vanishes and the change of loop size with the CDM halo parameter ˜R becomes more complicated.
+IV.
+SUMMARY
+We have studied the polarized image of an equatorial emitting ring around the rotating black hole surrounded
+by a CDM halo. Results show that the effects of the CDM halo density ρc on the polarization intensity and
+EVPA are similar to those of the characteristic radius ˜R of the halo. The changes of the polarization intensity
+and EVPA with ˜R also depend on the magnetic field configuration, the fluid velocity and the observed
+inclination angle in the spacetime of the rotating black hole surrounded by a CDM halo. When the magnetic
+field lies in the equatorial plane, with the increase of ˜R, the polarization intensity decreases, but in the case
+of the pure radial magnetic field, the EVPA only decreases and the change of the EVPA depends on the
+coordinate φ. When the angle χ changes from −90◦ to −180◦, the region where the EVPA increases with ˜R
+becomes gradually broader, so that the EVPA finally becomes a increasing function of ˜R. When the magnetic
+field is perpendicular to the equatorial plane, as the χ changes from −90◦ to −180◦, the region where the
+polarized intensity decreases with ˜R becomes broad, while the region where EVPA decrease with ˜R becomes
+narrow. However, in the case of χ = −90◦, the decreasing of EVPA with ˜R is dominated. With the increase of
+θo, the polarization intensity also decreases with ˜R, and the region where the EVPA decreases with ˜R becomes
+narrow.
+We also present the effects of ˜R on the Q−U loops, which depend heavily on the magnetic field configuration,
+the fluid velocity, the observation inclination angle and the spin parameter of the black hole. In the case with
+the lower observed inclination, the size of the two loops decreases with the increase of ˜R as the magnetic
+field lies in the equatorial plane, and the size of the outer loop decreases and the inner loop increases as the
+magnetic field is vertical to the equatorial plane. In the higher inclination angle case, the inner loop vanishes
+and the change of loop size with the CDM halo parameter ˜R becomes more complicated. Moreover, for the
+fixed ˜R or ρc, the dependence of polarization image on the magnetic field configuration, the fluid velocity and
+the observed inclination angle in rotating black hole surrounded by a CDM halo are similar to those in other
+rotating black hole cases.
+Finally, our results also indicate that the influence of the CDM halo on the polarized image of the emitting
+ring around the black hole (1) is generally minor, which is consistent with the effects of dark matter halo
+
+15
+on black hole shadows [38, 39]. This implies that the detection of these effects from the CDM halo is out of
+the reach of the current astronomical instruments. With the increasing accuracy and resolution of the future
+astronomical observations and the technological development, it is expected that these effects of the CDM
+halo on the polarize image of black holes can be detected.
+V.
+ACKNOWLEDGMENTS
+This work was supported by the National Natural Science Foundation of China under Grant No.12275078
+and 12035005.
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+
diff --git a/UNAzT4oBgHgl3EQflv0l/content/tmp_files/load_file.txt b/UNAzT4oBgHgl3EQflv0l/content/tmp_files/load_file.txt
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@@ -0,0 +1,1147 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf,len=1146
+page_content='Polarized image of a rotating black hole surrounded by a cold dark matter halo  Xin Qin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Songbai Chen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Zelin Zhang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Jiliang Jing1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2 †  1 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Key Laboratory of Low Dimensional  Quantum Structures and Quantum Control of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Synergetic Innovation Center for Quantum Effects and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Changsha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Hunan 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' People’s Republic of China  2Center for Gravitation and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Yangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Yangzhou 225009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' People’s Republic of China  Abstract  We have studied the polarized image of an equatorial emitting ring around a rotating black hole  surrounded by a cold dark matter (CDM) halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Results show that the CDM halo density has the  similar effects of the halo’s characteristic radius on the polarized image for the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The  effects of the CDM halo on the polarized image depend on the magnetic field configuration, the  fluid velocity and the observed inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' With the increase of the CDM halo parameters, the  observed polarization intensity decreases when the magnetic field lies in equatorial plane, but in the  case where the magnetic field is perpendicular to the equatorial plane, the change of the observed  polarization intensity with CDM halo also depends on the position of the emitting point in the  ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The change of the electric vector position angle (EVPA) with the CDM halo becomes more  complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Our results also show that the influence of the CDM halo on the polarized image is  generally small, which are consistent with the effects of dark matter halo on black hole shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  These results could help to further understand dark matter from black hole images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  PACS numbers:  04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='Cs, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='Mw, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='Lf  ∗ Corresponding author: csb3752@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='cn  † jljing@hunnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='cn  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  INTRODUCTION  The releasing of black hole images of M87* [1–6] and Sgr A*[7] by the Event Horizon Telescope (EHT)  collaboration, together with the corresponding polarized patterns [8, 9], means that the observational black  hole astronomy has been entered an new era of rapid progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' It is helpful to confirm the existence of black  holes, to test general relativity and to explore matter distribution around black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The polarized image of  M87* [8, 9] revealed that there is a strong magnetic field around the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' It is beneficial to understand  the process of accretion and the generation of jet in the vicinity of black holes in universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Therefore, a lot of  polarized images of black holes have been studied in both theoretical and experimental aspects [10–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  In general, numerical simulations are applied to obtain an accurate description of polarization image of a  black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' However, they are computationally expensive because of the broad parameter surveys and the  complicated couplings among astrophysical and relativistic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Recently, a simple model of an equatorial  ring of magnetized fluid has been proposed to study the polarized images of synchrotron emission around  Schwarzschild [10] and Kerr black hole [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Although only the emission from a single radius is considered in  this model, it can clearly reveal the dependence of the polarization signatures on the magnetic field config-  uration, the black hole spin and the observer inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' In addition, the studies [10, 11] also indicate that  the ring model image is broadly consistent with the polarization morphology of the EHT image, even though  there is a high fractional polarization after blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Thus, the simple ring model has been recently applied to  study the polarized image of an equatorial emitting ring in various spacetimes, such as, a 4D Gauss-Bonnet  black hole spacetime [12], a rotating black hole spacetime in the STVG-MOG theory [13], regular black hole  spacetimes [16], a Schwarzschilld-Melvin black hole spacetime [17], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Dark matter is a theoretical model to explain the discrepancy between the observed dynamics and the  amount of luminous matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' However, the nature of dark matter is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Dark matter is assumed to  be an invisible matter and has feeble couplings with the common visible matter at most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Despite extensive  observational data supporting its presence on a large scale [31–33], dark matter has not been directly detected  at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  The CDM model is one of the most popular dark matter models [34, 35], which has a good  consistency with the observation of large scale structure of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The spacetime metric of Schwarzschild  black hole and Kerr black hole surrounded by a CDM halo were studied in [36] and the effects of the CDM  halo on the quasinormal modes have also been studied in the spherical symmetric black hole spacetime [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  The studies of black hole shadows [38, 39] show that the effects of the CDM halo on shadows are generally  3  small and only become significant when the CDM parameter k = ρc ˜R3 increases to order of magnitude of 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  However, they are very important for understanding dark matter from black hole shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Along this line, we  here will study the polarized image of the equatorial emitting ring around a rotating black hole surrounded  by a CDM halo and probe the effects of the CDM halo on the polarization image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  The paper is organized as follows: Section II briefly introduces the rotating black hole surrounded by a CDM  halo and presents the calculation formulas for the observed polarization vector in the image of an emitting ring  in this spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Section III presents the polarization images of the synchrotron emitting ring and probes  the effects of the CDM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Finally, this paper ends with a summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  OBSERVED POLARIZATION FIELD IN A ROTATING BLACK HOLE SPACETIME  SURROUNDED BY DARK MATTER HALO  The metric analytical form of a rotating black hole surrounded by a CDM halo was obtained in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' In the  Boyer-Lindquist coordinates, it has the form  ds2 = −  �  1 − r2 − f(r)r2  ρ2  �  dt2 + ρ2  ∆ dr2 + 2  �  r2 − f(r)r2�  a sin2 θ  ρ2  dφdt+  ρ2dθ2 + sin θ2  ρ2  ��  r2 + a2�2 − a2∆ sin θ2�  dφ2,  (1)  with  f(r) =  �  1 + r  ˜R  �− 8πGρc ˜  R3  c2r  − 2GM  rc2 ,  ∆ = r2f(r) + a2,  ρ2 = r2 + a2 cos θ2,  (2)  where M is the mass of the black hole, G and c are Newtonian gravitational constant and the speed of light,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is the density of the halo at the moment when the dark matter halo collapsed and ˜R is the  corresponding characteristic radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' According to the observations on the Sgr A∗ [38], the best fit values of  the rotation curve of the Galaxy are ρc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙kpc−3 and ˜R = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='46 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The metric can reduce to  the Kerr metric in the limit ρc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here, we set G = 1 and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  In the spacetime of a rotating black hole surrounded by a CDM halo (1), the geodesic equation for photons  can be expressed as  ρ2  E pt = r2 + a2  ∆  �  r2 + a2 − aλ  �  + a  �  λ − a sin θ2�  ,  (3)  ρ2  E pφ = a  ∆  �  r2 + a2 − aλ  �  +  λ  sin θ2 − a,  ρ2  E pr = ±r  �  R(r),  ρ2  E pθ = ±θ  �  Θ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  4  The conserved quantities λ and η are the energy-rescaled angular momentum parallel to the axis of symmetry  and Carter constant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The radial potential R(r) and the angular potential Θ(θ) take the form  R(r) =  �  r2 + a2 − aλ  �2 − ∆  �  η + (a − λ)2�  ,  (4)  Θ(θ) = η + a2 cos θ2 − λ2 cot θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Utilizing the null geodesic Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (3), the coordinates (x, y) for the photon’s arrival position on the observer’s  screen can be obtained as  x = −  λ  sin θo  ,  y = ±o  �  Θ(θ),  (5)  where θo is the observed inclination angle from the normal direction of the accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The radial and  angle integrals of the photon trajectories from the initial position (rs, θs) to the final initial (ro, θo) can be  expressed as [11, 47]  Ir ≡  ro  rs  dr  ±r  �  R(r)  =  θo  θs  dθ  ±r  �  Θ(θ)  ≡ Gθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (6)  The slash in the path integral denotes that the sign of ±r or ±θ changes when the photon passes through  at the radial or angular turning point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' For a photon’s trajectory with m turning points, the angular path  integral can be written as  Gm  θ =  1  �  −u−a2  �  2mK  �u+  u−  �  − sign(β)Fo  �  ,  (7)  where  Fo = F  �  arcsin cos θo  √u+  ���u+  u−  �  ,  u± = ∆θ ±  �  ∆2  θ + η  a2 ,  ∆θ = 1  2  �  1 − η + λ2  a2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (8)  Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (5), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (6) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (7), one can numerically get the set of celestial coordinates (x, y) for  a photon emitted on an equatorial ring with a radius rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The four-momentum of the photon at the source  (rs, θs = π  2 ) is given by  pt = −1,  pr = ±r  �  R(rs)  ∆s  ,  pθ = ±s  √η,  pφ = λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (9)  Here, the sign of pθ at the source ±s = (−1)m±o and the sign of ±r can be computed by a semi-analytic  method [11, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The four-momentum pµ of photon at the source can be obtained as  pt = 1  r2s  �r2  s + a2  ∆s  �  r2  s + a2 − aλ  �  + a(λ − a)  �  ,  pr = ±r  1  r2s  �  R(rs),  (10)  pφ = 1  r2s  � a  ∆s  �  r2  s + a2 − aλ  �  + λ − a  �  ,  pθ = ±s  √η  r2s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  5  The emitted ring around a rotating black hole surrounded by a CDM halo (1) is assumed to lie in the equatorial  plane (θs = π  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' In the local orthonormal zero-angular-momentum-observer (ZAMO) frame of the point P,  the boosting velocity of the boosted emitter is assumed to be in the r-φ plane and has a form  ⃗β = βν  �  cos χ (ˆr) + sin χ(ˆφ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (11)  In the boosted orthonormal frame, the four-momentum p(a) of point emitter can be obtained by the four-  momentum pµ in a rotating black hole spacetime surrounded by a CDM halo, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=',  p(a) = Λ(a)  (b)η(b)(c)eµ  (c)pµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (12)  where η(b)(c) is the flat Minkowski metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The zero-angular-momentum-observer (ZAMO) tetrad eµ  (c) and  the Lorentz transformation Λ(a)  (b) are respectively given by  eµ  (c) =  �  �����  1  rs  �  Ξs  ∆s  0  ωs  rs  �  Ξs  ∆s  0  0  √∆s  rs  0  0  0  0  rs  √Ξs  0  0  0  0  − 1  rs  �  �����  ,  (13)  and  Λ(a)  (b) =  �  ��  γ  −βνγ cos χ  −βνγ sin χ  0  −βνγ cos χ (γ − 1) cos χ2 + 1 (γ − 1) sin χ cos χ 0  −βνγ sin χ (γ − 1) sin χ cos χ (γ − 1) sin χ2 + 1 0  0  0  0  1  �  �� ,  (14)  where γ =  1  √  1−β2 is the Lorentz factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' In the local frame, the temporal components of polarization vector  f (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Since the three-dimensional electric vector ⃗E of photon is along the direction of vector ⃗p × ⃗B, the  spatial components of polarization vector ⃗f can be expressed as  f (r) = p(φ) × B(θ)  |⃗p|  ,  f (φ) = p(θ) × B(r)  |⃗p|  ,  f (θ) = p(r) × B(φ)  |⃗p|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (15)  Thus, the photon polarization vector in the black hole spacetime can be obtained from the local polarization  vector by using an inverse transformation as  f µ = eµ  (c)Λ  (c)  (a) f (a),  (16)  where Λ  (c)  (a)  is the inverse matrix of Λ(a)  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' And the photon polarization vector satisfies the relationship as  f µfµ = sin ζ2| ⃗B|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (17)  Here, the ζ is the angle between photon momentum ⃗p and the magnetic field ⃗B and has the form  sin ζ = |⃗p × ⃗B|  |⃗p|| ⃗B|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (18)  6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 1: Effects of ρc on the polarized intensity and EVPA in the rotating black hole spacetime (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ˜R is  fixed to be 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='46kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here rs = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, θo = 20◦, βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, and χ = −90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 2: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is  fixed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here rs = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, θo = 20◦, βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, and χ = −90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  As the photon propagate along the null geodesic in the rotating black hole spacetime surrounded by a CDM  halo (1), the polarization vector f µ obeys  f µpµ = 0,  pµ∇µf ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (19)  Since the black hole spacetime (1) belong to a type-D spacetime, the conserved Penrose-Walker constant κ  R=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='46kpc,Br=1,Bμ=0,Bs=0  R=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='46kpc,Br=1,B=0,Bg=0  R=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='46kpc,Br=1,Bg=0,Be=0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='000 E  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='0E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5E  EVPA  EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='006  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='5  0  2  3  4  6  2  4  5  6  0  G  6  5  3  1  3  Φ  Φ  R=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='46kpc,Br=0,B=1,Bg=0  R=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='46kpc,Br=0,Bμ=1,Bg=0  R=17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='46kpc,Br=0,Bμ=1,Ba=0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='6F  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='010F  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='5  △EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='000  EVPA  三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='936x107Mo·kpc-3,Br=1,Bμ=0,Bo=0  Pc=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 Mo:kpc-3,Br=1,Bg=0,Be=0  R=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107Mokpc-3,Br=1,Bg=0,Be=0  =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107Mo:kpc-3,Br=1,Bg=0,Be=0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='0[  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='010E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='005E  EVPA  VPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='000  三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='015  79  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='73 /  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  =Φ  Φ=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='020  150  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2E  0  2  4  5  0  2  3  5  6  0  2  3  4  5  6  0  Φ  Φ  R=0 kpc  R=500 kpc  R=1000 kpc  R=1500 kpcPcPcPcPc7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 3: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for  different fluid direction angle χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is fixed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here rs = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, θo = 20◦,  βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, Br = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='87, Bφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5 and Bθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  can be expressed as  κ = pif j(linj − ljni − mi ¯mj + ¯mimj)Ψ(− 1  3)  2  ,  (20)  with  κ = κ1 + iκ2 = (A − iB)Ψ  − 1  3  2  ,  (21)  A = (ptf r − prf t) + a sin θ2(prf φ − pφf r),  B =  �  (r2 + a2)(pφf θ − pθf φ) − a(ptf θ − pθf t)  �  sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Here Ψ2 is the Weyl scalar with the form  Ψ2 =  1  r2 + a2 cos θ2  � r2  12h′′(r) −  �  r2 + 2iar cos θ  �  6 (r − ia cos θ) h′(r) +  �  r2 + 4iar cos θ − a2 cos θ2�  6(r − ia cos θ)2  h(r)  − 6Mr + r2 + a cos θ(6iM + 4ir − a cos θ)  6(r − ia cos θ)2  �  ,  (22)  where h(r) =  �  1 + r  ˜  R  �− 8πGρc ˜  R3  c2r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The Walker-Penrose constant build a bridge between polarization vectors at  1200  120  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='006 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='40b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='004  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  EVPA  AEVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='002  N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='25[  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='006  43元  269元  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  50 :  150 :  0  2  3  4  5  6  0  2  3  5  0  2  3  4  6  Φ  (=-150°  X=-150°  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='35F  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='015  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  △EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='25  EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='000  0  0  2  3  4  5  6  0  2  3  4  5  6  0  2  3  4  5  6  X  Φ  Φ  Φ  180°  180°  X=-180°  180°  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0[  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='25  EVPA  VPA  三  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='000  0  0  2  3  4  5  6  0  2  3  4  5  6  0  1  2  3  1  4  5  6  x  Φ  Φ  R=0 kpc  R=500 kpc  R=1000 kpc  R=1500 kpc8  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 4: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for  different observer inclination angle θo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is fixed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here rs = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3,  χ = −90◦, Br = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='87, Bφ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5 and Bθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  the source and at the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Using the celestial coordinates (x, y) and the Penrose-Walker constant κ at  the source, the polarization vector on the observer’s screen can be expressed as [41, 42, 49]  f x =  �  −1 − 5ξ  12 + ξ  2 ln  �R0  ˜R  �� 1  3 yκ2 − µκ1  µ2 + y2 ,  (23)  f y =  �  −1 − 5ξ  12 + ξ  2 ln  �R0  ˜R  �� 1  3 yκ1 + µκ2  µ2 + y2 ,  µ = −(x + a sin θo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Where the quantity ξ = − 8πGρc ˜  R3  c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' R0 is the distance between the centre of the black hole and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  The intensity of the linearly polarized synchrotron radiation from hot gas near the black hole to the observer  can be expressed as [10, 11]  |I| = g3+ανlp| ⃗B|1+αν sin ζ1+αν,  (24)  Here, g is the redshift factor of the photon travelling from the source to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The quantity lp is the  geodesic path length which the photon through the emitting material and has a form lp = p(t)  s  p(z)  s H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The height  of the disk H can be taken be a constant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The power αν of the magnetic field ⃗B is related to the properties  of the accretion disk and can be set to αν = 1 [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Then, the observed components of the polarization  vector can be expressed as  f x  obs =  �  lpg2|B| sin ζf x,  f y  obs =  �  lpg2|B| sin ζf y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (25)  0,=50°  =50  =50°  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='005  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='8  EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  山  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='020  101 元  _ 269 元:  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='025  100  150  0  2  3  4  6  0  3  5  6  0  2  3  4  5  6  Φ  Φ  800  0=800  0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='=80  080  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5[  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='06[  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='5  EVPA  AEVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='02  J=  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='02  101元  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  100  150  U  3  5  0  2  3  5  6  0  2  3  4  5  6  Φ  Φ  Φ  R=0 kpc  R=500 kpc  R=1000 kpc  R=1500 kpc。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='=50°o=50°o=80°o=80°A=8C9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 5: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for  different spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is fixed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content=' Here rs = 6, χ = −90◦, θo = 20◦,  βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='  The total polarization intensity and the EVPA in the observer’s screen can be expressed as  I = (f x  obs)2 + (f y  obs)2 ,  EVPA = 1  2 arctan U  Q,  (26)  where Q and U are the Stokes parameters  Q = (f y  obs)2 − (f x  obs)2 ,  U = −2f x  obsf y  obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (27)  For a rotating black hole surrounded by a CDM halo (1), the polarization intensity and EVPA in the pixel  related to the point source can be obtained by making use of the set of celestial coordinates (x, y) and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  a=0,Br=1,Bs=0,Be=0  a=0,Br=1,Bμ=0,Be=0  a=0,Br=1,Bμ=0,Be=0  a=0,Br=1,Bμ=0,B=0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='02  300  150  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2上  0  3  4  0  0  2  4  5  6  2  5  3  4  5  6  0  3  7  X  Φ  Φ  R=0 kpc  R=500kpc  R=1000 kpc  R=1500 kpc10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 6: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for  different χ in the case with the magnetic field owned only the vertical component Bθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is fixed to be  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here rs = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, θo = 20◦, βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, Br = 0, Bφ = 0 and Bθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  (20), (23), (25), (26) and (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The effects of dark matter halo on the total polarization image can be presented  through repeating similar operations along the emitting ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  EFFECTS OF THE CDM HALO ON THE POLARIZED IMAGE OF A BLACK HOLE  Now, we can probe the effects of the CDM halo on the polarization image of the equatorial emitting ring  with radius rs = 6 around the rotating black hole (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (1) and (2) show the effects of the density ρc and  the characteristic radius ˜R of CDM halo on the polarized image of the emitting ring in the case where the  magnetic field lies in the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' With increasing ρc or ˜R, the polarization intensity decreases, while  X==-90°  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='5  EVPA  △EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='10  三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  19元  289 元  150  150  6  0  3  4  5  6  0  2  3  4  5  3  4  5  2  6  Φ  Φ  K==120°  =-120°  120°  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5//  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0[  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='04 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='12  EVPA  AEVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='06  191元  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='12  100  5  0  0  1  2  5  6  0  2  3  4  6  0  2  3  5  6  x  Φ  Φ  Φ  X=-150°  X=-150°  X==150°  X=-150°  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='02日  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='16E  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='14F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='02  EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='06  31元  523元  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='16F  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='00  EVPA  AEVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='5E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='08F  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='04  143 /  151元  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5 E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='06  300  100  0  1  2  3  5  6  0  1  2  3  4  5  6  4  2  4  5  6  X  Φ  Φ  Φ  R=0 kpc  R=500kpc  R=1000kpc  R=1500kpc11  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 7: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for  different θo in the case with the magnetic field owned only the vertical component Bθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is fixed to be  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here rs = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, θo = 20◦, βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, Br = 0, Bφ = 0 and Bθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 8: Effects of ˜R on the polarized intensity and EVPA in the rotating black hole spacetime (1) for  different a in the case with the magnetic field owned only the vertical component Bθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' ρc is fixed to be  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here rs = 6, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, θo = 20◦, βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, Br = 0, Bφ = 0 and Bθ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  the EVPA decreases in the case with pure radial magnetic field and the change of the EVPA depends on the  coordinate φ in the case of the pure angular magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' In other words, the effects of the density ρc are  similar to those of the characteristic radius ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' It can be explained by a fact that the metric function f(r) in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (1) decreases with ρc and ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Thus, in the following parts, we only discuss the effects of the characteristic  。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='=50°  6=50°  0560°  0=560°  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='05F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5[  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='4  EVPA  EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='1  523元  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  Φ  00  300 :  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='05  0  2  3  5  6  0  2  3  4  5  6  0  2  3  4  5  6  Φ  Φ  Φ  0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='=800  0。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='=80°  @880°  @,880°  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='06  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='00  EVP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='02  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='06  59 元  131元  150  100  0  2  1  3  5  6  0  1  2  3  4  5  6  0  1  2  3  4  5  6  x  Φ  d  Φ  R=0 kpc  R=500 kpc  R=1000 kpc  R=1500 kpc。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='=50°=50°0=80°。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='=80°a=0,Br=0,Bμ=0,B=1  a=0,Br=0,Bμ=0,Be=1  a=0,Br=0,Bμ=0,Be=1  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='Br=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='B=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='Be=1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  △EVPA  EVPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='10  三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='20  19 /  ,191 元  d=  150  100  0  2  3  4  5  6  3  9  0  2  3  5  6  Φ  Φ  Φ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3,Br=0,B=0,B=1  a=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3,Br=0,Bμ=0,Be=1  a=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3,Br=0,Bμ=0,Bs=1  =-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3,Br=0,Bμ=0,Be=1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5  EVPA  VPA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='10  三 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='10  AEV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='20  19 元  _142 元  150  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  0  2  3  4  5  6  2  3  4  5  6  0  2  3  4  5  6  Φ  Φ  Φ  R=0 kpc  R=500 kpc  R=1000 kpc  R=1500 kpc12  radius ˜R of the CDM halo on the polarized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (3)-(5) present respectively the effects of the radius ˜R of the CDM halo on the polarized image in the  case the magnetic field lies in the equatorial plane for different fluid direction angles χ, observer inclination  angles θo and spin parameters of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' When χ changes from −120◦ to −180◦, the polarization  intensity still decreases with ˜R, the change of the EVPA becomes more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The region where the  EVPA increases with ˜R becomes gradually broader with χ, so that the EVPA finally becomes a increasing  function of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' This is also shown in the change of the quantity ∆EVPA ≡ EVPA − EVPAK, where EVPAK  denotes the corresponding EVPA in the Kerr black hole spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Similarly, with the increase of the observer  inclination angle θo, the polarization intensity decreases with ˜R and the region where the EVPA increases with  ˜R becomes gradually broader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' However, the EVPA finally does not become a increasing function of ˜R in the  high spin case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Moreover, one can find that in the rotating black hole case the dependence of the polarization  intensity and EVPA on the CDM halo radius ˜R is qualitatively similar to that in the non-rotating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (6)-(8) also present the effects of ˜R on the polarized image in the case where the magnetic field is  perpendicular to the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' For different χ, the changes of the polarization intensity are different  from those in the case with the pure equatorial magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' As the χ changes from −90◦ to −180◦, the  region where the polarized intensity decreases with ˜R becomes broad, while the region where EVPA decrease  with ˜R becomes narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' However, in the case of χ = −90◦, the decreasing of EVPA with ˜R is dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  With the increase of θo, the dependence of the polarization intensity on the angular coordinate φ is changed,  but the polarization intensity still decreases with ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The region where EVPA decrease with ˜R becomes narrow,  which is similar to that in the case where the χ changes from −90◦ to −180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (8) shows the changes of  the polarization intensity and EVPA with ˜R are also qualitatively similar for the different spin parameter a  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Finally, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' (9)-(10) show the effects of ˜R on the Q − U loops patterns in the image of the emitting  ring around a rotating black hole surrounded by the CDM halo, which depend heavily on the magnetic field  configuration, the fluid velocity, the observation inclination angle and the spin parameter of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  In the case with the lower observed inclination, the size of the two loops decreases with the increase of ˜R as  the magnetic field lies in the equatorial plane, while in the case of the magnetic field being perpendicular to  the equatorial plane, the size of the outer loop decreases and the inner loop increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' For the case of the  magnetic field lies in the equatorial plane, the sizes of two loops decrease with ˜R although the inner loop  dramatically shrinks in the higher observed inclination case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' For the case where the direction of magnetic field  13  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 9: Effects of ˜R on the Q − U diagram for different χ in the rotating black hole spacetime (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here  rs = 6, θo = 20◦, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, and βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' As ρc is fixed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The red solid line and the  blue dashed line correspond to the cases with the ˜R = 0 kpc and ˜R = 1500 kpc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Black  crosshairs indicate the origin of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 10: Effects of ˜R on the Q − U diagram for different θo in the rotating black hole spacetime (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Here  rs = 6, χ = −90◦, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3, and βν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' As ρc is fixed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='936×107 M⊙·kpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The red solid line and  the blue dashed line correspond to the cases with the ˜R = 0 kpc and ˜R = 1500 kpc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Black  crosshairs indicate the origin of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  X=-120°,Br=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content='2  8  6  2  0  Q  Q  Qo=80°,B,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='87,B=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='5,Bg=0o6o14  is perpendicular to the equatorial plane, the size of the outer loops decreases and the inner loops increases  with ˜R in the lower observed inclination θo = 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' However, in the higher inclination angle case, the inner  loop vanishes and the change of loop size with the CDM halo parameter ˜R becomes more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  SUMMARY  We have studied the polarized image of an equatorial emitting ring around the rotating black hole surrounded  by a CDM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Results show that the effects of the CDM halo density ρc on the polarization intensity and  EVPA are similar to those of the characteristic radius ˜R of the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The changes of the polarization intensity  and EVPA with ˜R also depend on the magnetic field configuration, the fluid velocity and the observed  inclination angle in the spacetime of the rotating black hole surrounded by a CDM halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' When the magnetic  field lies in the equatorial plane, with the increase of ˜R, the polarization intensity decreases, but in the case  of the pure radial magnetic field, the EVPA only decreases and the change of the EVPA depends on the  coordinate φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' When the angle χ changes from −90◦ to −180◦, the region where the EVPA increases with ˜R  becomes gradually broader, so that the EVPA finally becomes a increasing function of ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' When the magnetic  field is perpendicular to the equatorial plane, as the χ changes from −90◦ to −180◦, the region where the  polarized intensity decreases with ˜R becomes broad, while the region where EVPA decrease with ˜R becomes  narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' However, in the case of χ = −90◦, the decreasing of EVPA with ˜R is dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' With the increase of  θo, the polarization intensity also decreases with ˜R, and the region where the EVPA decreases with ˜R becomes  narrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  We also present the effects of ˜R on the Q−U loops, which depend heavily on the magnetic field configuration,  the fluid velocity, the observation inclination angle and the spin parameter of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' In the case with  the lower observed inclination, the size of the two loops decreases with the increase of ˜R as the magnetic  field lies in the equatorial plane, and the size of the outer loop decreases and the inner loop increases as the  magnetic field is vertical to the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' In the higher inclination angle case, the inner loop vanishes  and the change of loop size with the CDM halo parameter ˜R becomes more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Moreover, for the  fixed ˜R or ρc, the dependence of polarization image on the magnetic field configuration, the fluid velocity and  the observed inclination angle in rotating black hole surrounded by a CDM halo are similar to those in other  rotating black hole cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  Finally, our results also indicate that the influence of the CDM halo on the polarized image of the emitting  ring around the black hole (1) is generally minor, which is consistent with the effects of dark matter halo  15  on black hole shadows [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' This implies that the detection of these effects from the CDM halo is out of  the reach of the current astronomical instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' With the increasing accuracy and resolution of the future  astronomical observations and the technological development, it is expected that these effects of the CDM  halo on the polarize image of black holes can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by the National Natural Science Foundation of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='12275078  and 12035005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [1] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The Shadow of the  Supermassive Black Hole, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L1 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [2] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Array and Instru-  mentation, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L2 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [3] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Date Processing and  Calibration, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L3 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [4] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Imaging the Central  Supermassive Black Hole, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L4 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [5] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Physical origin of the  asymmetric ring, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L5 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [6] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The Shadow and  Mass of the Central Black Hole, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L6 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [7] The Event Horizon Telescope Collaboration, First Sagittarius A∗ Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' The Shadow  of the Supermassive Black Hole in the Center of the Milky Way, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 930, L12 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [8] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Polarization of the  Ring, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L7 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [9] The Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Magnetic Field  Structure near The Event Horizon, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' 875, L8 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='  [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content=' Himwich, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' Johnson, Polarized Image of Equatorial Emission in the Kerr  Geometry, Physical Review D 104 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='4, (2021) 044060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content=' arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
+page_content='09440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UNAzT4oBgHgl3EQflv0l/content/2301.01551v1.pdf'}
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diff --git a/V9FIT4oBgHgl3EQfhStk/content/tmp_files/2301.11287v1.pdf.txt b/V9FIT4oBgHgl3EQfhStk/content/tmp_files/2301.11287v1.pdf.txt
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+Simultaneous Asymmetric Quantum Remote State Preparation
+Scheme in Noisy Environments
+Binayak S. Choudhury1 ∗, Manoj Kumar Mandal1 †,
+Soumen Samanta1 ‡, Biswanath Dolai1,2 §
+1 Department of Mathematics, Indian Institute of Engineering Science and Technology,
+Shibpur B. Garden, Howrah - 711103, West Bengal, India
+2 Department of Physics, Bajkul Milani Mahavidyalaya, Bajkul,
+Purba Medinipur-721655, West Bengal, India
+Abstract In this paper we discuss a quantum multi-tasking protocol for preparation of known
+one-qubit and two-qubit states respectively in two different locations. The ideal remote state
+preparation protocol is discussed in the first place in which a five-qubit entangled state is utilized.
+The design for the preparation of such entanglement is also presented and run on IBM quantum
+composer platform. The effects of three types of noises on the protocol are discussed and in all
+these cases the reduced fidelities of the process are calculated. The variations of these fidelities
+with respect to different parameters are analysed.
+Keywords
+Multi-qubit entangled channel, Bell-like state, Cluster state, Quantum Measure-
+ment, Unitary operator, Noise, Fidelity.
+1
+Introduction
+Quantum teleportation (QT) introduced by Bennett et al. [1] in 1993 plays an important role
+in the history of quantum communication and information theory. It is a technique for transfer
+of unknown quantum information from one location to a distant location through a previously
+shared entanglement between the sender and the receiver.
+Since then, a series of modified
+quantum teleportation protocols such as controlled teleportation [2–9], bi-directional teleporta-
+tion [10–16], probabilistic teleportation [17–22], cyclic teleportation [23–28], multi-hop telepor-
+tation [29–33] and so on were developed in course of the last three decades which are applicable
+to a very wide variety of quantum states.
+There is another class of quantum communication scheme somewhat similar to quantum telepor-
+tation protocols in the sense that its aim is also to create a quantum state at a distant location
+but the prime difference is that the state to be remotely prepared is known to the sender. These
+types of quantum communication schemes are known as quantum remote state preparation (RSP)
+∗e-mail : binayak@math.iiests.ac.in, Fax: +91-33-26682916
+†e-mail: manojmandaliiest@gmail.com, Tel: +91-700395967
+‡corresponding author e-mail : s.samanta.math@gmail.com, Tel: +91-9804842707
+§e-mail : biswanathbmm@gmail.com, Tel: +91-9547360567
+1
+arXiv:2301.11287v1  [quant-ph]  25 Jan 2023
+
+protocols which were first introduced by Pati [34]. Following the work of Pati [34], several re-
+searchers developed various modified forms of quantum remote state preparation schemes such as
+controlled RSP [35–37], joint remote state preparation [38–40], cyclic RSP [41–43], bi-directional
+RSP [44–46], conference RSP [47], etc.
+However, in the process of quantum state transfer or creation of states in remote locations there
+will inevitably be interactions with the environment. As a result of this random coupling with
+the environment, the shared entangled channel is distorted which gives rise to quantum noisy
+channels. In a noisy channel quantum decoherence due to interaction with the environment is
+described by Kraus operators which are in general non-unitary and therefore irreversible. In
+recent times, some remote state preparation schemes through noisy environments are discussed
+in [48–52]. The conventional measure of effectiveness in quantum communication is the fidelity
+between the input state and the output state. The effect of noise is the reduction in fidelity.
+In this paper we describe a quantum communication scheme for remotely preparing a single
+qubit and a two-qubit quantum state respectively at the sites of two different parties. Both the
+states are known to the party intending to remotely create the two states. We first describe
+the ideal case where the task is performed through the utilization of a five-qubit cluster state in
+which there is no infiltration of noise. The preparation of the five-qubit entangled channel used
+here is also discussed. After that we consider the effect of noisy environment in our protocol.
+Specifically we discuss three types of noises Amplitude-damping, Phase-flip and Bit-flip noise.
+Due to the presence of noise the fidelity of the process which is a measure of effectiveness of
+the communication is decreased. We make a study of the variation of fidelity with respect to
+different parameters of the scheme.
+2
+Simultaneous quantum remote state preparation in ideal case
+Suppose there are three parties, namely, Alice, Bob and Candy who are situated at three different
+locations. Let us consider an arbitrary single-qubit and a two-qubit quantum state which are
+respectively given by
+|S1⟩ = (α|0⟩ + β|1⟩),
+(1)
+|S2⟩ = (γ|00⟩ + δ|11⟩),
+(2)
+where the coefficients α, β, γ and δ meet the normalization conditions, that is, |α|2 + |β|2 = 1
+and |γ|2 + |δ|2 = 1.
+A five-qubit cluster-state is given by
+|Φ⟩ = 1
+2(|00000⟩ + |01011⟩ + |10100⟩ − |11111⟩)12345,
+(3)
+which acts as the quantum channel.
+Let us consider the situation where Alice intends to remotely create the two states given in (1)
+and (2) in the locations of Bob and Candy respectively. Alice wants to prepare the quantum
+state |S1⟩ at Bob’s laboratory and at the same time she wants to create the quantum state |S2⟩ at
+Candy’s site. The sender Alice possesses the knowledge of the coefficients α, β, γ and δ. For this
+purpose Alice, Bob and Candy share a five-qubit cluster-state given in (3), in which the parti-
+cles (1, 2) belong to Alice, particle 3 belongs to Bob and particles (4, 5) are in the hands of Candy.
+2
+
+For our purpose we rewrite the quantum entangled channel given in (3) as
+|Φ⟩ = 1
+2(|00000⟩ + |01011⟩ + |10100⟩ − |11111⟩)a1a2b1c1c2.
+(4)
+Alice performs a 2-qubit measurement in the basis consisting of four linearly independent vectors
+given by
+|ξ1⟩ = (αγ|00⟩ + αδ|01⟩ + βγ|10⟩ − βδ|11⟩)a1a2,
+|ξ2⟩ = (αδ|00⟩ − αγ|01⟩ − βδ|10⟩ − βγ|11⟩)a1a2,
+|ξ3⟩ = (βγ|00⟩ + βδ|01⟩ − αγ|10⟩ + αδ|11⟩)a1a2,
+|ξ4⟩ = (βδ|00⟩ − βγ|01⟩ + αδ|10⟩ + αγ|11⟩)a1a2.
+(5)
+The choice of such a measurement base by Alice is possible since the coefficients α, β, γ, δ are
+known to Alice.
+We can express the above quantum channel given in (4) using the basis (5) as
+|Φ⟩ = 1
+2
+�
+|ξ1⟩a1a2 ⊗ (αγ|000⟩ + αδ|011⟩ + βγ|100⟩ + βδ|111⟩)b1c1c2
++ |ξ2⟩a1a2 ⊗ (αδ|000⟩ − αγ|011⟩ − βδ|100⟩ + βγ|111⟩)b1c1c2
++ |ξ3⟩a1a2 ⊗ (βγ|000⟩ + βδ|011⟩ − αγ|100⟩ − αδ|111⟩)b1c1c2
++ |ξ4⟩a1a2 ⊗ (βδ|000⟩ − βγ|011⟩ + αδ|100⟩ − αγ|111⟩)b1c1c2
+�
+= 1
+2
+�
+|ξ1⟩a1a2 ⊗ (α|0⟩ + β|1⟩)b1 ⊗ (γ|00⟩ + δ|11⟩)c1c2
++ |ξ2⟩a1a2 ⊗ (α|0⟩ − β|1⟩)b1 ⊗ (δ|00⟩ − γ|11⟩)c1c2
++ |ξ3⟩a1a2 ⊗ (β|0⟩ − α|1⟩)b1 ⊗ (γ|00⟩ + δ|11⟩)c1c2
++ |ξ4⟩a1a2 ⊗ (β|0⟩ + α|1⟩)b1 ⊗ (δ|00⟩ − γ|11⟩)c1c2
+�
+.
+(6)
+After the measurement Alice sends her outcome to Bob and Candy simultaneously using two
+separate classical communication channel. Receiving the measurement results from the sender
+Alice, the other two parties Bob and Candy make appropriate unitary operations on their re-
+spective particles to get the intended quantum states. That is the end of the protocol. The
+details of Alice’s measurement outcomes and corresponding appropriate unitary operations are
+given in Table 1.
+As an illustration, suppose Alice’s measurement outcome is |ξ2⟩ then the unitary operator ap-
+plied by Bob and Candy is (σz)b1 and (σx)c1 ⊗ (σxσz)c2 respectively.
+Table 1: Outcome results and corresponding unitary operators
+Measurement outcomes
+of Alice (|ξi)
+Bob’s unitary op-
+erator (U i)
+Candy’s
+unitary
+operator (V i)
+|ξ1⟩a1a2
+(I)b1
+(I)c1 ⊗ (I)c2
+|ξ2⟩a1a2
+(σz)b1
+(σx)c1 ⊗ (σxσz)c2
+|ξ3⟩a1a2
+(σxσz)b1
+(I)c1 ⊗ (I)c2
+|ξ4⟩a1a2
+(σx)b1
+(σx)c1 ⊗ (σxσz)c2
+3
+
+3
+Preparation of Entangled Channel
+The circuit for generating the quantum state |Φ⟩ in (3) is given in Fig. 1. It is generated by
+utilizing two Hadamard gates and four CNOT gates.
+Initially, a five-qubit state is prepared from a five-zero initial state
+|ψ1⟩ = |0⟩1 ⊗ |0⟩2 ⊗ |0⟩3 ⊗ |0⟩4 ⊗ |0⟩5.
+Now, first, one Hadamard gate is applied on qubit 1 and then a CNOT gate is applied with qubit
+1 as a control qubit and qubit 2 as the target qubit. Then the initial state |ψ1⟩ is converted to
+the state
+|ψ2⟩ =
+1
+√
+2
+�
+|00000⟩ + |11000⟩
+�
+12345.
+Again one Hadamard gate is applied on qubit 2 then the state |ψ2⟩ becomes
+|ψ3⟩ = 1
+2
+�
+|00000⟩ + |01000⟩ + |10000⟩ − |11000⟩
+�
+12345.
+In the next step, one CNOT gate is applied with qubit 1 as the control qubit and qubit 3 as the
+target qubit. After that two CNOT gates are applied with qubit 2 as the control qubit for each
+of qubits 4 and 5 respectively as target qubits. Then the state |ψ3⟩ is transferred to
+|Φ⟩ = 1
+2
+�
+|00000⟩ + |01011⟩ + |10100⟩ − |11111⟩
+�
+12345,
+which is the same as (3), that is, the entanglement resource we use here.
+We have executed this scheme of constructing the entangled state |Φ⟩ on IBM Quantum
+Composer and run over ibmq_qasm_simulator of 32 qubits. The explicit circuit and output
+results are given in Fig.1 and Fig.2 respectively.
+4
+Effect of noise on the proposed simultaneous remote state prepa-
+ration protocol
+In the practical field of quantum communication theory, mainly in the cases of quantum teleporta-
+tion and quantum remote state preparation schemes, quantum noise is an inevitable phenomenon
+and it is necessary to take noise into account in our present simultaneous remote state prepa-
+ration scheme. Here, we discuss the effects of three types of noises on our protocol which is a
+perfect communication scheme in the ideal situation where noise is neglected. We consider three
+different types of noises, namely, Amplitude-damping, Phase-flip and Bit-flip noisy environment.
+Suppose Alice creates the 5-qubit entangled channel in her laboratory, keeps her own qubits
+(a1, a2) with her and transmits the qubit b1 to Bob and the qubits (c1, c2) to Candy through
+noisy quantum channels. Thus only the qubits b1, c1, c2 are affected by the noise as these qubits
+4
+
+Figure 1: Quantum circuit
+are distributed through the noisy environment and the qubits a1, a2 are not influenced by any
+environmental noise since these are not transferred via noisy environment.
+In section 2, the proposed scheme is described by using a pure 5-qubit entangled state |Φ⟩ given
+in equation (4). The corresponding density matrix can be written as ρ = |Φ⟩⟨Φ|. However, after
+passing through noisy environments the corresponding density matrix ρ can be rewritten as
+ϵ(ρ) =
+�
+l,m,n
+�
+I ⊗ I ⊗ Xl ⊗ Xm ⊗ Xn
+�
+ρ
+�
+I ⊗ I ⊗ Xl ⊗ Xm ⊗ Xn
+�†
+(7)
+where Xl s, Xms, Xns are the Kraus operators satisfy �
+j X†
+j Xj = I and ‘†′ denotes conjugate
+transpose of a matrix.
+After that all the parties do the same job as in an ideal situation when we do not consider the
+effect of environmental noise in the said protocol. The intended remote state preparations are
+thereby completed.
+The final output state ρout
+i
+, where i ∈ {1, 2, 3, 4} can be calculated as
+Figure 2: Quantum circuit
+5
+
+[o] b
+<0I
+H
+q[1]
+<0I
+H
+q[2]
+10)
+q[3]
+<0
+q[4]
+<01
+:0
+c5
+1
+2250
+f 200
+e150
+q
+u
+1
+c 50
+y
+0
+00000
+01011
+10100
+11111
+Measurement outcomeρout
+i
+= Tra1a2{Mi[ϵ(ρ)]M†
+i },
+(8)
+where partial trace is taken over the qubit pair (a1, a2) and Mi is given by
+Mi = {Ia1a2 ⊗ (U i)b1 ⊗ (V i)c1c2}{|ξi⟩a1a2⟨ξi| ⊗ Ib1c1c2}
+(9)
+in which U i s and V i s are unitary operators given in Table 1.
+The fidelity corresponding to the output state ρout
+i
+can be computed as
+F = ⟨Ψ|ρout
+i
+|Ψ⟩,
+(10)
+where |Ψ⟩ represents the ideal output state which is the same as the input state. In our proposed
+scheme, it is given by
+|Ψ⟩ = (α|0⟩ + β|1⟩)b1 ⊗ (γ|00⟩ + δ|11⟩)c1c2.
+In the following we separately discuss the effects of three types of noises on our otherwise ideally
+noise-free protocol.
+4.1
+Amplitude-damping noisy environment
+The Kraus operator of an amplitude-damped noisy channel is expressed as:
+X0 =
+� 1
+0
+0
+√
+1 − λ
+�
+X1 =
+�
+0
+√
+λ
+0
+0
+�
+,
+where λ ∈ [0, 1] indicates the decoherence rate of amplitude-damping noise.
+Due to the random interaction with environment the energy of the quantum state will be dissi-
+pated. After the qubits b1, c1, c2 are transmitted through the amplitude-damping noisy environ-
+ment, the density matrix of the transformed quantum channel according to the formula (7) is
+given as (ignoring the constant factor)
+ϵAD(ρ) =
+��
+|00000⟩ + (1 − λ)|01011⟩ +
+√
+1 − λ|10100⟩ − (1 − λ)3/2|11111⟩
+�
+×
+�
+⟨00000|
++ (1 − λ)⟨01011| +
+√
+1 − λ⟨10100| − (1 − λ)3/2⟨11111|
+��
++
+���
+λ(1 − λ)|01010⟩
+−(1−λ)
+√
+λ|11110⟩
+�
+×
+��
+λ(1 − λ)⟨01010|−(1−λ)
+√
+λ⟨11110|
+��
++
+���
+λ(1 − λ)|01001⟩
+− (1 − λ)
+√
+λ|11101⟩
+�
+×
+��
+λ(1 − λ)⟨01001| − (1 − λ)
+√
+λ⟨11101|
+��
++
+��
+λ|01000⟩
+−λ
+√
+1 − λ|11100⟩
+�
+×
+�
+λ⟨01000|−λ
+√
+1 − λ⟨11100|
+��
++
+��√
+λ|10000⟩−(1−λ)
+√
+λ|11011⟩
+�
+×
+�√
+λ⟨10000| − (1 − λ)
+√
+λ⟨11011|
+��
++ (1 − λ)λ2
+�
+|11010⟩⟨11010| + |11001⟩⟨11001|
+�
++ λ3
+�
+|11000⟩⟨11000|
+�
+a1a2b1c1c2
+.
+(11)
+Now Alice performs her measurement with the measuring basis given in (5) and sends her mea-
+surement results to Bob and Candy classically. In order to get the desired state, Bob and Candy
+make appropriate unitary operations on their respective qubits.
+As an illustration, suppose Alice’s measurement outcome is |ξ2⟩. Therefore the density matrix of
+the final output state corresponding to the Alice’s measurement outcome |ξ2⟩ according to the
+formula (8) is given by
+6
+
+ρAD
+2
+=
+��
+αδ|011⟩ + (1 − λ)αγ|000⟩ +
+√
+1 − λβδ|111⟩ + (1 − λ)3/2βγ|100⟩
+�
+×
+�
+αδ⟨011|
++ (1 − λ)αγ⟨000| +
+√
+1 − λβδ⟨111| + (1 − λ)3/2βγ⟨100|
+��
++
+��
+−
+�
+λ(1 − λ)αγ|001⟩
+− (1 − λ)
+√
+λβγ|101⟩
+�
+×
+�
+−
+�
+λ(1 − λ)αγ⟨001| − (1 − λ)
+√
+λβγ⟨101|
+��
++
+���
+λ(1 − λ)αγ|010⟩ + (1 − λ)
+√
+λβγ|110⟩
+�
+×
+��
+λ(1 − λ)αγ⟨010| + (1 − λ)
+√
+λ
+βγ⟨110|
+��
++
+��
+− λαγ|011⟩ − λ
+√
+1 − λβγ|111⟩
+�
+×
+�
+− λαγ⟨011| − λ
+√
+1 − λβγ⟨111|
+��
++
+��
+−
+√
+λβδ|011⟩ − (1 − λ)
+√
+λβγ|000⟩
+�
+×
+�
+−
+√
+λβδ⟨011| − (1 − λ)
+√
+λβγ⟨000|
+��
++ (1 − λ)λ2βγ
+�
+|001⟩⟨001| + |010⟩⟨010|
+�
++ λ3βγ
+�
+|011⟩⟨011|
+�
+b1c1c2
+.
+(12)
+The fidelity of the final output state can be obtained according to the formula (10) as
+F1 = ⟨Ψ|ρAD
+2
+|Ψ⟩
+=
+�
+α2δ2 + (1 − λ)α2γ2 +
+√
+1 − λβ2δ2 + (1 − λ)3/2β2γ2
+�2
++
+�
+λα2γδ + λ
+√
+1 − λβ2γδ
+�2
++
+�√
+λαβδ2 + (1 − λ)
+√
+λαβγ2
+�2
++ λ3α2β2γ2δ2.
+4.2
+Phase-flip noisy environment
+The Kraus operators for the phase-flip noisy environment are given below:
+X0 =
+�
+1 − µ
+� 1
+0
+0
+1
+�
+X1 = õ
+� 1
+0
+0
+−1
+�
+,
+where µ ∈ [0, 1] defines the noise parameter in phase-flip noisy environment.
+Since the qubits b1, c1, c2 are transferred through the phase flip noisy environment, according to
+the equation (7) the density matrix of the quantum channel is transformed into
+ϵPF (ρ) = (1−µ)3
+�
+(|00000⟩+|01011⟩+|10100⟩−|11111⟩)×(⟨00000|+⟨01011|+⟨10100|−⟨11111|)
+�
++2(1−µ)2µ
+�
+(|00000⟩−|01011⟩+|10100⟩+|11111⟩)×(⟨00000|−⟨01011|+⟨10100|+⟨11111|)
+�
++(1−µ)µ2
+�
+(|00000⟩+|01011⟩+|10100⟩−|11111⟩)×(⟨00000|+⟨01011|+⟨10100|−⟨11111|)
+�
++(1−µ)2µ
+�
+(|00000⟩+|01011⟩−|10100⟩+|11111⟩)×(⟨00000|+⟨01011|−⟨10100|+⟨11111|)
+�
++2(1−µ)µ2
+�
+(|00000⟩−|01011⟩−|10100⟩−|11111⟩)×(⟨00000|−⟨01011|−⟨10100|−⟨11111|)
+�
++ µ3
+�
+(|00000⟩ + |01011⟩ − |10100⟩ + |11111⟩) × (⟨00000| + ⟨01011| − ⟨10100| + ⟨11111|)
+�
+.
+This can be written as
+ϵPF (ρ) =
+�
+(1 − µ)3 + (1 − µ)µ2
+�
+×
+�
+(|00000⟩ + |01011⟩ + |10100⟩ − |11111⟩) × (⟨00000|
+7
+
+(a)
+(b)
+(c)
+(d)
+Figure 3: (a): Variation of fidelity F1 with γ and λ when |S1⟩ = (
+√
+0.3|0⟩+
+√
+0.7|1⟩) (b): Variation
+of fidelity F1 with α and λ when |S2⟩ = (
+√
+0.3|00⟩ +
+√
+0.7|11⟩) (c): Variation of fidelity F1 with
+α and γ when ν = 0.5 (d) Variation of fidelity F1 with λ when |S1⟩ = (
+√
+0.4|0⟩ +
+√
+0.6|1⟩) and
+|S2⟩ = (
+√
+0.3|00⟩ +
+√
+0.7|01⟩).
++ ⟨01011| + ⟨10100| − ⟨11111|)
+�
++ 2(1 − µ)2µ
+�
+(|00000⟩ − |01011⟩ + |10100⟩
++ |11111⟩) × (⟨00000| − ⟨01011| + ⟨10100| + ⟨11111|)
+�
++
+�
+(1 − µ)2µ + µ3
+�
+×
+�
+(|00000⟩ + |01011⟩ − |10100⟩ + |11111⟩) × (⟨00000| + ⟨01011| − ⟨10100|
++ ⟨11111|)
+�
++ 2(1 − µ)µ2
+�
+(|00000⟩ − |01011⟩ − |10100⟩ − |11111⟩) × (⟨00000|
+− ⟨01011| − ⟨10100| − ⟨11111|)
+�
+a1a2b1c1c2
+.
+Now Alice makes the measurement with her measuring basis given in (5) and communicates
+with Bob and Candy classically. In order to obtain the intended state, Bob and Candy make
+appropriate unitary operations on their respective qubits.
+As an illustration, suppose Alice’s measurement outcome is |ξ2⟩. Therefore the density matrix
+of the final output states corresponding to Alice’s measurement outcome |ξ2⟩ according to the
+formula (8) is given by
+8
+
+0.9
+0.8
+0.8 --
+0.7
+Fidelity
+0.6
+0.6
+0.4 -
+0.5
+0.4
+0.2 ~
+0.3
+0 3
+0
+0.2
+0.2
+0.9
+0.4
+0.7
+0.8
+0.6
+0.1
+0.6
+0.5
+0.3
+0.4 
+0.8
+0.2
+Y
+0
+0.1
+10.9
+0.8
+0.8 -~-
+0.7
+Fidelity
+0.6
+0.6
+0.4
+0.5
+0.4
+0.2 ~
+0.3
+03
+0
+0.2
+0.2
+0.4 
+0.9
+0.7
+0.8
+0.6
+0.1
+0.6
+0.4
+0.5
+入
+0.8
+0.2
+0.3
+0.1
+α.0.9
+0.8
+0.8.
+0.7
+Fidelity
+0.6
+0.6
+0.4
+0.5
+0.2 
+0.4
+0 :
+0
+0.2
+0.3
+0.4
+0.8
+0.6 
+0.6
+0.2
+Y
+0.4
+0.8
+0.2
+1
+0
+α.0.9
+0.8
+0.7
+Fidelity
+0.6
+0.5
+0.4
+0.3
+0.2.
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9ρPF
+2
+=
+�
+(1 − µ)3 + (1 − µ)µ2
+�
+×
+�
+(αδ|011⟩ + αγ|000⟩ + βδ|111⟩ + βγ|100⟩) × (αδ⟨011|
++ αγ⟨000| + βδ⟨111| + βγ⟨100|)
+�
++ 2(1 − µ)2µ
+�
+(αδ|011⟩ − αγ|000⟩ + βδ|111⟩
+− βγ|100⟩) × (αδ⟨011| − αγ⟨000| + βδ⟨111| − βγ⟨100|)
+�
++
+�
+(1 − µ)2µ + µ3
+�
+×
+�
+(αδ|011⟩ + αγ|000⟩ − βδ|111⟩ − βγ|100⟩) × (αδ⟨011| + αγ⟨000| − βδ⟨111|
+− βγ⟨100|)
+�
++ 2(1 − µ)µ2
+�
+(αδ|011⟩ − αγ|000⟩ − βδ|111⟩ + βγ|100⟩) × (αδ⟨011|
+− αγ⟨000| − βδ⟨111| + βγ⟨100|)
+�
+b1c1c2
+.
+The fidelity of the output state with respect to Alice’s measurement result |ξ2⟩ can be obtained
+according to the formula (10) as
+F2 = ⟨Ψ|ρPF
+2
+|Ψ⟩
+=
+�
+(1 − µ)3 + (1 − µ)µ2
+�
++
+�
+2(1 − µ)2µ(δ2 − γ2)2
+�
++
+�
+(1 − µ)2µ + µ3
+�
+(α2 − β2)2
++ 2(1 − µ)µ2(α2 − β2)2(δ2 − γ2)2.
+4.3
+Bit-flip noisy environment
+The Kraus operators for the bit-flip noisy environment are given by
+X0 =
+� √1 − ν
+0
+0
+√1 − ν
+�
+=
+√
+1 − νI
+X1 =
+�
+0
+√ν
+√ν
+0
+�
+= √νσx,
+where ν ∈ [0, 1] indicates the noise intensity parameter in bit-flip noisy environment.
+According to the formula given in (7), the density matrix of the transformed entangled channel
+(as the qubits b1, c1, c2 pass through a bit-flip noisy environment channel) is given by
+ϵBF (ρ) = (1−ν)3
+�
+(|00000⟩+|01011⟩+|10100⟩−|11111⟩)×(⟨00000|+⟨01011|+⟨10100|−⟨11111|)
+�
++(1−ν)2ν
+�
+(|00001⟩+|01010⟩+|10101⟩−|11110⟩)×(⟨00001|+⟨01010|+⟨10101|−⟨11110|)
+�
++(1−ν)2ν
+�
+(|00010⟩+|01001⟩+|10110⟩−|11101⟩)×(⟨00010|+⟨01001|+⟨10110|−⟨11101|)
+�
++ν2(1−ν)
+�
+(|00011⟩+|01000⟩+|10111⟩−|11100⟩)×(⟨00011|+⟨01000|+⟨10111|−⟨11100|)
+�
++(1−ν)2ν
+�
+(|00100⟩+|01111⟩+|10000⟩−|11011⟩)×(⟨00100|+⟨01111|+⟨10000|−⟨11011|)
+�
++ν2(1−ν)
+�
+(|00101⟩+|01110⟩+|10001⟩−|11010⟩)×(⟨00101|+⟨01110|+⟨10001|−⟨11010|)
+�
++ν2(1−ν)
+�
+(|00110⟩+|01101⟩+|10010⟩−|11001⟩)×(⟨00110|+⟨01101|+⟨10010|−⟨11001|)
+�
++ ν3
+�
+(|00111⟩ + |01100⟩ + |10011⟩ − |11000⟩) × (⟨00111| + ⟨01100| + ⟨10011| − ⟨11000|)
+�
+.
+Now Alice performs the measurement on her two qubits a1, a2 with the basis given in (5) and
+9
+
+(a)
+(b)
+(c)
+(d)
+Figure 4: (a): Variation of fidelity F2 with γ and µ when |S1⟩ = (
+√
+0.3|0⟩+
+√
+0.7|1⟩) (b): Variation
+of fidelity F2 with α and µ when |S2⟩ = (
+√
+0.3|00⟩ +
+√
+0.7|11⟩) (c): Variation of fidelity F2 with
+α and γ when ν = 0.5 (d) Variation of fidelity F2 with µ when |S1⟩ = (
+√
+0.4|0⟩ +
+√
+0.6|1⟩) and
+|S2⟩ = (
+√
+0.3|00⟩ +
+√
+0.7|01⟩).
+sends the outcome results classically to Bob and Candy. In order to obtain the intended state,
+Bob and Candy make appropriate unitary operations on their respective qubits.
+As an illustration, suppose Alice’s measurement outcome is |ξ2⟩. Then Bob and Candy apply an
+appropriate unitary operations (σz)b1 and (σx)c1 ⊗ (σxσz)c2 on their respective qubits. There-
+fore, according to the formula given in (8) the reduced density matrix of the final output state
+corresponding to Alice’s measurement result |ξ2⟩ is given by
+ρBF
+2
+(ρ) = (1 − ν)3
+�
+(αδ|011⟩ + αγ|000⟩ + βδ|111⟩ + βγ|100⟩) × (αδ⟨011| + αγ⟨000| + βδ⟨111|
++βγ⟨100|)
+�
++(1−ν)2ν
+�
+(−αδ|010⟩−αγ|001⟩−βδ|110⟩−βγ|101⟩)×(−αδ⟨010|−αγ⟨001|
+− βδ⟨110| − βγ⟨101|)
+�
++ (1 − ν)2ν
+�
+(αδ|001⟩ + αγ|010⟩ + βδ|101⟩ + βγ|110⟩) × (αδ⟨001|
++ αγ⟨010| + βδ⟨101| + βγ⟨110|)
+�
++ ν2(1 − ν)
+�
+(−αδ|000⟩ − αγ|011⟩ − βδ|100⟩ − βγ|111⟩)
+×(−αδ⟨000|−αγ⟨011|−betaδ⟨100|−βγ⟨111|)
+�
++(1−ν)2ν
+�
+(−αδ|111⟩−αγ|100⟩−βδ|011⟩
+10
+
+1
+0.9
+0.8
+0.7
+Fidelity
+0.6
+0.5
+0.4
+E0
+0.2
+0.1
+0.5
+0.2 
+0.5
+0
+0.1
+0.4
+0.6
+0.7
+0.8
+0.9
+1
+μ0.9
+0.8
+0.8 ~
+Fidelity
+0.7
+0.6.
+0.6
+0.4 -
+0.5
+0.2 
+0:
+0.4
+0
+0.2
+0.3
+0.4
+0.8
+0.6
+0.6
+0.4
+0.2
+μ
+0.8
+0.2
+1
+0
+Y0.8 -3
+Fidelity
+0.6
+0.4
+0.2
+0 >
+0
+0.2
+0.4
+0.8
+0.6
+0.6
+0.4
+μ
+0.8
+0.2 
+0
+α.0.9
+1
+0.9
+0.8
+0.8
+Fidelity
+0.7
+0.7
+0.6 
+0.5
+0.6
+0.4 ~
+0.3
+0.5
+0.2
+0
+0.2
+0.4
+0.4
+0.8
+0.6
+0.6
+0.4
+EO
+Y
+0.8
+0.2
+1
+0(a)
+(b)
+(c)
+(d)
+Figure 5: (a): Variation of fidelity F3 with γ and µ when |S1⟩ = (
+√
+0.3|0⟩+
+√
+0.7|1⟩) (b): Variation
+of fidelity F3 with α and µ when |S2⟩ = (
+√
+0.3|00⟩ +
+√
+0.7|11⟩) (c): Variation of fidelity F3 with
+α and γ when ν = 0.5 (d) Variation of fidelity F3 with µ when |S1⟩ = (
+√
+0.4|0⟩ +
+√
+0.6|1⟩) and
+|S2⟩ = (
+√
+0.3|00⟩ +
+√
+0.7|01⟩).
+− βγ|000⟩) × (−αδ⟨111| − αγ⟨100| − βδ⟨011| − βγ⟨000|)
+�
++ ν2(1 − ν)
+�
+(αδ|110⟩ + αγ|101⟩
++ βδ|010⟩ + βγ|001⟩) × (αδ⟨110| + αγ⟨101| + βδ⟨010| + βγ⟨001|)
+�
++ ν2(1 − ν)
+�
+(−αδ|101⟩
+− αγ|110⟩ − βδ|001⟩ − βγ|010⟩) × (−αδ⟨101| − αγ⟨110| − βδ⟨001| − βγ⟨010|)
+�
++ ν3
+×
+�
+(αδ|100⟩+αγ|111⟩+βδ|000⟩+βγ|011⟩)×(αδ⟨100|+αγ⟨111|+βδ⟨1000|+βγ⟨011|)
+�
+.
+The fidelity of the final output state can be obtained according to the formula (10) as
+F3 = ⟨Ψ|ρBF
+2
+|Ψ⟩
+= (1 − ν)3 + 4ν2(1 − ν)γ2δ2 + 4(1 − ν)2να2β2 + 16ν3α2β2γ2δ2.
+5
+Discussion and Conclusion
+The speciality of the noisy environment considered here is that all the qubits which are transmit-
+ted experience noise individually, that is, there is no correlation regarding the noisy environment
+11
+
+0.9
+0.8
+0.8 ~
+0.7
+Fidelity
+0.6 -
+0.6
+0.4 -
+0.5
+0.4
+0.2 -
+0.3
+0:
+0
+0.2
+0.2
+0.4
+0.8
+0.6 
+0.6
+0. 1
+0.4
+V
+0.8
+0.2
+Y
+00.9
+0.8
+0.8 ~
+0.7
+Fidelity
+0.6
+0.6
+0.4 .
+0.5
+0.4
+0.2 
+0.3
+0:
+0
+0.2
+0.2
+0.4
+0.8
+0.6
+0.6
+0. 1
+0.4
+V
+0.8
+0.2
+0
+α.0.45
+0.5
+0.45
+0.4
+0.4 
+Fidelity
+0.35
+0.35
+0.3 
+0.25
+0.3
+0.2 -
+0.15
+0.25
+0.1 
+0
+0.2
+0.2
+0.4
+0.8
+0.6 
+0.6
+0.4
+0.15
+Y
+0.8
+0.2
+1
+0
+α.0.9
+0.8
+Fidelity
+0.7
+0.6
+0.5
+0.4.
+1
+1
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1between two qubits even when they are sent to the same party in the same prevalent noisy en-
+vironment. This is reflected in the expression of equation (7) where the ‘�’ is over the indices
+l, m and n. It is known that there are other types of noises than those considered here which
+can be modeled using the Kraus operator. There is no intrinsic reason for our consideration
+of only these three types of representative noises. Also it is known that the effect of noise can
+be minimized by executing weak and reversal measurements. In the present context we have
+not considered the effects of such noise-minimizing mechanisms. That would have warranted a
+separate work by themselves. Nonetheless, these are important aspects of communication that
+can be taken up in future works.
+It is interesting to see the variation of fidelity with the variation of other parameters given in
+Figures 3, Figure 4 and Figure 5. The common feature is that the fidelity in all cases go to unit
+value as the noise parameter tends to zero. This is as should be expected since with the noise
+parameter going to zero we approach the perfect remote state preparation protocol described in
+section 2 where the value of fidelity is 1.
+Further the communication scheme presented here is a multi-tasking protocol where more than
+one quantum information tasks are performed with the help of a single entanglement resource.
+With the increase in the complex nature of the jobs to be performed there is an increased necessity
+of entanglement with more number of qubits serving the communication scheme. Multi-partite
+entanglement is difficult to produce. Here our necessity is of a five-qubit entanglement. Prepa-
+ration of the entanglement resource is designed in section 3 and is run on IBM platform.
+Acknowledgement This work is supported by the Indian Institute of Engineering Science and
+Technology, Shibpur.
+Data availability statement Our manuscript has no associated data.
+Conflict of interest On behalf of all authors, the corresponding author states that there is no
+conflict of interest.
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+15
+
diff --git a/V9FIT4oBgHgl3EQfhStk/content/tmp_files/load_file.txt b/V9FIT4oBgHgl3EQfhStk/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf,len=1135
+page_content='Simultaneous Asymmetric Quantum Remote State Preparation  Scheme in Noisy Environments  Binayak S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Choudhury1 ∗, Manoj Kumar Mandal1 †,  Soumen Samanta1 ‡, Biswanath Dolai1,2 §  1 Department of Mathematics, Indian Institute of Engineering Science and Technology,  Shibpur B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Garden, Howrah - 711103, West Bengal, India  2 Department of Physics, Bajkul Milani Mahavidyalaya, Bajkul,  Purba Medinipur-721655, West Bengal, India  Abstract In this paper we discuss a quantum multi-tasking protocol for preparation of known  one-qubit and two-qubit states respectively in two different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The ideal remote state  preparation protocol is discussed in the first place in which a five-qubit entangled state is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  The design for the preparation of such entanglement is also presented and run on IBM quantum  composer platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The effects of three types of noises on the protocol are discussed and in all  these cases the reduced fidelities of the process are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The variations of these fidelities  with respect to different parameters are analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Keywords  Multi-qubit entangled channel, Bell-like state, Cluster state, Quantum Measure-  ment, Unitary operator, Noise, Fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  1  Introduction  Quantum teleportation (QT) introduced by Bennett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' [1] in 1993 plays an important role  in the history of quantum communication and information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' It is a technique for transfer  of unknown quantum information from one location to a distant location through a previously  shared entanglement between the sender and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Since then, a series of modified  quantum teleportation protocols such as controlled teleportation [2–9], bi-directional teleporta-  tion [10–16], probabilistic teleportation [17–22], cyclic teleportation [23–28], multi-hop telepor-  tation [29–33] and so on were developed in course of the last three decades which are applicable  to a very wide variety of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  There is another class of quantum communication scheme somewhat similar to quantum telepor-  tation protocols in the sense that its aim is also to create a quantum state at a distant location  but the prime difference is that the state to be remotely prepared is known to the sender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' These  types of quantum communication schemes are known as quantum remote state preparation (RSP)  ∗e-mail : binayak@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='iiests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='in, Fax: +91-33-26682916  †e-mail: manojmandaliiest@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='com, Tel: +91-700395967  ‡corresponding author e-mail : s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='samanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='com, Tel: +91-9804842707  §e-mail : biswanathbmm@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='com, Tel: +91-9547360567  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='11287v1  [quant-ph]  25 Jan 2023  protocols which were first introduced by Pati [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Following the work of Pati [34], several re-  searchers developed various modified forms of quantum remote state preparation schemes such as  controlled RSP [35–37], joint remote state preparation [38–40], cyclic RSP [41–43], bi-directional  RSP [44–46], conference RSP [47], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  However, in the process of quantum state transfer or creation of states in remote locations there  will inevitably be interactions with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' As a result of this random coupling with  the environment, the shared entangled channel is distorted which gives rise to quantum noisy  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' In a noisy channel quantum decoherence due to interaction with the environment is  described by Kraus operators which are in general non-unitary and therefore irreversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' In  recent times, some remote state preparation schemes through noisy environments are discussed  in [48–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The conventional measure of effectiveness in quantum communication is the fidelity  between the input state and the output state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The effect of noise is the reduction in fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  In this paper we describe a quantum communication scheme for remotely preparing a single  qubit and a two-qubit quantum state respectively at the sites of two different parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Both the  states are known to the party intending to remotely create the two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' We first describe  the ideal case where the task is performed through the utilization of a five-qubit cluster state in  which there is no infiltration of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The preparation of the five-qubit entangled channel used  here is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' After that we consider the effect of noisy environment in our protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Specifically we discuss three types of noises Amplitude-damping, Phase-flip and Bit-flip noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Due to the presence of noise the fidelity of the process which is a measure of effectiveness of  the communication is decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' We make a study of the variation of fidelity with respect to  different parameters of the scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  2  Simultaneous quantum remote state preparation in ideal case  Suppose there are three parties, namely, Alice, Bob and Candy who are situated at three different  locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Let us consider an arbitrary single-qubit and a two-qubit quantum state which are  respectively given by  |S1⟩ = (α|0⟩ + β|1⟩),  (1)  |S2⟩ = (γ|00⟩ + δ|11⟩),  (2)  where the coefficients α, β, γ and δ meet the normalization conditions, that is, |α|2 + |β|2 = 1  and |γ|2 + |δ|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  A five-qubit cluster-state is given by  |Φ⟩ = 1  2(|00000⟩ + |01011⟩ + |10100⟩ − |11111⟩)12345,  (3)  which acts as the quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Let us consider the situation where Alice intends to remotely create the two states given in (1)  and (2) in the locations of Bob and Candy respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Alice wants to prepare the quantum  state |S1⟩ at Bob’s laboratory and at the same time she wants to create the quantum state |S2⟩ at  Candy’s site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The sender Alice possesses the knowledge of the coefficients α, β, γ and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' For this  purpose Alice, Bob and Candy share a five-qubit cluster-state given in (3), in which the parti-  cles (1, 2) belong to Alice, particle 3 belongs to Bob and particles (4, 5) are in the hands of Candy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  2  For our purpose we rewrite the quantum entangled channel given in (3) as  |Φ⟩ = 1  2(|00000⟩ + |01011⟩ + |10100⟩ − |11111⟩)a1a2b1c1c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  (4)  Alice performs a 2-qubit measurement in the basis consisting of four linearly independent vectors  given by  |ξ1⟩ = (αγ|00⟩ + αδ|01⟩ + βγ|10⟩ − βδ|11⟩)a1a2,  |ξ2⟩ = (αδ|00⟩ − αγ|01⟩ − βδ|10⟩ − βγ|11⟩)a1a2,  |ξ3⟩ = (βγ|00⟩ + βδ|01⟩ − αγ|10⟩ + αδ|11⟩)a1a2,  |ξ4⟩ = (βδ|00⟩ − βγ|01⟩ + αδ|10⟩ + αγ|11⟩)a1a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  (5)  The choice of such a measurement base by Alice is possible since the coefficients α, β, γ, δ are  known to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='We can express the above quantum channel given in (4) using the basis (5) as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|Φ⟩ = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|ξ1⟩a1a2 ⊗ (αγ|000⟩ + αδ|011⟩ + βγ|100⟩ + βδ|111⟩)b1c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ |ξ2⟩a1a2 ⊗ (αδ|000⟩ − αγ|011⟩ − βδ|100⟩ + βγ|111⟩)b1c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ |ξ3⟩a1a2 ⊗ (βγ|000⟩ + βδ|011⟩ − αγ|100⟩ − αδ|111⟩)b1c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ |ξ4⟩a1a2 ⊗ (βδ|000⟩ − βγ|011⟩ + αδ|100⟩ − αγ|111⟩)b1c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|ξ1⟩a1a2 ⊗ (α|0⟩ + β|1⟩)b1 ⊗ (γ|00⟩ + δ|11⟩)c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ |ξ2⟩a1a2 ⊗ (α|0⟩ − β|1⟩)b1 ⊗ (δ|00⟩ − γ|11⟩)c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ |ξ3⟩a1a2 ⊗ (β|0⟩ − α|1⟩)b1 ⊗ (γ|00⟩ + δ|11⟩)c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ |ξ4⟩a1a2 ⊗ (β|0⟩ + α|1⟩)b1 ⊗ (δ|00⟩ − γ|11⟩)c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  (6)  After the measurement Alice sends her outcome to Bob and Candy simultaneously using two  separate classical communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Receiving the measurement results from the sender  Alice, the other two parties Bob and Candy make appropriate unitary operations on their re-  spective particles to get the intended quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' That is the end of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The  details of Alice’s measurement outcomes and corresponding appropriate unitary operations are  given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  As an illustration, suppose Alice’s measurement outcome is |ξ2⟩ then the unitary operator ap-  plied by Bob and Candy is (σz)b1 and (σx)c1 ⊗ (σxσz)c2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Table 1: Outcome results and corresponding unitary operators  Measurement outcomes  of Alice (|ξi)  Bob’s unitary op-  erator (U i)  Candy’s  unitary  operator (V i)  |ξ1⟩a1a2  (I)b1  (I)c1 ⊗ (I)c2  |ξ2⟩a1a2  (σz)b1  (σx)c1 ⊗ (σxσz)c2  |ξ3⟩a1a2  (σxσz)b1  (I)c1 ⊗ (I)c2  |ξ4⟩a1a2  (σx)b1  (σx)c1 ⊗ (σxσz)c2  3  3  Preparation of Entangled Channel  The circuit for generating the quantum state |Φ⟩ in (3) is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' It is generated by  utilizing two Hadamard gates and four CNOT gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Initially, a five-qubit state is prepared from a five-zero initial state  |ψ1⟩ = |0⟩1 ⊗ |0⟩2 ⊗ |0⟩3 ⊗ |0⟩4 ⊗ |0⟩5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Now, first, one Hadamard gate is applied on qubit 1 and then a CNOT gate is applied with qubit  1 as a control qubit and qubit 2 as the target qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Then the initial state |ψ1⟩ is converted to  the state  |ψ2⟩ =  1  √  2  �  |00000⟩ + |11000⟩  �  12345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Again one Hadamard gate is applied on qubit 2 then the state |ψ2⟩ becomes  |ψ3⟩ = 1  2  �  |00000⟩ + |01000⟩ + |10000⟩ − |11000⟩  �  12345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  In the next step, one CNOT gate is applied with qubit 1 as the control qubit and qubit 3 as the  target qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' After that two CNOT gates are applied with qubit 2 as the control qubit for each  of qubits 4 and 5 respectively as target qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Then the state |ψ3⟩ is transferred to  |Φ⟩ = 1  2  �  |00000⟩ + |01011⟩ + |10100⟩ − |11111⟩  �  12345,  which is the same as (3), that is, the entanglement resource we use here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  We have executed this scheme of constructing the entangled state |Φ⟩ on IBM Quantum  Composer and run over ibmq_qasm_simulator of 32 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The explicit circuit and output  results are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  4  Effect of noise on the proposed simultaneous remote state prepa-  ration protocol  In the practical field of quantum communication theory, mainly in the cases of quantum teleporta-  tion and quantum remote state preparation schemes, quantum noise is an inevitable phenomenon  and it is necessary to take noise into account in our present simultaneous remote state prepa-  ration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Here, we discuss the effects of three types of noises on our protocol which is a  perfect communication scheme in the ideal situation where noise is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' We consider three  different types of noises, namely, Amplitude-damping, Phase-flip and Bit-flip noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Suppose Alice creates the 5-qubit entangled channel in her laboratory, keeps her own qubits  (a1, a2) with her and transmits the qubit b1 to Bob and the qubits (c1, c2) to Candy through  noisy quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Thus only the qubits b1, c1, c2 are affected by the noise as these qubits  4  Figure 1: Quantum circuit  are distributed through the noisy environment and the qubits a1, a2 are not influenced by any  environmental noise since these are not transferred via noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  In section 2, the proposed scheme is described by using a pure 5-qubit entangled state |Φ⟩ given  in equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The corresponding density matrix can be written as ρ = |Φ⟩⟨Φ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' However, after  passing through noisy environments the corresponding density matrix ρ can be rewritten as  ϵ(ρ) =  �  l,m,n  �  I ⊗ I ⊗ Xl ⊗ Xm ⊗ Xn  �  ρ  �  I ⊗ I ⊗ Xl ⊗ Xm ⊗ Xn  �†  (7)  where Xl s, Xms, Xns are the Kraus operators satisfy �  j X†  j Xj = I and ‘†′ denotes conjugate  transpose of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  After that all the parties do the same job as in an ideal situation when we do not consider the  effect of environmental noise in the said protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The intended remote state preparations are  thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  The final output state ρout  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' where i ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 4} can be calculated as  Figure 2: Quantum circuit  5  [o] b  <0I  H  q[1]  <0I  H  q[2]  10)  q[3]  <0  q[4]  <01  :0  c5  1  2250  f 200  e150  q  u  1  c 50  y  0  00000  01011  10100  11111  Measurement outcomeρout  i  = Tra1a2{Mi[ϵ(ρ)]M†  i },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  (8)  where partial trace is taken over the qubit pair (a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' a2) and Mi is given by  Mi = {Ia1a2 ⊗ (U i)b1 ⊗ (V i)c1c2}{|ξi⟩a1a2⟨ξi| ⊗ Ib1c1c2}  (9)  in which U i s and V i s are unitary operators given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  The fidelity corresponding to the output state ρout  i  can be computed as  F = ⟨Ψ|ρout  i  |Ψ⟩,  (10)  where |Ψ⟩ represents the ideal output state which is the same as the input state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' In our proposed  scheme, it is given by  |Ψ⟩ = (α|0⟩ + β|1⟩)b1 ⊗ (γ|00⟩ + δ|11⟩)c1c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  In the following we separately discuss the effects of three types of noises on our otherwise ideally  noise-free protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1  Amplitude-damping noisy environment  The Kraus operator of an amplitude-damped noisy channel is expressed as:  X0 =  � 1  0  0  √  1 − λ  �  X1 =  �  0  √  λ  0  0  �  ,  where λ ∈ [0, 1] indicates the decoherence rate of amplitude-damping noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Due to the random interaction with environment the energy of the quantum state will be dissi-  pated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' After the qubits b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' c2 are transmitted through the amplitude-damping noisy environ-  ment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' the density matrix of the transformed quantum channel according to the formula (7) is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='given as (ignoring the constant factor)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='ϵAD(ρ) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|00000⟩ + (1 − λ)|01011⟩ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λ|10100⟩ − (1 − λ)3/2|11111⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='⟨00000|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ (1 − λ)⟨01011| +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λ⟨10100| − (1 − λ)3/2⟨11111|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)|01010⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='−(1−λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ|11110⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)⟨01010|−(1−λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ⟨11110|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)|01001⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='− (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ|11101⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)⟨01001| − (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ⟨11101|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ|01000⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='−λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λ|11100⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ⟨01000|−λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λ⟨11100|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ|10000⟩−(1−λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ|11011⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ⟨10000| − (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ⟨11011|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ (1 − λ)λ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|11010⟩⟨11010| + |11001⟩⟨11001|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ λ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|11000⟩⟨11000|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='a1a2b1c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  (11)  Now Alice performs her measurement with the measuring basis given in (5) and sends her mea-  surement results to Bob and Candy classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' In order to get the desired state, Bob and Candy  make appropriate unitary operations on their respective qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  As an illustration, suppose Alice’s measurement outcome is |ξ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Therefore the density matrix of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='the final output state corresponding to the Alice’s measurement outcome |ξ2⟩ according to the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='formula (8) is given by  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='ρAD  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='αδ|011⟩ + (1 − λ)αγ|000⟩ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λβδ|111⟩ + (1 − λ)3/2βγ|100⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='αδ⟨011|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ (1 − λ)αγ⟨000| +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λβδ⟨111| + (1 − λ)3/2βγ⟨100|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)αγ|001⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='− (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λβγ|101⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)αγ⟨001| − (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λβγ⟨101|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)αγ|010⟩ + (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λβγ|110⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ(1 − λ)αγ⟨010| + (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='βγ⟨110|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='− λαγ|011⟩ − λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λβγ|111⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='− λαγ⟨011| − λ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1 − λβγ⟨111|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λβδ|011⟩ − (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λβγ|000⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λβδ⟨011| − (1 − λ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='λβγ⟨000|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ (1 − λ)λ2βγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|001⟩⟨001| + |010⟩⟨010|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ λ3βγ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='|011⟩⟨011|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='b1c1c2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  (12)  The fidelity of the final output state can be obtained according to the formula (10) as  F1 = ⟨Ψ|ρAD  2  |Ψ⟩  =  �  α2δ2 + (1 − λ)α2γ2 +  √  1 − λβ2δ2 + (1 − λ)3/2β2γ2  �2  +  �  λα2γδ + λ  √  1 − λβ2γδ  �2  +  �√  λαβδ2 + (1 − λ)  √  λαβγ2  �2  + λ3α2β2γ2δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  Phase-flip noisy environment  The Kraus operators for the phase-flip noisy environment are given below:  X0 =  �  1 − µ  � 1  0  0  1  �  X1 = √µ  � 1  0  0  −1  �  ,  where µ ∈ [0, 1] defines the noise parameter in phase-flip noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Since the qubits b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' c2 are transferred through the phase flip noisy environment,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' according to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='the equation (7) the density matrix of the quantum channel is transformed into  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='ϵPF (ρ) = (1−µ)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00000⟩+|01011⟩+|10100⟩−|11111⟩)×(⟨00000|+⟨01011|+⟨10100|−⟨11111|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+2(1−µ)2µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00000⟩−|01011⟩+|10100⟩+|11111⟩)×(⟨00000|−⟨01011|+⟨10100|+⟨11111|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+(1−µ)µ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00000⟩+|01011⟩+|10100⟩−|11111⟩)×(⟨00000|+⟨01011|+⟨10100|−⟨11111|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+(1−µ)2µ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00000⟩+|01011⟩−|10100⟩+|11111⟩)×(⟨00000|+⟨01011|−⟨10100|+⟨11111|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+2(1−µ)µ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00000⟩−|01011⟩−|10100⟩−|11111⟩)×(⟨00000|−⟨01011|−⟨10100|−⟨11111|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ µ3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00000⟩ + |01011⟩ − |10100⟩ + |11111⟩) × (⟨00000| + ⟨01011| − ⟨10100| + ⟨11111|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  This can be written as  ϵPF (ρ) =  �  (1 − µ)3 + (1 − µ)µ2  �  ×  �  (|00000⟩ + |01011⟩ + |10100⟩ − |11111⟩) × (⟨00000|  7  (a)  (b)  (c)  (d)  Figure 3: (a): Variation of fidelity F1 with γ and λ when |S1⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3|0⟩+  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7|1⟩) (b): Variation  of fidelity F1 with α and λ when |S2⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3|00⟩ +  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7|11⟩) (c): Variation of fidelity F1 with  α and γ when ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='5 (d) Variation of fidelity F1 with λ when |S1⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='4|0⟩ +  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6|1⟩) and  |S2⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3|00⟩ +  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7|01⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  + ⟨01011| + ⟨10100| − ⟨11111|)  �  + 2(1 − µ)2µ  �  (|00000⟩ − |01011⟩ + |10100⟩  + |11111⟩) × (⟨00000| − ⟨01011| + ⟨10100| + ⟨11111|)  �  +  �  (1 − µ)2µ + µ3  �  ×  �  (|00000⟩ + |01011⟩ − |10100⟩ + |11111⟩) × (⟨00000| + ⟨01011| − ⟨10100|  + ⟨11111|)  �  + 2(1 − µ)µ2  �  (|00000⟩ − |01011⟩ − |10100⟩ − |11111⟩) × (⟨00000|  − ⟨01011| − ⟨10100| − ⟨11111|)  �  a1a2b1c1c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Now Alice makes the measurement with her measuring basis given in (5) and communicates  with Bob and Candy classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' In order to obtain the intended state, Bob and Candy make  appropriate unitary operations on their respective qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  As an illustration, suppose Alice’s measurement outcome is |ξ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Therefore the density matrix  of the final output states corresponding to Alice’s measurement outcome |ξ2⟩ according to the  formula (8) is given by  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='8 --  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7  Fidelity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='4 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2 ~  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3  0 3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  Y  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='8 -~-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7  Fidelity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='9ρPF  2  =  �  (1 − µ)3 + (1 − µ)µ2  �  ×  �  (αδ|011⟩ + αγ|000⟩ + βδ|111⟩ + βγ|100⟩) × (αδ⟨011|  + αγ⟨000| + βδ⟨111| + βγ⟨100|)  �  + 2(1 − µ)2µ  �  (αδ|011⟩ − αγ|000⟩ + βδ|111⟩  − βγ|100⟩) × (αδ⟨011| − αγ⟨000| + βδ⟨111| − βγ⟨100|)  �  +  �  (1 − µ)2µ + µ3  �  ×  �  (αδ|011⟩ + αγ|000⟩ − βδ|111⟩ − βγ|100⟩) × (αδ⟨011| + αγ⟨000| − βδ⟨111|  − βγ⟨100|)  �  + 2(1 − µ)µ2  �  (αδ|011⟩ − αγ|000⟩ − βδ|111⟩ + βγ|100⟩) × (αδ⟨011|  − αγ⟨000| − βδ⟨111| + βγ⟨100|)  �  b1c1c2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  The fidelity of the output state with respect to Alice’s measurement result |ξ2⟩ can be obtained  according to the formula (10) as  F2 = ⟨Ψ|ρPF  2  |Ψ⟩  =  �  (1 − µ)3 + (1 − µ)µ2  �  +  �  2(1 − µ)2µ(δ2 − γ2)2  �  +  �  (1 − µ)2µ + µ3  �  (α2 − β2)2  + 2(1 − µ)µ2(α2 − β2)2(δ2 − γ2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3  Bit-flip noisy environment  The Kraus operators for the bit-flip noisy environment are given by  X0 =  � √1 − ν  0  0  √1 − ν  �  =  √  1 − νI  X1 =  �  0  √ν  √ν  0  �  = √νσx,  where ν ∈ [0, 1] indicates the noise intensity parameter in bit-flip noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  According to the formula given in (7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' the density matrix of the transformed entangled channel  (as the qubits b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' c2 pass through a bit-flip noisy environment channel) is given by  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='ϵBF (ρ) = (1−ν)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00000⟩+|01011⟩+|10100⟩−|11111⟩)×(⟨00000|+⟨01011|+⟨10100|−⟨11111|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+(1−ν)2ν  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00001⟩+|01010⟩+|10101⟩−|11110⟩)×(⟨00001|+⟨01010|+⟨10101|−⟨11110|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+(1−ν)2ν  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00010⟩+|01001⟩+|10110⟩−|11101⟩)×(⟨00010|+⟨01001|+⟨10110|−⟨11101|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ν2(1−ν)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00011⟩+|01000⟩+|10111⟩−|11100⟩)×(⟨00011|+⟨01000|+⟨10111|−⟨11100|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+(1−ν)2ν  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00100⟩+|01111⟩+|10000⟩−|11011⟩)×(⟨00100|+⟨01111|+⟨10000|−⟨11011|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ν2(1−ν)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00101⟩+|01110⟩+|10001⟩−|11010⟩)×(⟨00101|+⟨01110|+⟨10001|−⟨11010|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ν2(1−ν)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00110⟩+|01101⟩+|10010⟩−|11001⟩)×(⟨00110|+⟨01101|+⟨10010|−⟨11001|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ ν3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(|00111⟩ + |01100⟩ + |10011⟩ − |11000⟩) × (⟨00111| + ⟨01100| + ⟨10011| − ⟨11000|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Now Alice performs the measurement on her two qubits a1, a2 with the basis given in (5) and  9  (a)  (b)  (c)  (d)  Figure 4: (a): Variation of fidelity F2 with γ and µ when |S1⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3|0⟩+  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7|1⟩) (b): Variation  of fidelity F2 with α and µ when |S2⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3|00⟩ +  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7|11⟩) (c): Variation of fidelity F2 with  α and γ when ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='5 (d) Variation of fidelity F2 with µ when |S1⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='4|0⟩ +  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6|1⟩) and  |S2⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='7|01⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  sends the outcome results classically to Bob and Candy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' In order to obtain the intended state,  Bob and Candy make appropriate unitary operations on their respective qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  As an illustration, suppose Alice’s measurement outcome is |ξ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Then Bob and Candy apply an  appropriate unitary operations (σz)b1 and (σx)c1 ⊗ (σxσz)c2 on their respective qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' There-  fore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' according to the formula given in (8) the reduced density matrix of the final output state  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='corresponding to Alice’s measurement result |ξ2⟩ is given by  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='ρBF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(ρ) = (1 − ν)3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(αδ|011⟩ + αγ|000⟩ + βδ|111⟩ + βγ|100⟩) × (αδ⟨011| + αγ⟨000| + βδ⟨111|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+βγ⟨100|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+(1−ν)2ν  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(−αδ|010⟩−αγ|001⟩−βδ|110⟩−βγ|101⟩)×(−αδ⟨010|−αγ⟨001|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='− βδ⟨110| − βγ⟨101|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ (1 − ν)2ν  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(αδ|001⟩ + αγ|010⟩ + βδ|101⟩ + βγ|110⟩) × (αδ⟨001|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='+ αγ⟨010| + βδ⟨101| + βγ⟨110|)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='+ ν2(1 − ν)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='(−αδ|000⟩ − αγ|011⟩ − βδ|100⟩ − βγ|111⟩)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='5 (d) Variation of fidelity F3 with µ when |S1⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='4|0⟩ +  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='6|1⟩) and  |S2⟩ = (  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='3|00⟩ +  √  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='7|01⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  − βγ|000⟩) × (−αδ⟨111| − αγ⟨100| − βδ⟨011| − βγ⟨000|)  �  + ν2(1 − ν)  �  (αδ|110⟩ + αγ|101⟩  + βδ|010⟩ + βγ|001⟩) × (αδ⟨110| + αγ⟨101| + βδ⟨010| + βγ⟨001|)  �  + ν2(1 − ν)  �  (−αδ|101⟩  − αγ|110⟩ − βδ|001⟩ − βγ|010⟩) × (−αδ⟨101| − αγ⟨110| − βδ⟨001| − βγ⟨010|)  �  + ν3  ×  �  (αδ|100⟩+αγ|111⟩+βδ|000⟩+βγ|011⟩)×(αδ⟨100|+αγ⟨111|+βδ⟨1000|+βγ⟨011|)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  The fidelity of the final output state can be obtained according to the formula (10) as  F3 = ⟨Ψ|ρBF  2  |Ψ⟩  = (1 − ν)3 + 4ν2(1 − ν)γ2δ2 + 4(1 − ν)2να2β2 + 16ν3α2β2γ2δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  5  Discussion and Conclusion  The speciality of the noisy environment considered here is that all the qubits which are transmit-  ted experience noise individually, that is, there is no correlation regarding the noisy environment  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='2  0  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='9  1between two qubits even when they are sent to the same party in the same prevalent noisy en-  vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' This is reflected in the expression of equation (7) where the ‘�’ is over the indices  l, m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' It is known that there are other types of noises than those considered here which  can be modeled using the Kraus operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' There is no intrinsic reason for our consideration  of only these three types of representative noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Also it is known that the effect of noise can  be minimized by executing weak and reversal measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' In the present context we have  not considered the effects of such noise-minimizing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' That would have warranted a  separate work by themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Nonetheless, these are important aspects of communication that  can be taken up in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  It is interesting to see the variation of fidelity with the variation of other parameters given in  Figures 3, Figure 4 and Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' The common feature is that the fidelity in all cases go to unit  value as the noise parameter tends to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' This is as should be expected since with the noise  parameter going to zero we approach the perfect remote state preparation protocol described in  section 2 where the value of fidelity is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Further the communication scheme presented here is a multi-tasking protocol where more than  one quantum information tasks are performed with the help of a single entanglement resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  With the increase in the complex nature of the jobs to be performed there is an increased necessity  of entanglement with more number of qubits serving the communication scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Multi-partite  entanglement is difficult to produce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Here our necessity is of a five-qubit entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Prepa-  ration of the entanglement resource is designed in section 3 and is run on IBM platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Acknowledgement This work is supported by the Indian Institute of Engineering Science and  Technology, Shibpur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Data availability statement Our manuscript has no associated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  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/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  References  [1] Bennett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Brassard, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': Bidirectional quantum teleportation of an arbitrary number  of qubits by using four qubit cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': Asymmetric bidirectional quantum state exchange between  Alice and Bob through a third party.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': Probabilistic teleportation of three qubit entangled state via five-qubit  cluster state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': Probabilistic teleportation via a non-maximally entangled GHZ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=': Probabilistic quantum teleportation of shared quantum secret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': Probabilistic resumable quantum tele-  portation for an arbitrary two-qubit entangled state with different quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=', Prakash, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=': Probabilistic Quantum  Teleportation via 3-Qubit Non-maximally Entangled GHZ State by Repeated Generalized  Measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': Deterministic multi-hop teleportation of arbitrary single qubit  state via partially entangled GHZ-type state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=' : Bidirectional Controlled Remote State Prepa-  ration of an Arbitrary Two-Qubit State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 58(7), 2228-2234 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=': Deterministic controlled bidirectional remote state preparation in  dissipative environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Communica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Samanta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': A bidirectional hybrid quantum  communication scheme for a known and an unknown qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Quantum Studies: Mathematics  and Foundations Doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='1007/s40509-022-00284-y (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  [47] Choudhury, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Samanta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=': A Controlled Asymmetric Quantum Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+page_content=': Effects of noise on joint remote state preparation of an arbitrary  equatorial two-qubit state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 56, 720-728 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  [49] Sun, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Qu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=': The effect of quantum noise on two dif-  ferent deterministic remote state preparation of an arbitrary three-particle state protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 17, 1-18, (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  [50] Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Zhou, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Xu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Luo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=': Cyclic hybrid double-channel quantum com-  munication via Bell-state and GHZ-state in noisy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' IEEE Access 7,8737687,  80530-80541 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  [51] Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' : Enhancing remote state preparation via five-qubit cluster state in noisy envi-  ronments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Optical and Quantum Electronics 53(2),104 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  [52] Gong, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Chang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' : Multi-party controlled cyclic hybrid  quantum communication protocol in noisy environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Quantum Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content=' 21(11),375  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
+page_content='  15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FIT4oBgHgl3EQfhStk/content/2301.11287v1.pdf'}
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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+1
+Quadratic Matrix Factorization with Applications
+to Manifold Learning
+Zheng Zhai, Hengchao Chen, and Qiang Sun
+Abstract—Matrix factorization is a popular framework for modeling low-rank data matrices. Motivated by manifold learning problems,
+this paper proposes a quadratic matrix factorization (QMF) framework to learn the curved manifold on which the dataset lies. Unlike
+local linear methods such as the local principal component analysis, QMF can better exploit the curved structure of the underlying
+manifold. Algorithmically, we propose an alternating minimization algorithm to optimize QMF and establish its theoretical convergence
+properties. Moreover, to avoid possible over-fitting, we then propose a regularized QMF algorithm and discuss how to tune its
+regularization parameter. Finally, we elaborate how to apply the regularized QMF to manifold learning problems. Experiments on a
+synthetic manifold learning dataset and two real datasets, including the MNIST handwritten dataset and a cryogenic electron
+microscopy dataset, demonstrate the superiority of the proposed method over its competitors.
+Index Terms—Quadratic matrix factorization, alternating minimization, convergence property, manifold learning.
+!
+1
+INTRODUCTION
+M
+ATRIX factorization has achieved many successes in
+various applications, including factor models [1],
+clustering [2, 3], recommendation system [4], graph and
+representation learning [5]. The key idea behind matrix
+factorization is that any data matrix admitting a low-rank
+structure can be represented as a product of two matrices
+with smaller dimensions. In a general form, matrix factor-
+ization solves
+min
+U∈RD×r,V ∈Rr×m
+U,V ∈C
+∥X − UV ∥2
+F.
+(1)
+where X ∈ RD×m is the data matrix with dimension D
+and sample size m, U ∈ RD×r and V ∈ Rr×m are factors
+of rank r, C is the feasible set encoding additional structural
+information, and ∥·∥F is the Frobenius norm. Building upon
+(1), many well-known algorithms in the literature can be
+obtained by taking specific feasible sets C. For example, if
+C = {V : V V T = Ir} enforces orthonormal constraints on
+the rows of V , then (1) reduces to principal component anal-
+ysis (PCA). If C enforces non-negative constraints on both
+U and V , or either U or V , then (1) becomes nonnegative
+matrix factorization (NMF) [6] or semi-NMF [7] respectively.
+NMF also shares a strong connection to spectral clustering
+[8].
+Although matrix factorization (1) has been widely stud-
+ied in various forms, it is less explored in the nonlinear case:
+some rows of V might be nonlinear functions of other rows
+of V . Nonlinear matrix factorization naturally arises in man-
+ifold learning problems, where data {xi}m
+i=1 ⊆ RD are as-
+sumed to concentrate near an unknown d-dimensional man-
+•
+Zheng Zhai, Hengchao Chen, and Qiang Sun are with the Department of
+Statistical Sciences, University of Toronto
+•
+Corresponding author: Qiang Sun. E-mail: qiang.sun@utoronto.ca
+Manuscript received April 19, 2005; revised August 26, 2015.
+ifold M and the goal is to recover M from {xi}m
+i=1 [9, 10].
+Mathematically, assume {xi}m
+i=1 are given by
+xi = f(τi) + ϵi,
+i = 1, . . . , m,
+where f is a bijective smooth mapping from an open set T ⊆
+Rd to M ⊆ RD, τi ∈ T is the d-dimensional representation
+of xi, and ϵi is the i-th approximation error. Here f and τi
+are non-identifiable in the sense that f(τi) = f ′(τ ′
+i), where
+f ′ = f ◦ g and τ ′
+i = g−1(τi) for any diffeomorphism g on T ,
+but f(τi) is uniquely defined. To find f(τi), we propose to
+minimize the residual sum of squares with respect to f and
+τi:
+min
+f∈F,{τi}m
+i=1⊆Rd
+m
+�
+i=1
+∥xi − f(τi)∥2
+F.
+(2)
+Here F = {f : Rd �→ RD} is a prediction function class.
+Let X = (x1, . . . , xm), Φ = (τ1, . . . , τm), and Ξ(f, Φ) =
+(f(τ1), . . . , f(τm)). Then (2) can be rewritten as
+min
+f∈F,Φ∈Rm×d ∥X − Ξ(f, Φ)∥2
+F.
+(3)
+If F consists of linear functions only, i.e., F = {f |
+f(τ) = Aτ, A ∈ RD×d}, then the optimization problem (3)
+reduces to the matrix factorization problem (1) with U = A,
+V = Φ, and C = RD×r × Rr×m. This is referred to as linear
+matrix factorization (LMF). The linear assumption on f can
+be too restrictive for manifold learning problems because it
+does not take the curved structure into account and thus is
+only applicable to model flat manifolds, i.e., manifolds that
+are locally isometric to the Euclidean spaces. For general
+manifolds, it is better to consider f as quadratic functions.
+Higher-order polynomial functions for f are also possible
+but tend to overfit noisy data, rendering poor generalization
+performance. For the reasons above, this paper focuses on
+the quadratic function class
+F = {f(τ) = c + Aτ + B(τ, τ) : c ∈ RD, A ∈ RD×d,
+symmetric tensor B ∈ RD×d×d}.
+(4)
+arXiv:2301.12965v1  [cs.LG]  30 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+2
+For any quadratic function f, there exists a unique matrix
+R ∈ RD×(2+3d+d2)/2 such that f(τ) = Rξ(τ), where
+ξ(τ) = [1, τ T , ψ(τ)T ]T
+ψ(τ) = [τ 2
+[1], τ[1]τ[2], ..., τ[1]τ[d], τ 2
+[2], ..., τ 2
+[d]]T ∈ R(d2+d)/2.
+(5)
+Here τ[i] denotes the i-th coordinate of τ and ψ(·) maps
+τ to a vector consisting of all quadratic and interaction
+terms of {τ[i]}d
+i=1. Let T(Φ) = (ξ(τ1), . . . , ξ(τm)). Then
+the optimization problem (3) with the quadratic function
+class (4) reduces to minR,Φ ∥X − RT(Φ)∥2
+F. To make Φ
+identifiable, we propose to solve the following optimization
+problem
+min
+R,Φ
+ΦΦT =Id,Φ1m=0
+∥X − RT(Φ)∥2
+F.
+(6)
+This is again a special case of the general matrix factoriza-
+tion problem (1), and we emphasize that T(Φ) encodes an
+implicit constraint that the last (d2 + d)/2 rows of T(Φ) are
+quadratic functions of its second-to-(1 + d)-th rows. Thus,
+we refer to (6) as quadratic matrix factorization (QMF).
+LMF has been widely studied [6, 11, 12, 13], but the
+proposed QMF is much more difficult and has received
+little attention. In this paper, we propose to optimize the
+QMF problem (6) with respect to Φ and R alternatively.
+Specifically, with Φ fixed, optimizing (6) with respect to
+R is equivalent to a linear regression problem. With R
+fixed, minimizing (6) over Φ is a non-convex quadratic
+projection problem, i.e., projecting a target point onto the
+quadratic surface determined by R. This quadratic pro-
+jection problem can be efficiently solved by an alternat-
+ing minimization algorithm when ∥RJ∥F is small, where
+J = (0 I(d2+d)/2)T ∈ R(2+3d+d2)/2×(d2+d)/2. This motivates
+us to add a regularizer λ∥RJ∥2
+F to (6) and solve the regular-
+ized QMF problem:
+min
+R,Φ
+ΦΦT =Id,Φ1m=0
+ℓλ(R, Φ) = ∥X − RT(Φ)∥2
+F + λ∥RJ∥2
+F.
+(7)
+To solve (7), an alternating minimization algorithm is pro-
+posed in Section 4. We also discuss how to tune λ properly.
+Our contributions are four-fold. First, motivated by man-
+ifold learning problems, we introduce the quadratic matrix
+factorization model and propose an alternating minimiza-
+tion algorithm for solving (6). Second, we establish the
+theoretical convergence property of the QMF algorithm.
+Third, motivated by the theoretical analysis of the QMF al-
+gorithm, we propose a regularized QMF algorithm and give
+an adaptive parameter tuning method. Finally, we apply the
+regularized QMF algorithm to solve general manifold learn-
+ing problems. Numerically, we examine the performance
+of the proposed method on a synthetic manifold learning
+dataset and two real-world datasets, including the MNIST
+handwritten dataset and a cryogenic electron microscopy
+dataset, and demonstrate the superiority of the proposed
+method over its competitors.
+1.1
+Related Work and Paper Organization
+Our QMF model is different from the problems studied
+in [14, 15], which are also referred to as quadratic matrix
+factorization. They consider approximating a matrix by a
+product of multiple low-rank matrices, and some factor ma-
+trix appears twice in the approximation, that is, X ≈ UU T
+or X ≈ AUBU T C with U being an unknown factor. They
+focus on estimating U. In contrast, our paper is motivated
+by manifold learning problems and focuses on solving (1)
+with quadratic constraints: some rows of V are quadratic
+functions of the other rows of V . Their algorithms cannot be
+applied to solve our QMF problem (6).
+Many manifold learning methods, such as LLE [16],
+Isomap [17], Laplacian eigenmaps [18] and diffusion maps
+[19], aim to find a lower-dimensional representation of the
+dataset while preserving certain geometric structures. In
+contrast, our target is to recover the underlying manifold
+structure on which the dataset lies. Most algorithms with
+the same purpose are based on tangent space estimation
+[20, 21, 22, 23, 24]. These methods share one common
+limitation that they do not take the higher-order smoothness
+into account. Compared with the aforementioned methods,
+the local polynomial approximation algorithm, which takes
+higher-order smoothness into account, could achieve a bet-
+ter convergence rate as shown in [9]. However, [9] and [10]
+mainly study the statistical properties of the local polyno-
+mial fitting algorithms, leaving several important computa-
+tional issues, such as the algorithmic convergence properties
+untouched. Our paper addresses these computational issues
+by providing algorithmic convergence properties and a reg-
+ularized QMF algorithm that potentially avoids over-fitting.
+The rest of the paper proceeds as follows. In Section 2, we
+describe the alternating minimization algorithm for solving
+the QMF problem (6). As a key component, we propose an
+alternating minimization algorithm to solve the quadratic
+projection problem and present its theoretical analysis. We
+establish the algorithmic convergence property of QMF in
+Section 3. In Section 4, we develop a regularized QMF
+algorithm and discuss the tuning method. Applications to
+manifold learning algorithms are given in Section 5, and
+numerical experiments are carried out in Section 6. We
+conclude this paper with several remarks in Section 7 and
+leave technical proofs in the Appendix.
+1.2
+Notation
+Throughout this paper, we denote by ∥ · ∥ and ∥ · ∥F the
+spectral norm and the Frobenius norm respectively. For
+a vector v ∈ RD, we use ∥v∥1 = �D
+i=1 |vi| to denote
+its ℓ1 norm. For a matrix M ∈ Rr×m, denote by σi(M),
+σmin(M) = σmin{r,m}(M), and ∥M∥2,1 = �r
+i=1 ∥Mi·∥ the i-
+th largest singular value, the smallest singular value, and the
+ℓ2,1 norm of M respectively. Here Mi· is the i-th row of M.
+Also, denote by M † the Moore-Penrose inverse of M and by
+PM = M T (MM T )†M ∈ Rm×m the projection matrix corre-
+sponding to the row subspace of M, i.e., the subspace in Rm
+spanned by the rows of M. Let B ∈ RD×d×d be a tensor, then
+B(τ, η) ∈ RD for any τ, η ∈ Rd. We use Bk ∈ Rd×d to denote
+the k-th slice of B, i.e., B(τ, η)k = τ T Bkη = ⟨Bk, ητ T ⟩ for
+all τ, η ∈ Rd and k = 1, . . . , D. We refer to B as a symmetric
+tensor if {Bk}D
+k=1 are all symmetric. For any η ∈ Rd, define
+Bη as the action of B on the vector η:
+Bη = [B1η, . . . , BDη]T ∈ RD×d,
+(8)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+3
+-2
+0
+2
+-2
+-1
+0
+1
+2
+3
+linear
+-2
+0
+2
+-2
+-1
+0
+1
+2
+3
+quadratic
+Fig. 1. A comparison between the linear and quadratic fitting in the swiss
+roll fitting problem. The above figures display the fitted curves using
+linear and quadratic functions, respectively. The dots represent the raw
+data and the dashed lines represent the underlying truth.
+where Bk denotes the k-th slice of B. Thus, B(τ, η) = Bητ =
+Bτη when B is symmetric. Denote by B∗(·) : RD �→ Rd×d
+the adjoint operator of B:
+B∗(c) =
+D
+�
+k=1
+ckBk ∈ Rd×d,
+∀c ∈ RD,
+where Bk is the k-th slice of B. Let 1m = [1, . . . , 1]T ∈
+Rm and 0 be an all-zero matrix, whose size depends on
+the context. In addition, let Id be the identity matrix of size
+d×d. Given two matrices A, D ∈ Rd×d, we use A ⪰ D (resp.
+A ⪯ D) to indicate that A − D (resp. D − A) is a positive
+semi-definite matrix.
+2
+QUADRATIC MATRIX FACTORIZATION
+This section presents an alternating minimization algorithm
+for solving quadratic matrix factorization. Given {xi}m
+i=1 ⊆
+RD, the goal is to solve
+min
+f∈F,{τi}m
+i=1⊆Rd
+m
+�
+i=1
+∥xi − f(τi)∥2
+F,
+(9)
+where F is the quadratic function class (4). Before proceed-
+ing to the algorithm, let us first illustrate the advantages of
+the quadratic function class over the linear function class
+in a swiss roll fitting example. As in Figure 1, we generate
+noisy data points near a swiss roll, fit such data using linear
+and quadratic functions, and compare the fitted curves with
+the underlying truth. It turns out that linear fitting tends
+to return a polygon, while quadratic fitting could produce
+curved lines that recover the underlying truth better. The
+difference between linear and quadratic fitting becomes
+more significant in the central region, where the curvature
+is large. This coincides with the intuition that quadratic
+fitting performs better because it takes the curvature into
+consideration, while linear fitting does not.
+To formally present the algorithm, we need some no-
+tations. Recall that ψ(·) and ξ(·) are given by (5). For any
+symmetric tensor B ∈ RD×d×d, there exists a unique matrix
+Q ∈ RD×(d2+d)/2 such that
+B(τ, τ) = Qψ(τ),
+∀τ ∈ Rd.
+(10)
+In particular, the k-th row Qk· of Q relates to the k-th slice
+of B via the equality Qk·ψ(τ) = τ T Bkτ. We shall refer to
+such Q as the matrix representation of the symmetric tensor
+B. Similarly, for any quadratic function f(τ) = c + Aτ +
+B(τ, τ), there exists a unique matrix R ∈ RD×(2+3d+d2)/2
+such that f(τ) = Rξ(τ), ∀τ ∈ Rd. Indeed, R = [c, A, Q] with
+Q being the matrix representation of the symmetric tensor
+B. Let X = [x1, . . . , xm] ∈ RD×m, Φ = [τ1, ..., τm] ∈ Rd×m,
+and
+Ψ(Φ) = [ψ(τ1), ..., ψ(τm)] ∈ R
+d2+d
+2
+×m.
+(11)
+Then the optimization problem (9) is equivalent to
+min
+R,Φ ℓ(R, Φ) = ℓ(c, A, Q, Φ) = ∥X − RT(Φ)∥2
+F,
+(12)
+where
+R = R(c, A, Q) = [c, A, Q],
+T(Φ) = [ξ(τ1), . . . , ξ(τm)] =
+�
+1m, ΦT , ΨT (Φ)
+�T
+.
+(13)
+This can be viewed as a matrix factorization problem, with
+constraints given by (13). The last (d2 + d)/2 rows of T(Φ)
+in the constraints (13) are determined by the second-to-(1 +
+d)th rows of T(Φ) through the mapping ψ. Recall that ψ(τ)
+collects all quadratic and interaction terms of {τ[i]}d
+i=1, thus
+these constraints specify all possible quadratic constraints.
+Problem (12) is thus referred to as QMF. LMF is a special
+case of QMF with additional constraints that Q = 0.
+However, (12) suffers from non-identifiability issues.
+This is because the minima of (12) are determined by T(Φ)
+only through its row space. To see this, we fix Φ and
+consider minimizing ℓ(R, Φ) with respect to R only. The
+minimizer �R and the product �RT(Φ) are given by
+�R = argmin
+R
+ℓ(R, Φ) = XT(Φ)T (T(Φ)T(Φ)T )†,
+�RT(Φ) = XT(Φ)T (T(Φ)T(Φ)T )†T(Φ) = XPT (Φ),
+(14)
+where M † denotes the Moore–Penrose inverse of M and
+PT (Φ) = T(Φ)T (T(Φ)T(Φ)T )†T(Φ) ∈ Rm×m. Thus the
+loss of �RT(Φ) only depends on the row subspace of T(Φ).
+Substituting (14) into (12), we obtain
+min
+Φ min
+R ℓ(R, Φ) = min
+Φ ∥X − XPT (Φ)∥2
+F.
+In particular, if Φ1 and Φ2 satisfy PT (Φ1) = PT (Φ2), then
+minR ℓ(R, Φ1) = minR ℓ(R, Φ2) and thus it is impossible
+to distinguish Φ1 and Φ2 when optimizing (12). Proposi-
+tion 1 provides concrete transformations on Φ such that
+minR ℓ(R, Φ) stays the same.
+Proposition 1 Suppose Φ′ = ZΦ + u1T
+m for some invertible
+matrix Z ∈ Rd×d and some vector u ∈ Rd. Then for any R, there
+exists R′ such that ℓ(R′, Φ′) = ℓ(R, Φ). In particular, we have
+minR ℓ(R, Φ′) = minR ℓ(R, Φ).
+The proof of Proposition 1 is left in the Appendix.
+To overcome non-identifiability issues, we add additional
+constraints ΦΦT = Id and Φ1m = 0 to (12) and solve
+min
+R,Φ
+ΦΦT =Id,Φ1m=0
+ℓ(R, Φ) = ∥X − RT(Φ)∥2
+F,
+(15)
+where R = R(c, A, Q) and T(Φ) are defined in (13). Several
+remarks follow. First, by Proposition 1, the minima of the
+constrained problem (15) and the unconstrained problem
+(12) are the same. Thus these two problems are equivalent.
+Second, by introducing constraints ΦΦT = Id, we reduce
+the solution space from Rd×m to the Stiefel manifold, which
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+4
+Algorithm 1: An alternating minimization algo-
+rithm for quadratic matrix factorization.
+Data: X = [x1, . . . , xm] ∈ RD×m
+Result: Φ = [τ1, . . . , τm] ∈ Rd×m, quadratic function
+f(τ) = Rtξ(τ).
+1 initialize Φ0 ∈ Rd×m as the top d eigenvectors of
+G = (X − ¯x1T
+m)T (X − ¯x1T
+m);
+2 while ∥ΦT
+t Φt − ΦT
+t−1Φt−1∥ > ϵ do
+3
+update Rt = argminR ℓ(R, Φt−1) as in (14);
+4
+for i = 1 to m do
+5
+solve the i-th projection problem
+�τi,t = argminτ∈Rd ∥xi − Rtξ(τ)∥2;
+6
+end
+7
+set �Φt = [�τ1,t, . . . , �τm,t];
+8
+update Φt = Zt �Φt(Im − 1m1T
+m/m) with Zt
+given by (17);
+9 end
+potentially helps speed up the optimization procedure [25].
+Third, we still do not have any restrictions on R in (15), so
+optimizing (15) over R with fixed Φ is still a regression prob-
+lem with its solution given by (14). Fourth, the constraint
+ΦΦT = Id enforces the scale of Φ to be neither too large
+nor too small. Thus, the configuration of the approximation
+RT(Φ) largely depends on R and can be controlled by
+regularizing R properly. We will explore this last point in
+Section 4.
+Now we present our first main algorithm. We adopt an
+alternating minimization strategy to solve (15). To begin
+with, we initialize Φ0 ∈ Rd×m as the top d eigenvectors
+of the gram matrix G = (X − ¯x1T
+m)T (X − ¯x1T
+m), where
+¯x =
+1
+m
+�m
+i=1 xi. During the t-th loop, we first fix Φ = Φt−1
+and update Rt = argminR ℓ(R, Φt−1) as in (14). Next,
+we fix R = Rt and update �Φt = argminΦ ℓ(Rt, Φ). This
+is a separable problem in the sense that �Φt is given by
+�Φt = [�τ1,t, . . . , �τm,t] with
+�τi,t = argmin
+τ∈Rd ∥xi − Rtξ(τ)∥2,
+(16)
+where ξ(τ) is defined in (5). We refer to (16) as a quadratic
+projection problem because it finds the closest point
+Rtξ(�τi,t) to xi on the quadratic surface ft(τ) = Rtξ(τ). At
+the end of the t-th loop, we set Φt = Zt �Φt(Im − 1m1T
+m/m)
+with Zt given by
+Zt = (�Φt(Im − 1T
+m1m/m)�ΦT
+t )−1/2.
+(17)
+This ensures that the constraints ΦtΦT
+t = Id and Φt1m = 0
+hold. Given a precision level ϵ > 0, we will repeat the above
+iterations until the stopping criteria ∥ΦT
+t Φt−ΦT
+t−1Φt−1∥ ≤ ϵ
+is met, i.e., the row subspace of Φt stabilizes. Algorithm 1
+summarizes the details. Till now, the only issue we have not
+yet addressed is how to solve (16), which will be discussed
+in the following subsection.
+2.1
+Quadratic Projection
+Since the parameters Rt and xi are fixed when solving (16),
+we omit the subscripts i and t for simplicity. We rewrite the
+loss function in (16) as
+h(τ) = ∥x − c − Aτ − B(τ, τ)∥2,
+(18)
+where c, A, B are determined by R via (10) and (13). Mini-
+mizing (18) with respect to τ is non-convex, and we consider
+minimizing the following surrogate loss instead:
+min
+τ,η g(τ, η) = 1
+2∥x − c − Aτ − B(τ, η)∥2
++1
+2∥x − c − Aη − B(τ, η)∥2.
+Note that g is symmetric in τ and η, and g(τ, τ) = h(τ).
+Starting from τ0 = η0 = 0 ∈ Rd, we update {τs, ηs}
+iteratively in the following manner:
+� τs = argminτ g(τ, ηs−1),
+ηs = argminη g(τs, η).
+(19)
+Upon convergence such that (τ ∗, η∗) = argminτ,η g(τ, η),
+we take τ ∗ to be the solution to (16). In what follows, we
+show how (19) can be efficiently solved and prove that
+(τs, ηs) converges to (τ ∗, τ ∗) for some stationary point τ ∗
+of h under certain conditions.
+The update rule (19) can be implemented efficiently as
+all iterates admit closed-form solutions. Let us fix η = ηs−1
+and consider optimizing g(τ, ηs−1) over τ. Recall Bη is given
+by (8) for any η ∈ Rd and B(τ, η) = Bτη = Bητ. Then
+g(τ, ηs−1) can be rewritten as
+g(τ, ηs−1) = 1
+2∥x − c − (A + Bηs−1)τ∥2
++1
+2∥x − c − Aηs−1 − Bηs−1τ∥2.
+This is a quadratic function of τ, so τs = argminτ g(τ, ηs−1)
+admits a closed-form solution
+τs = Γ−1
+ηs−1ζηs−1,
+where Γηs−1 and ζηs−1 are
+Γηs−1 = (A + Bηs−1)T (A + Bηs−1) + BT
+ηs−1Bηs−1,
+(20)
+ζηs−1 = (A + Bηs−1)T (x − c) + BT
+ηs−1(x − c − Aηs−1).
+(21)
+Since g(τ, η) is symmetric in τ and η, the dual problem ηs =
+argminη g(τs, η) can be solved similarly as
+ηs = Γ−1
+τs ζτs,
+where Γτs and ζτs are given by (20) and (21) with ηs−1
+replaced by τs.
+Now we provide conditions under which (τs, ηs) con-
+verges such that τ ∗ = lims τs = lims ηs, and (τ ∗, τ ∗) is
+a first-order stationary point of g. This implies that τ ∗ is
+a stationary point of h. The key is to analyze the Hessian
+matrix of g:
+Hg(τ, η) =
+� Γη
+Hητ
+Hτη
+Γτ
+�
+,
+where Γη and Γτ are given by (20) with parameter η and τ
+respectively and Hητ and Hτη are given by
+Hητ = HT
+ητ = − 2B∗(x − c − A(τ + η)/2 − B(τ, η))
++ BT
+η (A + Bτ) + (A + Bη)T Bτ.
+(22)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+5
+Here Bη and Bτ are defined in (8), and B∗(·) represents the
+adjoint operator of B. Γη is always positive definite when
+σd(A) > 0 because
+Γη = 1
+2AT A + ( 1
+√
+2A +
+√
+2Bη)T ( 1
+√
+2A +
+√
+2Bη) ⪰ 1
+2AT A.
+(23)
+The following theorem shows that if (τs, ηs) is bounded,
+then under certain conditions on B, we have (τs, ηs) con-
+verges, τ ∗ = lims→∞ τs = lims→∞ ηs, and (τ ∗, τ ∗) is a
+stationary point of g.
+Theorem 2 Suppose σd(A) > 0 and define Sα = {τ ∈ Rd |
+∥τ∥ ≤ α} for some α > 0. Denote by Bk the k-th slice of B for
+k = 1, . . . , D. If b = maxk σ1(Bk) satisfies
+(2∥x − c∥1 + 4α∥A∥2,1)b + 3Dα2b2 ≤ σ2
+d(A)/4,
+(24)
+then Hg(τ, η) is positive definite and σmin(Hg(τ, η))
+≥
+σ2
+d(A)/4 for all τ, η ∈ Sα. If the sequence (τs, ηs) obtained
+by (19) falls into the region Sα × Sα for sufficiently large s,
+then (τs, ηs) converges, τ ∗ = lims→∞ τs = lims→∞ ηs, and
+(τ ∗, τ ∗) is a stationary point of g.
+Proof Since σd(A) > 0, by (23), Γη is positive definite with
+σmin(Γη) ≥ σ2
+d(A)/2. By symmetry, such property holds for Γτ
+as well. As for Hητ and Hτη, we can use (24) to show that
+σ1(Hητ) = σ1(Hτη) ≤ σ2
+d(A)/4,
+(25)
+holds for any τ, η ∈ Sα. In particular, for any τ, η ∈ Sα, we have
+σ1(B∗(B(τ, η))) ≤ Dα2b2,
+σ1(BT
+η A) ≤ α∥A∥2,1b,
+σ1(AT Bτ) ≤ α∥A∥2,1b,
+σ1(BT
+η Bτ) ≤ Dα2b2,
+Similarly,
+σ1(B∗(x − c − A(τ + η)/2))
+≤(∥x − c∥1 + ∥A(τ + η)/2∥1)b ≤ (∥x − c∥1 + α∥A∥2,1)b.
+Combining these inequalities with (22), we obtain
+σ1(Hητ) = σ1(Hτη)
+≤ (2∥x − c∥1 + 4α∥A∥2,1)b + 3Dα2b2.
+Then (25) follows from the condition (24). To proceed, we decom-
+pose Hg(τ, η) into the following summation of a positive definite
+matrix and a symmetric matrix:
+Hg(τ, η) =
+�Γη
+0
+0
+Γτ
+�
++
+� 0
+Hητ
+Hτη
+0
+�
+.
+(26)
+For the second term, it follows from (25) that
+σ1
+�� 0
+Hητ
+Hτη
+0
+��
+= σ1(Hητ) ≤ σ2
+d(A)/4,
+for any τ, η
+∈
+Sα. Since min{σmin(Γτ), σmin(Γη)}
+≥
+σ2
+d(A)/2, the first term in (26) is positive definite, Hg(τ, η) is
+positive definite, and
+σmin(Hg(τ, η)) ≥ min{σmin(Γτ), σmin(Γη} − σ1(Hητ)
+≥σ2
+d(A)/4,
+∀τ, η ∈ Sα.
+(27)
+Since the region Sα × Sα is convex, (27) implies the strong
+convexity of g over Sα × Sα.
+Suppose that the sequence (τs, ηs) generated by (19) falls into
+the region Sα × Sα for sufficiently large s. Since g is smooth
+and strongly convex over the region Sα × Sα, it is well-known
+Fig. 2.
+An illustration of the effect of b on the convergence behav-
+iors of {(τs, ηs)}∞
+s=1 generated by (19). The left column corresponds
+to the setting (x, c, A, 20B) and the right column corresponds to the
+setting (x, c, A, 30B), where the parameters are set as follows: A =
+[−0.8979, 1.0086, −0.5422]T , B
+=
+[0.7817, −1.4908, −0.3679], c
+=
+[0.4171, 0.9176, 0.1759]T , and x = [0.2561, 0.7500, 0.0099]T . The first
+row displays g(τs, ηs) with the background surface representing the
+function g. The dotted line stands for the case τ = η. The second row
+displays h(τs) under the two settings.
+that (τs, ηs) would converge to the unique first-order stationary
+point of g in Sα × Sα [26]. Next, we will prove lims→∞ τs =
+lims→∞ ηs by contradiction. In specific, define τ ∗ = lims→∞ τs
+and η∗ = lims→∞ ηs and assume τ ∗ ̸= η∗. Since (τ ∗, η∗) is
+a first-order stationary point of g in Sα × Sα, by symmetry,
+we know (η∗, τ ∗) is also a first-order stationary point of g in
+Sα × Sα. However, this contradicts the uniqueness of the first-
+order stationary point of g in Sα × Sα.
+Suppose (τs, ηs) ⊆ Sα × Sα for sufficiently large s.
+Theorem 2 proves the convergence of (τs, ηs) by establishing
+the strong convexity of g over Sα ×Sα under condition (24).
+Condition (24) holds when both α and ∥x − c∥1 are small or
+b = maxk σ1(Bk) is small. In particular, condition (24) holds
+if b ≤ b0, where
+b0 =−∥x − c∥1 − 2α∥A∥2,1
+3Dα2
++
+�
+(∥x − c∥1 + 2α∥A∥2,1)2 + 3Dα2σ2
+d(A)/4
+3Dα2
+.
+It trivially holds when b = 0, which corresponds to LMF. In
+general, by the relationship (10) between B and Q, we have
+σ1(Bk) ≤ ∥Bk∥F ≤ ∥Qk·∥, where Qk· is the k-th row of Q.
+Thus, b is small if ∥Q∥F is small.
+To conclude this subsection, we use an example with
+D = 3, d = 1 to illustrate the effect of b on the convergence
+of {(τs, ηs)}∞
+s=1. We consider two different settings of g and
+display the sequence {(τs, ηs)}∞
+s=1 in Figure 2. The left panel
+has a smaller b than the right panel. The first row of Figure 2
+shows that (τs, ηs) converges with lim τs = lim ηs in the left
+panel while (τs, ηs) converges with lim τs ̸= lim ηs in the
+right panel. In addition, the second row shows that τs in the
+left panel converges to the global minimum of h while τs in
+the right panel does not. The differences between these two
+cases lie in the different values of b. In particular, if b is too
+large as in the right panel, (τs, ηs) would not converge to
+
+0.5
+0.5
+g(T,n)
+(u")6
+0
+0
+0.120-0.08.08.04
+0.15
+0.15
+0.1
+0.1
+0.05
+0.05
+0.040.060.080.1 0.12
+n
+T
+n
+T
+100
+0.4
+0.3
+0.2
+0.1
+10-1
+0
+0.05
+0.1
+0.15
+0.2
+0.05
+0.1
+0.15
+T
+TJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+6
+the same limiting point, that is lim τs ̸= lim ηs. In this case,
+ηs is not guaranteed to converge to the global minimum of
+h in general.
+3
+CONVERGENCE PROPERTIES
+This section establishes the convergence property for Algo-
+rithm 1. In particular, we provide conditions under which a
+limit point of the update sequence is a stationary point of
+(12).
+Theorem 3 Denote by {(Rt, �Φt, Φt)}∞
+t=1 the update sequence
+generated by Algorithm 1. For any sub-sequence {tj}∞
+j=1 such
+that limj Φtj = Φ∗, limj Φtj−1 = Φ∗∗, limj �Φtj = �Φ∗∗, if
+σmin(T(Φ∗)T(Φ∗)T ) > 0 and σmin(T(Φ∗∗)T(Φ∗∗)T ) > 0,
+then Rtj+1 and Rtj also converge. Denote by R∗ = limj Rtj+1
+and R∗∗ = limj Rtj. Also, we have �Φ∗∗ = argminΦ ℓ(R∗∗, Φ).
+If we assume
+ℓ(R∗∗, Φ∗∗) − ℓ(R∗∗, �Φ∗∗) ≥ γ∥Φ∗∗ − �Φ∗∗∥2
+F
+(28)
+holds for some constant γ > 0, then Φ∗∗ = �Φ∗∗ = Φ∗ and
+R∗∗ = R∗ and the accumulation point (R∗, Φ∗) satisfies the first
+order Karush-Kuhn-Tucker (KKT) condition of the problem (12).
+Proof By Algorithm 1, the update sequence {(Rt, �Φt, Φt)}∞
+t=1
+satisfies
+�
+�
+�
+�
+�
+Rt = argminR ℓ(R, Φt−1),
+�Φt = argminΦ ℓ(Rt, Φ),
+Φt = Zt �Φt(Im − 1m1T
+m/m),
+where Zt is defined by (17). It implies that ℓ(Rt+1, Φt) is
+monotone decreasing:
+ℓ(Rt+1, Φt) = min
+R ℓ(R, Φt) = min
+R ℓ(R, �Φt)
+≤ℓ(Rt, �Φt) ≤ ℓ(Rt, Φt−1),
+(29)
+where the second equality follows from Proposition 1. By the
+monotone convergence theorem and the fact that ℓ(Rt+1, Φt) ≥
+0, both ℓ(Rt, Φt−1) and ℓ(Rt, �Φt) converge to the same value.
+Consider the sub-sequence {tj}∞
+j=1 such that limj Φtj =
+Φ∗
+and
+σmin(T(Φ∗)T(Φ∗)T )
+=
+γ1
+>
+0.
+Then
+σmin(T(Φ)T(Φ)T ) ≥ γ1/2 > 0 for any Φ ∈ S∗
+ϵ0 = {Φ |
+∥Φ − Φ∗∥F ≤ ϵ0} with some sufficiently small ϵ0. Thus,
+XT(Φ)T (T(Φ)T(Φ)T )−1 is well-defined for Φ ∈ S∗
+ϵ0 and is
+a continuous function of Φ over S∗
+ϵ0. Since limj Φtj = Φ∗, we
+know Φtj ∈ S∗
+ϵ0 for sufficiently large j. By continuity, we have
+lim
+j Rtj+1 = lim
+j XT(Φtj)T (T(Φtj)T(Φtj)T )†
+=XT(Φ∗)T (T(Φ∗)T(Φ∗)T )−1.
+where we use the definition (14) of Rtj+1. Denote by R∗ =
+limj Rtj+1.
+Let us further assume that the sub-sequence {tj}∞
+j=1
+satisfies
+limj Φtj−1
+=
+Φ∗∗,
+limj �Φtj
+=
+�Φ∗∗,
+and
+σmin(T(Φ∗∗)T(Φ∗∗)T ) > 0. Then Rtj converges by the same
+argument and denote by R∗∗ = limj Rtj. Moreover, we claim
+that �Φ∗∗ = argminΦ ℓ(R∗∗, Φ). Otherwise, there exists Φ′ such
+that ℓ(R∗∗, Φ′) < ℓ(R∗∗, �Φ∗∗). By continuity, there exists some
+j0 such that ℓ(Rtj0 , Φ′) < ℓ(R∗∗, �Φ∗∗), which contradicts to
+the relationship (29). Furthermore, we assume (28) holds for some
+constant γ > 0.
+Now we are in a position to prove that Φ∗∗ = �Φ∗∗ = Φ∗,
+R∗∗ = R∗, and (R∗, Φ∗) satisfies the KKT condition. By
+continuity, we have
+ℓ(R∗∗, Φ∗∗) = lim
+j ℓ(Rtj, Φtj−1)
+= lim
+j ℓ(Rtj, �Φtj) = ℓ(R∗∗, �Φ∗∗),
+where the second equality follows from (29). By (28), we have
+Φ∗∗ = �Φ∗∗, or equivalently ∥Φtj−1 − �Φtj∥2
+F → 0 as j →
+∞. Since Φtj = argminΦΦT =Id,Φ1m=0 ∥Φ − �Φtj∥2
+F, we have
+∥Φtj − �Φtj∥2
+F ≤ ∥Φtj−1− �Φtj∥2
+F and thus ∥Φtj − �Φtj∥2
+F → 0 as
+l → ∞. Then the fact Φtj → Φ∗ implies that Φ∗∗ = �Φ∗∗ = Φ∗.
+By (2) and its R∗∗ version, we have R∗∗ = R∗. Finally, by taking
+the limit on the following optimality conditions:
+�
+�
+�
+�
+�
+�
+�
+∇Φℓ(Rtj, Φ)|Φ=�Φtj = 0,
+Φtj = Ztj �Φtj(Im − 1m1T
+m/m),
+∇Rℓ(R, Φtj)|R=Rtj +1 = 0,
+we prove that {R∗, Φ∗} satisfies the KKT conditions of the
+problem (12).
+Let us discuss the assumptions in Theorem 3. First, the
+feasible set G = {Φ ∈ Rd×m | ΦΦT = Id, Φ1m = 0} is
+compact, so there always exists a sub-sequence {tj}∞
+j=1 such
+that {Φtj}∞
+j=1 and {Φtj−1}∞
+j=1 converge. If we assume �Φt is
+bounded, then we can choose {tj}∞
+j=1 such that �Φtj also con-
+verges. Second, the assumption σmin(T(Φ∗)T(Φ∗)T ) > 0
+and σmin(T(Φ∗∗)T(Φ∗∗)T ) > 0 are mild, especially when m
+is much larger than d. Finally, the assumption (28) charac-
+terizes the γ-strong convexity of ℓ(R∗∗, Φ) with respect to Φ
+surrounding the minimizer �Φ∗∗. Theorem 2 provides condi-
+tions under which g(τ, η) is strongly convex over a bounded
+set, which implies the strong convexity of ℓ(R∗∗, Φ) with
+respect to Φ over a bounded set.
+4
+REGULARIZED QUADRATIC MATRIX FACTORIZA-
+TION
+Motivated by the discussion at the end of Section 2.1,
+we propose a regularized quadratic matrix factorization
+(RQMF) method. Specifically, we add a regularizer λ∥Q∥F
+to the original problem (15) to prevent ∥Q∥F from being too
+large. This leads to the following RQMF problem:
+min
+R,Φ
+ΦΦT =Id,Φ1m=0
+ℓλ(R, Φ) = ∥X − RT(Φ)∥2
+F + λ∥RJ∥2
+F, (30)
+where R and T(Φ) are given by (13),
+J =
+�0
+I(d2+d)/2
+�T ∈ R
+2+3d+d2
+2
+× d2+d
+2
+, and RJ = Q. (31)
+Since the columns of X are assigned equal weights in
+(30), we denote it as RQMF-E in the experiment section to
+distinguish it from its kernel version. In the following, we
+can still denote it as RQMF when not causing confusion.
+As shall be seen later, RQMF with a proper λ avoids
+over-fitting and thus enjoys a better generalization perfor-
+mance. In the rest of this section, we will first provide an
+alternating minimization algorithm to solve (30) and then
+elaborate how to tune λ properly. Also, we show that no
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+7
+Algorithm 2: Regularized Quadratic Matrix Factor-
+ization
+Data: X = [x1, . . . , xm] ∈ RD×m
+Result: Φ = [τ1, . . . , τm] ∈ Rd×m, quadratic function
+f(τ) = Rtξ(τ).
+1 initialize Φ0 ∈ Rd×m as the top d eigenvectors of
+G = (X − ¯x1T
+m)T (X − ¯x1T
+m);
+2 while ∥ΦT
+t Φt − ΦT
+t−1Φt−1∥ > ϵ do
+3
+update Rt = argminR ℓλ(R, Φt−1) as in (32);
+4
+for i = 1 to m do
+5
+solve the i-th projection problem
+�τi,t = argminτ∈Rd ∥xi − Rtξ(τ)∥2;
+6
+end
+7
+set �Φt = [�τ1,t, . . . , �τm,t];
+8
+update Φt = Zt �Φt(Im − 1m1T
+m/m) with Zt
+given by (17);
+9 end
+matter how λ is chosen, RQMF always outperforms LMF in
+terms of memorization properties.
+To solve (30), we adopt the same alternating minimiza-
+tion strategy described in Algorithm 1. When R is fixed,
+minimizing ℓλ(R, Φ) with respect to Φ is equivalent to
+minimizing ℓ(R, Φ) with respect to Φ, since the regular-
+izer λ∥RJ∥2
+F is independent of Φ. It thus reduces to the
+quadratic projection problem discussed in Section 2.1. On
+the other hand, when Φ is fixed, minimizing ℓλ(R, Φ) with
+respect to R is a ridge regression problem, and the solution
+can be given in closed form as
+�R = argmin
+R
+ℓλ(R, Φ)
+= XT(Φ)T (T(Φ)T(Φ)T + λJJT )−1.
+(32)
+Here we use the observation that T(Φ)T(Φ)T + λJJT is
+invertible as shown in Lemma 4 below. Therefore, to solve
+(30), it suffices to replace (14) in Algorithm 1 by (32), which
+gives us Algorithm 2, the RQMF algorithm.
+Lemma 4 Suppose ΦΦT = Id and Φ1m = 0. If λ > 0,
+then T(Φ)T(Φ)T + λJJT is positive definite. Furthermore,
+JT (T(Φ)T(Φ)T + λJJT )−1J is also positive definite.
+The proof of Lemma 4 is left in the Appendix. The
+following proposition shows that no matter how λ is chosen,
+RQMF memorizes the data better than LMF. Recall that
+LMF corresponds to RQMF with λ → ∞, or equivalently
+Q = RJ = 0.
+Proposition 5 For any λ > 0, the RQMF that solves (30)
+memorizes the data better than LMF in the following sense:
+∥X − R∗T(Φ∗)∥2
+F ≤ ∥X − R′T(Φ′)∥2
+F,
+(33)
+where (R′, Φ′) = argminR∈Ω,Φ ℓλ(R, Φ) with Ω = {R | R =
+[c, A, Q], Q = 0} is the solution of LMF and (R∗, Φ∗) =
+argminR,Φ ℓλ(R, Φ) is the solution of RQMF.
+The proof of Proposition 5 is left in the Appendix.
+4.1
+Tuning Parameter Selection
+This section discusses how to tune λ. Before presenting our
+new tuning method, let us first illustrate the effect of λ in
+Figure 3. We generate 240 data points uniformly on the
+unit circle and then manually add normal noises obeying
+N(0, 0.12I). For each target sample, we fit a curve around
+the target data using the nearest 40 data points and then
+project the target data onto the fitted curve. To fit the curve,
+we use the RQMF algorithm with λ = 0.1, 0.01, and 0.
+Figure 3 shows that the RQMF algorithm with λ = 0.01
+achieves the best performance. When λ = 0.1 is too large,
+the RQMF algorithm behaves like linear matrix factorization
+and tends to use straight lines as the fitted curves. When
+λ = 0 is too small, the RQMF algorithm tends to overfit data
+with excessively curved lines. Therefore, it is important to
+pick a proper λ.
+In what follows, we will describe a new adaptive tuning
+method. Recall that when Φ is fixed, �R in (32) is a function of
+λ. Define s(λ) = ∥ �R(λ)J∥2
+F and denote by s′(λ) and s′′(λ)
+the corresponding first and second derivatives.
+Proposition 6 shows that s′(λ) < 0 and s′′(λ) > 0 when
+s(λ) > 0 and λ > 0. This implies that s(λ) is a decreasing
+function of λ while s′(λ) is a strictly increasing function of
+λ. In particular, the root λ = s′−1(−δ) is unique and can be
+easily found via the bisection method. We propose to pick
+λ = s′−1(−δ) for a prescribed δ > 0.
+The proposed tuning method is reasonable in the follow-
+ing sense. Recall that s(λ) is the quantity that the regularizer
+∥RJ∥2
+F aims to control and s′(λ) measures the sensitivity
+of the target quantity s(λ) with respect to λ. Thus, our
+tuning method chooses λ corresponding to a prespecified
+sensitivity level δ.
+To illustrate the advantage of tuning λ via s′−1(·), we
+implement RQMF with 50 different λ evenly spaced in
+[0, 0.1]. We use samples drawn from the sine curve, that
+is, (ti, sin(ti)) + ϵi with {ti}21
+i=1 evenly distributed in [ π
+3 , 2π
+3 ]
+and ϵi
+i.i.d.
+∼ N(0, 0.032I). For each λ, we implement RQMF
+and compute the error, that is, the average distance between
+the fitted data points and the underlying truth. Also, we
+calculate the values of s(λ) and δ(λ) = −s′(λ). Figure 4
+displays the error against different λ, s(λ), and δ(λ). It
+shows that the error versus δ(λ) curve is the flattest near the
+optimal error. To achieve a prespecified error, say 1.2×10−3,
+the feasible choice of δ(λ) has a much wider range than that
+of λ or s(λ). Thus, it is easier to achieve good performances
+of RQMF by choosing δ(λ) rather than λ or s(λ). In practice,
+we choose δ > 0 as a constant smaller than −s′(0), where Φ
+determining the function s(·) is given by LMF.
+Proposition 6 Suppose Φ is fixed with ΦΦT = Id and Φ1m =
+0. Define �R(λ) by (32) and s(λ) = ∥ �R(λ)J∥2
+F. Then s′(λ) ≤ 0
+and s′′(λ) ≥ 0. Furthermore, the strict inequalities s′(λ) < 0
+and s′′(λ) > 0 hold if s(λ) > 0 and λ > 0.
+The proof of Proposition 6 is collected in the Appendix.
+5
+APPLICATIONS TO MANIFOLD LEARNING
+This section applies the RQMF algorithm to manifold learn-
+ing problems. Assume data {xi}m
+i=1 ⊆ RD are generated
+near an unknown smooth manifold M of intrinsic dimen-
+sion d. Here we no longer assume all data belong to the
+same local chart. Instead, we assume data belong to a union
+of several local charts. To recover the underlying manifold,
+we can apply the RQMF algorithm for each local chart.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+8
+-1
+-0.5
+0
+0.5
+1
+-1
+-0.5
+0
+0.5
+1
+noisy data
+-1
+-0.5
+0
+0.5
+1
+=0.1
+-1
+-0.5
+0
+0.5
+1
+=0.01
+-1
+-0.5
+0
+0.5
+1
+=0
+Fig. 3. Illustration of the effect of λ for fitting a circle. This figure displays the generated data and the locally fitted curves with λ = 0.1, 0.01, 0 (from
+left to right). In the last three figures, the rhombuses stand for the target samples, and the asterisks represent the place where the target samples
+are projected to fitted curves.
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+1.1
+1.2
+1.3
+1.4
+1.5
+1.6
+1.7
+1.8
+MSE
+10-3
+-s'( )
+s( )
+Fig. 4. The fitting error of RQMF against different λ, δ(λ) = −s′(λ), and
+s(λ). Here we take λ ∈ [0, 0.1] and compute δ(λ) and s(λ). To compare
+these three curves in the same horizontal axis [0, 0.1], we shift and re-
+scale δ(λ) and s(λ) via the function: f(x) = 0.1 ·
+x−min(x)
+max(x)−min(x).
+The performance of this divide-and-conquer strategy
+depends on choices of specific local charts. Besides, the
+quality of the fitted points on a single chart cannot be
+guaranteed uniformly: recovering the central region tends
+to be of higher quality than recovering the marginal re-
+gion. Also, for a data point belonging to multiple local
+charts, the fitted points in different charts are different
+and it is hard to determine which one is the best. To
+address these challenges, we propose an improved divide-
+and-conquer strategy. This strategy constructs a local chart
+for each data point y by finding its nearest K samples or by
+N(y, a) = {xi | ∥xi − y∥ ≤ a} for some a > 0. Then for
+each target sample, we denoise this particular data point by
+applying the RQMF algorithm to the corresponding chart.
+Compared with the original divide-and-conquer strategy,
+our strategy treats each data point as an individual problem
+and improves the accuracy.
+In a more general form, we could use a kernel function
+Kh(·, ·) to assign a closer point with higher importance,
+where h is the bandwidth. For a target sample y, we modify
+the loss function in (30) as
+min
+R,Φ
+ΦΦT =I,Φ1T
+m=0
+ℓλ,y,h(R, Φ) =∥(X − RT(Φ))W 1/2
+h
+(y)∥2
+F
++ λ∥RJ∥2
+F,
+(34)
+where X = (x1, . . . , xm) is the global data matrix and
+W 1/2
+h
+(y) ∈ Rm×m is a diagonal weight matrix with the
+i-th diagonal element equal to K1/2
+h
+(xi, y). If we choose
+Kh(x, y) = 1∥x−y∥≤a, then (34) reduces to the improved
+divide-and-conquer strategy mentioned above. It is also
+possible to use other kernel functions, such as the Gaussian
+kernel. To distinguish the kernel RQMF model from the
+previous equal-weight RQMF, we use RQMF-E to represent
+equal-weigth RQMF and RQMF-K to represent RQMF with
+weights determined by a kernel.
+We also discuss how to tune λ for each sub-problem (34).
+Picking the same λ for all sub-problems is not desirable
+since the best λ depends on the weights W 1/2
+h
+(y) and the
+curvature of the underlying truth, which vary as y changes.
+Instead, we suggest using the tuning method proposed
+in Section 4.1, which picks λ = s′−1(−δ) for the same
+prescribed sensitivity level δ > 0 for all sub-problems. The
+same δ would result in different λ’s for different charts, and
+this strategy often leads to better fitting accuracies in our
+experience.
+6
+NUMERICAL EXPERIMENTS
+This section presents numerical experiments on a synthetic
+manifold learning dataset and two real datasets, including
+the MNIST handwritten dataset and a cryogenic electron
+microscopy dataset, to examine the finite-sample perfor-
+mance of the proposed method. Our goal is to reconstruct
+the underlying manifold from noisy data and compare our
+method with five commonly used methods. We first briefly
+describe these five competitors.
+•
+Local PCA For any x, it first finds the K nearest data
+points {xi1, ..., xiK} and then computes the covari-
+ance matrix M =
+1
+K
+�K
+k=1(xik − cx)(xik − cx)T ∈
+RD×D, where cx = �K
+k=1 xik/K is the center of
+these samples. Denote by P ∈ RD×D the projection
+matrix corresponding to the space spanned by the d
+principle eigenvectors of M. The denoised point of x,
+a point on the estimated manifold “projected” from
+x, is given by xnew = cx + P(x − cx).
+•
+KDE & LOG-KDE Ridge Estimation Both methods
+are special cases of the nonparametric ridge esti-
+mation method [21]. Let ˆp(x) = �
+i Kh(x, xi) be
+the kernel density estimation (KDE) with the kernel
+function Kh(·, ·) and the bandwidth h. KDE ridge
+estimator estimates the ridge:
+ridge := {x |Π⊥(∇2ˆp(x))∇ˆp(x) = 0,
+λd+1(∇2ˆp(x)) < 0},
+(35)
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+9
+TABLE 1
+Comparisons of different methods on the synthetic spherical dataset in terms of MSE and SD (in bracket) with varying K.
+K
+7
+10
+13
+16
+19
+22
+25
+28
+RQMF-E
+0.0243 (0.0325)
+0.0165 (0.0227)
+0.0122 (0.0172)
+0.0115 (0.0164)
+0.0148 (0.0256)
+0.0130 (0.0222)
+0.0149 (0.0344)
+0.0156 (0.0356)
+RQMF-K
+0.0188 (0.0281)
+0.0170 (0.0270)
+0.0155 (0.0231)
+0.0148 (0.0233)
+0.0153 (0.0242)
+0.0159 (0.0252)
+0.0170 (0.0266)
+0.0185 (0.0280)
+Local PCA
+0.0437 (0.0521)
+0.0437 (0.0522)
+0.0435 (0.0520)
+0.0432 (0.0519)
+0.0434 (0.0521)
+0.0434 (0.0518)
+0.0434 (0.0517)
+0.0434 (0.0517)
+KDE
+0.0333 (0.0480)
+0.0298 (0.0469)
+0.0302 (0.0483)
+0.0307 (0.0482)
+0.0323 (0.0493)
+0.0342 (0.0499)
+0.0369 (0.0492)
+0.0389 (0.0501)
+LOG-KDE
+0.0278 (0.0417)
+0.0192 (0.0350)
+0.0159 (0.0344)
+0.0155 (0.0348)
+0.0168 (0.0349)
+0.0192 (0.0370)
+0.0230 (0.0395)
+0.0275 (0.0415)
+Mfit
+0.0392 (0.0463)
+0.0333 (0.0424)
+0.0262 (0.0347)
+0.0215 (0.0314)
+0.0183 (0.0302)
+0.0160 (0.0299)
+0.0154 (0.0303)
+0.0185 (0.0376)
+Moving LS
+0.0420 (0.0844)
+0.0673 (0.1155)
+0.1017 (0.1474)
+0.1506 (0.1666)
+0.1898 (0.1819)
+0.2264 (0.2033)
+0.2544 (0.2060)
+0.2642 (0.2072)
+Numbers in bold and underlined are the best and second-best results for each column’s setting, respectively.
+where Π⊥(∇2ˆp(x)) = I − UU T with U ∈ RD×d
+given by the top p principal eigenvectors of ∇2ˆp(x).
+Although the ridge in (35) does not admit close-
+form solutions, we may use the subspace constrained
+mean shift (SCMS) algorithm to find the denoised
+point of any point x and thus the ridge [20]. Similarly,
+the LOG-KDE ridge estimation is merely KDE ridge
+estimation with ˆp(x) replaced by log ˆp(x).
+•
+Mfit Mfit, proposed by [22], estimates the following
+manifold:
+�
+x | Πx
+� �
+i
+α(x, xi)Πi(x − xi)
+� = 0
+�
+,
+where α(·, ·) is a weight function, Πi is the projection
+matrix onto the approximate normal space at xi,
+and Πx is the projection matrix corresponding to
+the top D − d principal eigenvectors of the matrix
+�
+i α(x, xi)Πi. Again, we may use the SCMS algo-
+rithm to solve this problem.
+•
+Moving LS The moving least square (LS) consists of
+two steps [27]. For any x ∈ RD, we first find the
+d-dimensional hyperplane H in RD minimizing the
+following quantity
+L1(H) =
+min
+q∈H,x−q⊥H
+�
+i
+α(q, xi)ρ2(xi, H),
+where α(·, ·) is a weight function and ρ(xi, H) is
+the distance between xi and H. We can construct a
+coordinate system on H with origin q, where q is
+the projection point of x on H. Using this coordinate
+system, we can obtain the d-dimensional configura-
+tion x′
+i of xi by projecting xi to H. Second, we fit a
+polynomial function p : Rd → RD of a given degree
+by minimizing the following weighted squares
+L2(p) =
+�
+i
+α(q, xi)∥p(x′
+i) − xi∥2
+2.
+The denoised point of x is then given by p(0). In our
+experiments, we fix the degree of p as two.
+6.1
+A Synthetic Example
+In this subsection, we compare RQMF-E and RQMF-K with
+the above five competitors in a synthetic spherical fitting ex-
+periment. We simulate the noisy data {xi}240
+i=1 by generating
+240 points uniformly from the unit sphere S in R3 first and
+then adding independent noises following N(0, σ2I) with
+σ = 0.2. All algorithms take in the noisy data {xi} and then
+output the denoised data {�xi}. To measure the performance
+Fig. 5. An illustration of the impact of δ and K for spherical fitting for
+RQMF-E (left) and RQMF-K(right).
+Fig. 6. An illustration of local fitted surface under the impact of δ for
+RQMF-E when K = 18.
+of different algorithms, we use the mean squared error
+(MSE) and standard derivation (SD):
+MSE =
+m
+�
+i=1
+∥�xi − PS(�xi)∥2
+2/m,
+SD =
+�
+�
+�
+� 1
+m
+m
+�
+i=1
+(∥�xi − PS(�xi)∥2
+2 − MSE)2,
+where PS(·) is the projector onto the sphere.
+We briefly discuss how RQMF-E and RQMF-K are im-
+plemented. RQMF-E denoises each data point y using the
+K nearest neighbors of y, where K is a tuning parameter.
+To avoid overfitting, we require K to be larger than the rank
+(d2+3d+2)/2 of RT(Φ). For RQMF-K, we use the Gaussian
+kernel and set the bandwidth as h = dK/3 + 3, where dK
+is the distance from y to its K-th nearest neighbor. For both
+RQMF algorithms, we set the regularization parameter λ
+as λ = s′−1(−δ), where δ is a tuning parameter. To select
+K and δ for both RQMFs, we compute the MSEs of both
+RQMF-E and RQMF-K with different (K, δ) as shown in Fig-
+ure 5. When K is small, the local data approximates a plane
+and thus a smaller δ (larger λ) yields a better performance
+for RQMF-E. When K is large, the local data exhibits nonlin-
+ear structures, thus it is better to use a larger δ (smaller λ) for
+RQMF-E. The effect of δ is visualized in Figure 6. It shows
+that smaller δ tends to fit flatter planes in comparison with
+
+0.04
+0.03
+MSE
+0.02
+0.01 
+0
+7 10 13 16 19
+500
+.22
+100
+25
+28
+0.1
+K0.04
+0.03
+MSE
+0.02
+0.01
+0
+3
+5
+7
+9
+500
+100
+11
+a
+13
+K
+0.18=1
+S=10
+S=100
+1
+0
+0
+0
+0
+0
+-1
+.1JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+10
+Fig. 7. The first row displays 12 examples from the original MNIST dataset. The second to the sixth rows collect the results for the RQMF-K, KDE,
+LOG-KDE, Mfit, and Moving LS algorithms, respectively.
+TABLE 2
+Comparison of the smoothness measured by the average of the nonzero of the absolute value of I ∗ w corresponding to 12 example images
+Image ID
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+Original
+1.0637
+1.2264
+1.0306
+0.8760
+1.0332
+0.9499
+1.0246
+1.0261
+1.0068
+1.0267
+0.7178
+1.0326
+RQMF-K
+0.4848
+0.5795
+0.5168
+0.5785
+0.5440
+0.4851
+0.5733
+0.5319
+0.6011
+0.5388
+0.5327
+0.5470
+KDE
+1.0875
+0.8876
+0.7574
+0.9382
+0.8387
+0.9246
+0.7489
+0.6942
+0.6687
+0.9070
+0.6320
+0.7359
+LOG-KDE
+1.1395
+0.8861
+0.7732
+0.9460
+0.8606
+0.9329
+0.7650
+0.7523
+0.6979
+0.8812
+0.6361
+0.7764
+Mfit
+1.1769
+0.9158
+0.8567
+0.9026
+0.9520
+0.9705
+0.7956
+0.8274
+0.9107
+0.8562
+0.6189
+0.7976
+Moving LS
+1.1926
+1.3185
+1.4126
+1.0072
+1.2117
+1.1529
+1.4050
+1.3821
+1.4752
+1.2451
+0.9490
+1.4394
+Numbers in bold and underlined are the best and second-best results for each column’s setting, respectively.
+larger δ, which coincides with the phenomenon in Figure 5.
+To characterize the good performance region for δ and K
+in Figure 5, we choose δ = max{1, 8K − 125} to determine
+δ based on K for RQMF-E in this experiment. On the other
+hand, the performance of RQMF-K is relatively robust to the
+choice of δ, thus we fix δ = 100 for different choices of K.
+Now we compare RQMF-E and RQMF-K with their
+competitors. The results are collected in Table 1. The results
+indicate that RQMF-E and RQMF-K outperform other meth-
+ods for a wide range of K. When K = 7, RQMF-K achieves
+the best performance and when 10 < K < 28, RQMF-E
+achieves the best performance among all methods. If we
+focus on the best performance of different algorithms, the
+RQMF-E is still favored with the minimal MSE = 0.0115
+when K = 16. The superior performances of RQMF demon-
+strate the benefits of using the curvature information in the
+denosing procedure. It is also worth noting that RQMF out-
+performs Moving LS, which also fits a quadratic polynomial
+in its second step. This is possibly due to fact that RQMF it-
+eratively updates the local representations Φ of data points,
+while Moving LS only uses the local coordinates learned in
+its first step. The estimation error of the local coordinates
+learned in Moving LS could lead to a degradation of the
+final fitting accuracy.
+6.2
+An Application to the MNIST Handwritten Digit
+Dataset
+This subsection compares RQMF-K and its competitors on
+the MNIST handwritten digit dataset [28]. Each image in the
+dataset consists of 28 × 28 pixels. We use g(a, b) to denote
+the grey value of an image at pixel (a, b) and each image is
+determined by such a function g. Only pixels with nonzero
+grey values are considered, so the dataset for each image is
+given by {xi = (ai, bi) ∈ R2 | g(ai, bi) > 0}. In this way,
+each image can be viewed as a perturbed one-dimensional
+manifold in R2 and our goal is to recover the underlying
+manifold, which is also referred to as the principal curve
+[20].
+The pixel closer to the principal curve tends to have
+a larger grey value. Thus, it is natural to use the grey
+values as weights in (34). Specifically, for each y, we set
+the diagonal weight matrix Wh(y) in (34) by (Wh(y))ii =
+Kh(y, xi)g(xi)/s for some constant s, where Kh(·, ·) is
+a Gaussian kernel and the bandwidth h is given by the
+distance of y to y’s K-th nearest pixel. We use δ = 100
+to tune the parameter λ = s′−1(δ).
+Since the smoooth curve contains the most significant
+signal and the isolated points can be thought as noise, we
+measure the smoothness of the image using the convolution
+of the original image with the Laplace operator
+w =
+�
+�
+0
+1
+0
+1
+−4
+1
+0
+1
+0
+�
+� .
+The obtained matrix I ∗w is a discrete version of the Laplace
+operator defined for function f, i.e., ∆2f(x, y) =
+∂2f
+∂x2 +
+∂2f
+∂y2 . The average value of I ∗ w represents the degree of
+smoothness of the image I. We report the average of the
+nonzero of the absolute value in I ∗ w in Table 2.
+For each image, we apply all six manifold learning
+algorithms to recover the principal curve, and the results
+are displayed in Figure 7 and Table 2. It turns out that
+
+457822233.LT223215223.23F1123JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+11
+Noiseless Images
+Noisy Images
+RQMF-E
+Local PCA
+KDE
+LOG-KDE
+Mfit
+Moving LS
+Fig. 8. Visualization the fitting and denoised results of 16 randomly chosen images by RQMF-E and five related manifold learning methods with
+K = 60.
+the denoised images by RQMF-K are smoother than the
+original images and images output by KDE, LOG-KDE,
+Mfit. While Moving LS also produces smoother images, it
+tends to twist the original images too much; see images of
+6,8,9. In contrast, RQMF preserves the major contour of the
+original images much better.
+6.3
+An Application to Cryo-EM
+This subsection compares RQMF-E and RQMF-K with its
+competitors on the Cryo-EM dataset [29]. This dataset con-
+sists of n = 2000 images with shape 64 × 64. Each image
+is modeled as a vector in R4096 with elements given by
+the grey values on all 4096 pixels. The whole dataset is
+then represented as {ιi}m
+i=1 ⊆ R4096, where ιi denotes
+the i-th image. The dataset inherently resides on a lower-
+dimensional manifold in R4096 due to the image generation
+and processing of Cryo-EM, such as rotation, projection,
+and blurring by convolution [22]. In our experiment, we
+take the original data as the underlying manifolds, add
+noises to the original data, apply all six manifold learning
+algorithms to the noisy data, and finally compare their
+recovery accuracies.
+It is computationally expensive to directly fitting the
+manifold in R4096 using any manifold learning algorithm.
+To reduce the dimensionality, we approximate the original
+dataset by a D-dimensional subspace such that ιi ≈ Uxi,
+where U ∈ R4096×D denotes D principal eigenvectors of
+S =
+1
+m
+�
+i ιiιT
+i and xi = U ⊤ιi ∈ RD. We could fit a d-
+dimensional manifold in RD and map such a manifold to
+the pixel space R4096 via U. In what follows, we fix D = 20.
+We construct the noisy dataset as {ι′
+i = Ux′
+i}m
+i=1 with x′
+i
+given by
+x′
+i = xi + ϵi,
+ϵi ∼ N(0, σ2ID),
+i = 1, . . . , m.
+0.1
+1
+5
+10
+50
+100
+300
+1.3
+1.35
+1.4
+1.45
+1.5
+1.55
+1.6
+1.65
+MSE
+K=28
+K=32
+K=36
+K=40
+K=44
+K=48
+K=52
+K=56
+K=60
+Fig. 9. The performance of RQMF-E with different K and δ for Cryo-EM.
+Next, we recover a 5-dimensional manifold in RD by apply-
+ing all six manifold learning algorithms to {x′
+i}m
+i=1. For each
+method and each sample, we use the nearest K samples
+to fit the local manifold. In addition, we use the Gaussian
+kernel in (34) and for each y, we set the bandwidth h as
+∥y − xiK∥2, where xiK is the K-th nearest neighbour of y.
+Figure 8 visualizes 16 denoised images using these manifold
+learning methods with K = 60.
+To evaluate the performance, we use the following mean
+squared error and standard deviation of the error between
+
+0008
+000O
+8
+00000JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015
+12
+TABLE 3
+Comparisons of six manifold learning algorithms on the Cryo-EM dataset. We report MSE and SD (in bracket) with varying K.
+K
+40
+44
+48
+52
+56
+60
+RQMF-E
+1.5757(0.7065)
+1.4203(0.6669)
+1.3482(0.6013)
+1.3791(0.6363)
+1.4134(0.6465)
+1.3764(0.6623)
+RQMF-K
+2.1407(1.1057)
+2.1008(1.0804)
+2.0718(1.0612)
+2.0310(1.0385)
+1.9889(1.0064)
+1.9399(0.9618)
+Local-PCA
+1.8296(0.7531)
+1.8091(0.7211)
+1.7364(0.7662)
+1.7423(0.7476)
+1.7822(0.7958)
+1.8130(0.7553)
+KDE
+4.5179(1.5050)
+4.4408(1.4786)
+4.3820(1.4143)
+4.3258(1.4975)
+4.3213(1.4728)
+4.2659(1.4698)
+LOG-KDE
+1.9323(0.9683)
+1.8667(0.9296)
+1.8461(0.9188)
+1.8018(0.8514)
+1.7974(0.8288)
+1.7955(0.8242)
+Mfit
+2.4032(1.1298)
+2.3861(1.1216)
+2.3477(1.0988)
+2.2902(1.1229)
+2.2852(1.0957)
+2.2225(1.0391)
+Moving-LS
+1.5518(1.0340)
+1.4685(0.7865)
+1.5185(0.9541)
+1.5515(1.2535)
+1.5083(1.1097)
+1.4567(0.9147)
+Numbers in bold and underlined are the best and second-best results for each column’s setting, respectively.
+the solution and the real image:
+MSE = 1
+m
+m
+�
+i=1
+∥ιi − �ιi∥2
+2,
+SD =
+�
+�
+�
+� 1
+m
+m
+�
+i=1
+(∥ιi − �ιi∥2
+2 − MSE)2,
+where ˆιi = U ˆxi for all i and ˆxi is the i-th fitted point in
+RD. Figure 9 displays the MSE of RQMF-E with different K
+and δ. It shows that RQMF-E achieves the best performance
+when K = 48 and δ = 50.
+To compare different methods, we collect the MSEs of
+all methods with varying K in Table 3. It can be seen that
+RQMF-E outperforms other methods in terms of MSE and
+SD in most settings. If we focus on the best performance
+of each method, RQMF-E is again favored with an error
+1.3482 when K = 48. Therefore, by taking the curvature
+information into account, RQMF-E exhibits a stronger ex-
+pressive ability than other methods and thus achieves better
+denoising performance on this dataset.
+7
+CONCLUDING REMARKS
+This paper proposes a quadratic matrix factorization frame-
+work to learn the structure of the observed data. We develop
+an alternating minimization algorithm to solve the non-
+convex quadratic matrix factorization problem as well as
+a regularized version. Theoretical convergence properties
+are established. We also present a novel transformation-
+based parameter-tuning method for regularized quadratic
+matrix factorization and intuitively argue its advantages
+over naively tuning the original regularization parameter.
+Furthermore, we apply the proposed methods to manifold
+learning problems. We demonstrate the superiority of the
+proposed method numerically in a synthetic manifold learn-
+ing dataset and two real datasets, i.e., the MNIST handwrit-
+ten dataset and a cryogenic electron microscopy dataset.
+There are several interesting directions for future re-
+search. First, our work and most related works assume the
+intrinsic dimension d is known a priori, while this informa-
+tion is often not available in practice. Thus, it remains an
+important question to estimate the intrinsic dimensionality
+d under the quadratic matrix factorization framework. It
+would also be interesting to characterize the impact if d
+is misspecified. Second, the noises in the signal-plus-noise
+model may be heavy-tailed or even adversarial, so it is im-
+portant to develop robust algorithms. Third, non-negative
+constraints are widely used in linear matrix factorization
+[6, 12]. It is interesting to study how non-negative con-
+straints can be used in QMF to enhance performance.
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+
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+1
+Parsimonious System Identification from
+Fragmented Quantized Measurements
+Omar M.Sleem and Constantino M. Lagoa
+Dep. of Electrical Engineering, Pennsylvania State University, State College, PA 16801, USA.
+Email: { oms46@psu.edu, cml18@psu.edu }
+Abstract
+Quantization is the process of mapping an input signal from an infinite continuous set to a countable
+set with a finite number of elements. It is a non-linear irreversible process, which makes the traditional
+methods of system identification no longer applicable. In this work, we propose a method for parsimo-
+nious linear time invariant system identification when only quantized observations, discerned from noisy
+data, are available. More formally, given a priori information on the system, represented by a compact
+set containing the poles of the system, and quantized realizations, our algorithm aims at identifying
+the least order system that is compatible with the available information. The proposed approach takes
+also into account that the available data can be subject to fragmentation. Our proposed algorithm relies
+on an ADMM approach to solve a ℓp, (0 < p < 1), quasi-norm objective problem. Numerical results
+highlight the performance of the proposed approach when compared to the ℓ1 minimization in terms of
+the sparsity of the induced solution.
+Index Terms
+System identification, Sparsity, Quantization, ADMM
+I. INTRODUCTION
+A. Motivation
+Quantization is the division of a quantity into a discrete number of small parts, often assumed
+to be integral multiple of a common quantity [1], [2]. A classical example of quantization by
+rounding off, for the application of estimating densities of histograms, was analyzed in [3]. Since
+the processing of signals, i.e. speech and image, requires a digital environment, quantization plays
+an important role in bridging the analog and digital worlds [4]. On one hand, quantization led to
+a new research area in control theory called network controlled systems (NCS) [5]. NCS deals
+arXiv:2301.12043v1  [eess.SY]  28 Jan 2023
+
+2
+with the idea of controlling a process when the input and output signals are transmitted via a
+communication channel. On the other hand, it revealed the incompetence of the classical theory
+of system identification in considering quantized measurements [5].
+When a signal is subject to quantization, the quantization noise can no longer be modeled as
+a filtered white (zero mean and independent over time) noise and is signal dependent. Hence, in
+[6], [7] and references therein, the traditional theory of system identification was suggested to
+be modified to tackle the fact that the measurements are subject to quantization. Moreover, from
+[8] (section 10.1), the classical identification procedures are not suitable for robust identification,
+when the signal is subject to quantization, because they identify a set of parameters of a fixed
+mathematical structure, where a fixed system order must be assumed.
+Inspired by this, various works –which will be discussed in the next section in more detail–
+explored the problem of system identification given quantized realizations. However, in this
+paper we aim to present a new approach to the problem of (Linear Time Invariant) LTI system
+identification from quantized outputs. This approach allows for the use of a priori information
+on the system and fragmented measurements of the output. In addition, our approach aims to
+recover the least order system that is compatible with the data by minimizing an ℓp, (0 < p < 1),
+quasi-norm objective.
+The paper is organized as follows; in the remaining of the introduction, we provide a compre-
+hensive discussion of the previous related work and our contribution. Section II introduces the
+notations that are used throughout the paper. In section III, we thoroughly describe the system
+model used. The parsimonious system identification problem is formally provided in section
+IV. The proposed (Alternating Direction Method of Multipliers) ADMM algorithm based on ℓp
+quasi-norm approximation is described in section V. We validate our approach with an extensive
+suite of numerical simulations in section VI. Finally, the paper is concluded in section VII.
+B. Related work
+The problem of simple representation of signals using quantization dates back to the 1940’s
+and is one of the main threads of information theory [9]. However, rigorous analysis did not
+begin until the 1980’s. In [10], [11], considering digital feedback control systems, the authors
+proposed a way in which one can specify system structures that alleviate the adverse effects of
+quantization. The works in [12], [13] demonstrate that quantization can induce a chaotic behavior
+in digital feedback systems. The results in [14], [15] are recognized as a quantum leap because
+
+3
+the author was able to analyze the behavior of control systems in detail. The circumstances
+under which a discrete unstable LTI system can be stabilized, by choosing feedback control that
+depends on the quantized measurements, are studied. In [16], the authors proposed a control
+design methodology, assuming a quantizer with variable sensitivity along with system state, that
+stabilizes LTI control systems with quantized measurements. In [17], the coarsest quantizer that
+stabilizes a single input LTI system is shown to be a logarithmic one and can be obtained by
+solving a linear quadratic regulator problem. Abundant other works investigated the problem of
+the stabilization of NCS in different situations, e.g., [18]–[20].
+Despite that ample research activity in the stabilization and state estimation, quantization
+in system identification problems was still not properly pondered [21]. In [22], the authors
+studied the effect of quantization on I/O data used for system identification in a controlled
+plant whose parameters may change during the operation. They derived the optimal quantization
+scheme and showed that it is coarse near the origin of the signals and dense at a distance
+from it. This result is opposite to the case of stabilization in [17] and reveals duality between
+system identification and stabilization. Similar properties of the optimal quantizer were concluded
+in [23], where the author considered a least square error objective function –for parameter
+estimation– subject to a constraint on the number of subsections of the quantized signals or the
+expectation of the optimal code length for either high or low resolution. In [24], the problem of
+system identification using uniformly quantized realizations was considered, where, the proposed
+formulation is a least square minimization of the difference equation errors over all time samples
+with the system parameters as optimization variables. Regardless of the high accuracy in the
+estimation of the unknown information in the I/O data, the proposed method stills suffers the
+drawback of high computational complexity and noise neglection. The work in [25] aimed to
+solve these drawbacks by exploiting statistical properties instead of deterministic treatment. In
+particular, an identification method for a linear system based on quantized measurements was
+derived. Using traditional equi-spaced quantizer, an instrumental level identification approach
+was proposed to enhance the estimation accuracy. The authors of [26] took this approach a step
+further where a variation for the equi-spaced quantizer was considered. They showed that using
+a generalized noise shaping coder improves the accuracy of the estimates.
+Another line of research includes the identification using a general class of quantized observa-
+tions that allows the segmentation of the output range into a collection of subsets that may have
+unequal, fixed lengths or even design variables such as quantization design in communication
+
+4
+systems and NCS [27]. This serves in favor of understanding the potency of systems with
+limited sensor information, which in turns rapports the gap between resource limitations and
+identification complexity in sensor and communication networks. In particular, the work in
+[28] considered the identification of a gain system by exploiting the information from multiple
+thresholds sensor and the convex combination of these thresholds. The results were extended
+to the case of a noisy communication channel through which the sensor output information
+is transmitted. The authors prove that their estimator is asymptotically efficient achieving the
+Cramer-Rao lower bound. Furthermore, the results were extended to a finite impulse response
+and transfer function models for periodic bounded input signals. In [29], the authors focused
+on relationships between the identification space and time complexities. They showed that
+the asymptotic efficiency of empirical measure based algorithms yield to a tight bound on
+identification accuracy. This in turns aids to derive a separation principle of the complexities
+(time and space). The gained insights aim to provide a feasible approach for optimal utility
+of communication bandwidth resources in magnifying the identification accuracy. The role of
+dithering noise –adding artificial noise to the observed signal before quantization in order to
+mitigate the effects of quantization– at the sensor was studied in [30]. The authors asserted that
+tailored dithering noise can considerably simplify the derivation of optimal estimators in the
+expense of a decreased signal to noise ratio.
+C. Contributions
+The different methods reviewed in the previous part aim to either stabilize the system, find an
+optimal quantization scheme or solve a system identification problem. In this work, we focus on
+the latter problem where, to the best of our knowledge, none of the proposed methods address the
+problem of identifying the system of least order that is compatible with collected information.
+The problem of identifying systems using collected measurements can also include several
+other challenges including; 1) One can be faced with fragmented data due to the misplacement
+of sensors or external disturbances that can possibly make the collected data unreliable. 2) The
+ability to handle prior information on the system, e.g., constraints on the locations of the poles.
+In this paper, we aim to develop an algorithm that tackles the challenges mentioned above.
+More precisely, we consider a system in which the a priori information can be described
+by constraining the locations of the system’s poles to be in a known compact set. Then, by
+exploiting “simple representations” of transfer functions, we develop an efficient algorithm that
+
+5
+aims at finding the lowest order system that is compatible with fragmented quantized output
+measurements. This algorithm is based on an ADMM approach to the problem of ℓp quasi-norm
+optimization. To validate our analysis, we consider two different numerical examples; 1) System
+identification with randomly generated data set. 2) Identification with actual data collected from
+the motion of flexible robotic arm [31]. The numerical results in both examples show that our
+method is competitive against the ℓ1 convex relaxation objective in terms of both the detected
+system order and accuracy of the recovered realizations.
+A preliminary version of part of this work was presented in [32]. This journal version includes
+a generalized formulation where a non continuous input data stream can be handled (input data
+is composed of independent chunks). Moreover, we do not assume a continuous measurement
+of the chunks’ outputs from the quantizer, i,e,. output data is subject to fragmentation. Unlike
+[32], we assume a generalized quantizer whose input is prone to noise and show that it plays
+an important role in the sparsity of the induced solution. We provide a new experiment to
+demonstrate the superior performance of our method in a more practical scenario.
+II. NOTATIONS
+Unless otherwise specified, we denote scalars with non boldface letters, e.g., x, vectors with
+lowercase boldface letters, e.g., x, with i-th entry as xi, while matrices are in uppercase, e.g.,
+X, with (i, j)-th entry as xi,j. Xj,: specifies the j-th row of the matrix X. R and C are the sets
+of real and complex numbers respectively. For a vector x and matrix X, |.| is an element-wise
+absolute value of the applied variable. However, for a set X, |.| operator stands for the cardinality
+of the set. We use ⪯ for element wise inequality of vectors. For any constant c > 0, we define
+Ic
+∆= [−c, c]. For a positive integer n, we let [n]
+∆= {1, . . . , n}. The p-th norm of a vector x ∈ Rn
+is defined such that
+∥x∥p
+∆= (
+n
+�
+i=1
+|xi|p)
+1
+p.
+(1)
+It is important to note that when 0 < p < 1, the expression in (1) is a quasi-norm satisfying the
+same axioms of the norm except the triangular inequality making it a non-convex function. For
+a complex number x, we use ¯x to denote the complex conjugate of that number. We let 1 be a
+vector of all entries equal to 1, 0 is a vector of zeros and 1X(.) be the indicator function to the
+set X, i.e., it evaluates to zero if its argument belongs to the set X and is +∞ otherwise. The
+compact set formed by the union of the interior and boundary of a unit circle, i.e. unit desk, on
+
+6
+Fig. 1: System model.
+the complex domain centered around the origin is denoted by D. Finally, for a matrix X, we let
+vec(X) be the vector formed by stacking its rows.
+III. SYSTEM DESCRIPTION
+We consider the system shown in figure 1, where a discrete time input u(k) on a finite time
+horizon is applied to a linear time invariant (LTI) system G. In control systems, the technology
+used to sense the process variable (output of the controlled process) often introduces noise,
+e.g., noise in electrical signals is due to interference from other electrical sources. We let a
+measurement noise n(k) ∈ Iϵ be added to the system output y(k). The noisy output ˆy(k) is then
+measured by the effect of a sensor that quantizes its input to discrete samples z(k). In the next
+part, we describe each component in figure 1 thoroughly.
+A. LTI system G
+We consider a stable finite dimensional LTI system G with poles that are contained in the
+compact set D. The transfer function of the system, in the z-domain, can then represented as
+H(z) = r +
+�
+q∈D
+aq
+z − q,
+(2)
+with r ∈ R and aq ∈ C being the coefficient that is associated with pole q. For systems with
+repeated poles, an approximation by systems with transfer functions as in (2) can be made with
+an arbitrary small precision level.
+B. Input data
+The system G models the relationship between the input u(k) and the output signal y(k).
+Besides boundness, we impose no constraints on the values of the samples of u(k). The stability
+of the system G ensures that the output y(k) is bounded as long as u(k) is bounded as well.
+
+G7
+Fig. 2: Input/output data example. The circle indicates that the data is missing at that instance.
+As mentioned before, and without loss of generality, we assume discrete time data with a
+sampling time of 1 unit. Moreover, we do not require continuous measurement of data. More
+precisely, input data is divided into multiple sets where continuous measurements are available.
+We refer to these sets as ”chunks”.
+The upper part of figure 2 provides an input stream example, where T different input data
+chunks, with size ni for chunk i ∈ [T], are presented. For chunk i, the input sample at instance
+kj is denoted by u(k(i)
+j ) where j ∈ [ni]. Separations between different chunks as well as their
+sizes are arbitrary.
+C. Quantizer and output data
+We assume a general quantizer, Q, that consists of the set of intervals S = {Si, i ∈ I}, with
+the index set I as ordinarily a collection of consecutive integers beginning with 1, together with
+a set of quantization levels L = {Li, i ∈ I}, so that the overall quantizer is defined by Q(x) = Li
+for x ∈ Si. The sets Si partition the real line. That is, the cells are disjoint and exhaustive [9].
+Without loss of generality, we assume a symmetric quantizer, where L|I| = −L1 as the saturation
+level of the quantizer. Figure 3 provides an example for a uniform symmetric quantizer with
+2m-levels, m = 3, saturation value of 1 and a quantization step ∆ =
+1
+2m−1−0.5 = 0.2857. A
+cosine signal S(t) is applied to the quantizer to produce the discrete signal ¯S(t).
+In addition, we do not assume that the all the output data stream is available within a chunk,
+i.e., the data is subject to fragmentation. This arises in cases when intermittent measurements
+are collected from the sensor or failure in communication occurs. The second part of Figure 2
+
+8
+Fig. 3: Sensor operation example.
+provides an output example of a uniform 2m levels sensor, with m = 2 and a saturation level of
+1.5, where the output for chunk i ∈ [T] at instance kj is denoted by z(k(i)
+j ) with j ∈ [ni].
+As mentioned in the previous section, the input data stream (and correspondingly the output)
+chunks’ separations are arbitrary and hence, we assume that the data from different chunks is
+independent. This, along with the time invariance assumption of the system, makes it reasonable
+to assume that the data chunks starting instances are the same, i.e., k(i)
+1 = 1 for all i ∈ [T]. For
+ease of notation, we drop the subscript and let u(i)(k) and z(i)(k) represent the input and output
+samples respectively of chunk i ∈ [T] where k ∈ [ni].
+IV. PROBLEM STATEMENT
+Given input/output data u(i)(k) and z(i)(k), we aim to reconstruct the least order system that
+is compatible with the input output information and a priori assumptions on the system. More
+formally, the problem we aim to address can be stated as follows
+Problem. Given
+• Set D that contains the poles of the LTI system G.
+• Input data chunks u(i)(k), k ∈ [ni], i ∈ [T], which are applied to the system G.
+• A range Iϵ which includes the measurement noise n(i)(k), k ∈ [ni] and i ∈ [T].
+• Measurements of the fragmented sensor output realizations z(i)(k) for k ∈ Ki ⊆ [ni].
+
+S(t)
+s(t)
+Quantizer
+ue
+0.5
+0.5
+0.5
+ signal vali
+0
+screte
+-0.5
+-0.5
+-0.5
+1
+0
+5
+10
+1
+0
+1
+0
+5
+10
+Time
+Continous sianal yalue
+Time9
+find the most parsimonious system that is compatible with the a priori assumptions and a
+posteriori data mentioned above.
+Remark. The formulation above assumes only the following a priori information, which are;
+1)the system is stable and 2)the noise is bounded in Iϵ. However, any other a priori information
+on the system G that can be translated to constraints on the position of the poles (such as
+settling time), is compatible with the approach presented in this paper.
+A. Parsimonious identification as a block sparsification problem
+From the definition of linear systems, the output at instance k ∈ [ni] within chunk i ∈ [T],
+y(i)(k), can be decomposed as,
+y(i)(k) = y(i)
+zi (k) + y(i)
+zs (k),
+(3)
+where, y(i)
+zi (k) is the zero input response at instance k of chunk i, i.e., the response due to
+the initial conditions of the system before the input is applied, while y(i)
+zs (k) is the zero state
+response. From [33], the zero input response can be written as,
+y(i)
+zi (k) =
+�
+q∈D
+b(i)
+q qk−1,
+∀k ∈ [ni],
+∀i ∈ [T],
+(4)
+such that, similar to (2), b(i)
+q
+∈ C is the coefficient that is associated to pole q for chunk i.
+The zero state response is obtained by convolving the input sequence with the system’s impulse
+response,
+y(i)
+zs (k) =
+k
+�
+m=0
+u(i)(m)h(k − m),
+∀k ∈ [ni],
+∀i ∈ [T],
+(5)
+where h(k)
+∆= Z−1
+z [H](k) is the system’s impulse response and Z−1
+z [H](k) is the inverse z-
+transform of H(z) with index k. By taking the inverse z-transform of (2), the impulse response
+can be easily found to be
+h(k) = δ(k)r +
+�
+q∈D
+aqqk−1step(k − 1),
+(6)
+where δ(k) is the dirac delta functional and step(·) is the step function defined as,
+step(k) =
+�
+�
+�
+�
+�
+1
+if k ≥ 0
+0
+if k < 0
+.
+
+10
+Since system complexity and order are always related with the number of poles used to describe
+the system, we aim to reconstruct the system and the associated noise realization n(k) for each
+sample, given only the quantized realizations z(k), that can be depicted by the least number
+of poles. First, we let Υ : D → CT+1 be the mapping from every pole q to the corresponding
+coefficients aq and b(i)
+q , i.e., Υ(q) =
+�
+aq
+b(1)
+q
+. . .
+b(T)
+q
+�⊤
+. The problem mentioned earlier can
+then be formulated such that, for all k ∈ Ki, i ∈ [T] and q ∈ D, we solve;
+min
+aq,b(i)
+q ,r,n(i)(k)
+Cardinality{q ∈ D : Υ(q) ̸= 0},
+(7a)
+s.t.
+y(i)(k) = y(i)
+zi (k) + y(i)
+zs (k),
+(7b)
+y(i)
+zi (k) =
+�
+q∈D
+b(i)
+q qk−1,
+(7c)
+y(i)
+zs (k) =
+k
+�
+m=0
+u(i)(m)h(k − m),
+(7d)
+h(k) = δ(k)r +
+�
+q∈D
+aqqk−1step(k − 1),
+(7e)
+ˆy(i)(k) = y(i)(k) + n(i)(k),
+(7f)
+z(i)(k)=Q(ˆy(i)(k)),
+(7g)
+n(i)(k) ∈ Iϵ,
+(7h)
+aq = ¯a¯q,
+b(i)
+q = ¯b(i)
+¯q .
+(7i)
+Constraint (7i) implies that the coefficients that are associated with complex conjugate poles
+have to be complex conjugate as well.
+V. PROPOSED SOLUTION
+Theoretically, we aim to solve the problem in (7). However, this is not feasible because the
+unit circle contains an infinite number of poles which makes the computational complexity of
+the problem intractable. We aim to implement an approximation of the above problem which
+is based on using a grid of the unit circle of size n. The denser the grid, the more accurate
+the approximation is to the original problem. However, a trade-off could exist as it increases
+the problem’s computational complexity. First, we define the vector q⊤= [q1, . . . qn], which is
+composed of complex conjugates and real poles resulted from the gridding effect, the vector of
+the associated zero state coefficients a⊤=[aq1, . . . aqn] and the matrix of zero input coefficients
+
+11
+B ∈ CT×n, where Bi,:= [b(i)
+q1 , . . . b(i)
+qn]. We also let ni ∈ R|Ki| be the vector of noise realizations
+n(i)(k) for k ∈ Ki with chunk i ∈ [T].
+Second, we aim to equalize the energy contribution of all the poles and hence, we let the
+scaling factor α ∈ Rn be defined as,
+αm =
+1 − |qm|2
+1 − |qm|2N+2
+∀m ∈ [n].
+(8)
+The scaling factor α aims to make the Hankel matrix formed by the system’s impulse response
+has a nuclear norm equal to 1. For more information on α and its proper choice, the interested
+reader is referred to [34]. A good approximation for the problem in (7) can then be defined such
+that, for all k ∈ Ki, i ∈ [T] and j ∈ [n], we aim to solve;
+min
+a,B,r,ni,d ∥d∥0 ,
+(9a)
+s.t.
+y(i)(k) = y(i)
+zi (k) + y(i)
+zs (k),
+(9b)
+y(i)
+zi (k) =
+�
+j∈[n]
+αjb(i)
+qj qk−1
+j
+,
+(9c)
+y(i)
+zs (k) =
+k
+�
+m=0
+u(i)(m)h(k − m),
+(9d)
+h(k) = δ(k)r +
+�
+q∈D
+aqqk−1step(k − 1),
+(9e)
+ˆy(i)(k) = y(i)(k) + n(i)(k),
+(9f)
+z(i)(k)=Q(ˆy(i)(k)),
+(9g)
+n(i)(k) ∈ Iϵ,
+(9h)
+aqj = ¯a¯qj,
+bqj = ¯b¯qj,
+(9i)
+|a| ⪯ d,
+|Bi,:| ⪯ d.
+(9j)
+The auxiliary variable d ∈ Rn
++ ensures block sparsity of the zero state and zero input coefficients,
+i.e., a and B. A proper choice of the vector α, defined in (8), and the use of (9a) and (9j) allow
+the identification of the system with the least number of poles, i.e., least order system. However,
+the ℓ0 pseudo-norm is an NP hard problem and hence, using notions of sparsity [35], the objective
+function is relaxed using the ℓp(0 < p < 1) quasi-norm, i.e., ∥d∥0 in (9a) is replaced with ∥d∥p
+p
+defined as in (1).
+
+12
+For notation simplicity, we define the vector w ∈ C1+n(T+1)+�
+i∈[T ] |Ki|, which is the concate-
+nation of the variables r, a, vec(B) and ni for i ∈ [T]. Let the set D ⊆ C1+n(T+1)+�
+i∈[T ] |Ki|×Rn
++
+as the set of doubles (w, d) where constraints (9b) to (9j) are satisfied. Hence, the problem in
+(9), after the objective function relaxation, will have the compact representation in the form;
+min
+w,d
+∥d∥p
+p ,
+(10a)
+s.t.
+w, d ∈ D.
+(10b)
+As discussed, we aim to recover the lowest order system and hence, we consider the case
+when 0 < p < 1, which lead to a non-convex objective in (10). In our anaylysis, we consider an
+ADMM approach that utilizes the structure of the problem in order to divide the optimization
+over the variables via iteratively solving simpler sub-problems. Starting with the epi-graph form
+of (10) through introducing the auxiliary variable t ∈ Rn, where,
+min
+w,d,t
+1⊤t,
+(11)
+s.t.
+ti ≥ |di|p,
+i ∈ [n],
+w, d ∈ D.
+Let the non-convex set X ⊂ R2 be the epigraph of the scalar function |d|p, i.e., X = {(d, t) ∈
+R2 : t ≥ |d|p}. Then, (11) can be cast as
+min
+w,d,t
+�
+i∈[n]
+1X(di, ti) + 1⊤t,
+(12)
+s.t.
+w, d ∈ D.
+In order to write (12) in an ADMM form, we introduce the variables s ∈ C1+n(T+1)+�
+i∈[T ] |Ki|,
+f and z ∈ Rn, and hence, an equivalent ADMM formulation can be then given by:
+min
+w,d,t,s,f,z
+�
+i∈[n]
+1X(di, ti) + gD(s, f) + 1⊤z,
+(13)
+s.t.
+w = s :
+λ1,
+d = f :
+λ2,
+t = z :
+θ.
+
+13
+The dual variables associated with the constraints w = s, d = f and t = z are λ1, λ2 and θ,
+respectively. Hence, the Lagrangian function corresponding to (13) augmented with a quadratic
+penalty on the violation of the equality constraints with penalty parameter ρ > 0, is given by:
+Lρ(d, t, s, f, w, z, λ1, λ2, θ) =
+�
+i∈[n]
+1X(di, ti) + gD(s, f)+
+1⊤z + λ⊤
+1(w − s) + λ⊤
+2(d − f) + θ⊤(t − z) + ρ
+2(∥w − s∥2
+2
++ ∥d − f∥2
+2 + ∥t − z∥2
+2).
+(14)
+Considering the three block variables Q1 = (d, t), Q2 = (s, f) and Q3 = (w, z), ADMM
+[36] consists of the following iterations, where l is the iteration number:
+Q{l+1}
+1
+=argmin
+d,t
+Lρ(Q1, Q{l}
+2 , Q{l}
+3 ,λ{l}
+1 ,λ{l}
+2 ,θ{l}),
+(15)
+Q{l+1}
+2
+=argmin
+s,f
+Lρ(Q{l+1}
+1
+,Q2,Q{l}
+3 ,λ{l}
+1 ,λ{l}
+2 ,θ{l}),
+(16)
+Q{l+1}
+3
+=argmin
+w,z
+Lρ(Q{l+1}
+1
+,Q{l+1}
+2
+,Q3,λ{l}
+1 ,λ{l}
+2 ,θ{l}),
+(17)
+λ{l+1}
+1
+=λ{l}
+1
++ ρ(w{l+1} − s{l+1}),
+(18)
+λ{l+1}
+2
+=λ{l}
+2
++ ρ(d{l+1} − f {l+1}),
+(19)
+θ{l+1}
+1
+=θ{l}
+1
++ ρ(t{l+1} − z{l+1}).
+(20)
+A. (d, t) update
+From the expression of the augmented Lagrangian in (14) and by completing the square, the
+update of d and t in (15) can be found by solving the following optimization,
+min
+d,t
+∥d − (f {l} − λ{l}
+2
+ρ )∥2
+2 + ∥t − (z{l} − θ{l}
+ρ )∥2
+2,
+s.t.
+(di, ti) ∈ X
+∀i ∈ [n].
+(21)
+It can be realized that the problem in (21) enjoys a separable structure and hence is amenable
+to decentralization. However, it is a non-convex problem due to the nature of the set X. In [37],
+the authors considered a similar problem and it was shown that the element-wise optimization
+of (21) boils down to finding the roots, a∗
+i , of the scalar 2v polynomial;
+a2v
+i + u
+v
+�
+a2u
+i − ˜tiau
+i
+�
+− ˜xiav
+i ,
+(22)
+where ˜xi = f {l}
+i
+−
+λ{l}
+i,2
+ρ , ˜ti = z{l}
+i
+− θ{l}
+i
+ρ
+and u, v ∈ Z+ such that p = u/v. They showed that, in
+proposition 1, the entry-wise solution of (21) is given by (d∗
+i , t∗
+i ) = (a∗v
+i , a∗u
+i ) for all i ∈ [n].
+
+14
+B. (s, f) update
+By fixing all the remaining variables, the (s, f) update in (16) can be easily shown to be the
+solution of the following optimization problem;
+min
+s,f
+∥s−(w{l}+ λ{l}
+1
+ρ )∥2
+2+∥f −(d{l+1} + λ{l}
+2
+ρ )∥2
+2,
+s.t.
+(s, f) ∈ D.
+(23)
+The problem in (23) is clearly a convex optimization one that can be solved by various methods
+including sub-gradient projection [38], interior point and ellipsoid methods [39], [40].
+C. (w, z) update
+From the Lagrangian expression in (14), the w update can be found by solving;
+w{l+1} = argmin
+w
+∥w − (s{l+1} − λ{l}
+1
+ρ )∥2
+2
+= s{l+1} − λ{l}
+1
+ρ ,
+(24)
+while that of z is given by;
+z{l+1} =argmin
+z
+1⊤z+θ{l}⊤(t{l+1}−z)+ ρ
+2∥t{l+1}−z∥2
+2
+=t{l+1}+ θ{l}−1
+ρ
+.
+(25)
+The steps of the ADMM algorithm described in the previous sections can then be summarized
+as in algorithm 1.
+VI. NUMERICAL RESULTS
+In this section, we validate the ability of algorithm 1 in solving problem (10). For comparison
+purposes, we use a convex relaxation of (9), using the ℓ1 norm in the objective, as a baseline.
+We did not include any other solution methods discussed in the literature due to the lack of their
+ability to handle the stability of the system when data fragmentation takes place. Our numerical
+results consists mainly of two parts; 1) System identification with random data. 2) Identification
+with real data from a flexible robot arm. In the next parts, we assume that p = 0.5, i.e., ℓ0.5.
+With this selection of p, the algorithm converges more quickly and the polynomial root finding
+problem in (22) is easier to solve. Numerical experiments were carried out for various values
+of p, i.e., p ∈ {1
+3, 1
+4}, however, they were not found to outperform the ℓ0.5 case. Therefore, they
+are not included in the numerical results section and still under investigation.
+
+15
+Algorithm 1 ADMM algorithm
+1: Initialize: w, z, s, f, λ1, λ2, θ, ρ, k = 0, v = 1, u = 2.
+2: repeat
+3:
+for i ∈ [n] do
+4:
+solve a2v
+i + u
+v
+�
+a2u
+i − ˜tiau
+i
+�
+− ˜xiav
+i = 0
+5:
+(d{l+1}
+i
+, t{l+1}
+i
+) = (a∗v
+i , a∗u
+i )
+6:
+ˆd = d{l+1}+ λ{l}
+2
+ρ , ˆw = w{l}+ λ{l}
+1
+ρ
+7:
+(s{l+1}, f {l+1})=argmin
+s,f∈D
+∥s − ˆw∥2
+2 + ∥f − ˆd∥2
+2
+8:
+w{l+1} = s{l+1} − λ{l}
+1
+ρ
+9:
+z{l+1} = t{l+1} + θ{l}−1
+ρ
+10:
+λ{l+1}
+1
+= λ{l}
+1
++ ρ(w{l+1} − s{l+1})
+11:
+λ{l+1}
+2
+= λ{l}
+2
++ ρ(d{l+1} − f {l+1})
+12:
+θ{l+1}
+1
+= θ{l}
+1
++ ρ(t{l+1} − z{l+1})
+13:
+l = l + 1
+14: until convergence
+A. System identification with random data
+We consider four data chunks, T = 4, with 50 samples per chunk, where the samples of
+each chunk are drawn independently from a symmetric uniform distribution on the interval I5.
+Input chunks are applied to a randomly generated stable LTI system with a known order, where
+the initial conditions of the zero input response for each chunk are initialized through samples
+of zero mean Gaussian distribution with standard deviation σ = 10−2. We assume a uniform
+gridding of the unit circle into n = 146 points. As mentioned before in section V, the denser
+the grid of the the unit circle is, the better the system is represented but the more complex it
+will be. From [41], our choice is a good approximation. Realization noise is added to the LTI
+system’s output, where samples of the noise, n(k), are drawn independently from a uniform
+distribution on the interval I0.25. We assume a symmetric 2m-levels, m = 3, uniform quantizer
+that maps the entire domain I∞ to 23 levels equally spaced on the interval I3 with quantization
+step ∆ = 0.8571. 5 samples per chunk, (10%) of the chunk size, are missing from the quantizer
+output, where the instances of the missing chunks are random and independent from each other.
+It is important to highlight that the chunks’ sizes and number of missing samples per chunk
+
+16
+could be arbitrary and different among chunks, however, we only assumed that these quantities
+are equal among chunks to simplify the implementation.
+All the other parameters in step 1 of algorithm 1 are initialized through samples from a
+Gaussian distribution of zero mean and 10−1 standard deviation. The value of ρ is set to 20. We
+define a threshold ¯ϵ as the value below which a vector entry is considered zero. The value of
+the threshold ¯ϵ is chosen such that it is less than 0.5% of the maximum value of the optimal
+vector d, which makes ¯ϵ = 10−3 a good choice. The algorithm stops if either ∥d − f∥2 ≤ 10−2
+or an iteration budget of 100 iterations is consumed. This budget value was determined through
+a process of trial and error across several repetitions of the experiment. In some cases, the
+algorithm’s output of ∥d − f∥2 converges to a value that is only slightly greater than 10−2, but
+very close to it. Figure 4a shows the convergence of ∥d−f∥2
+∥f∥2
+with respect to the iteration number
+for a single run. It can be realized that a budget of around 80 iterations is enough for the
+algorithm to converge. We perform two different experiments: 1) A single system is considered
+and different properties from ℓ1 and ℓ0.5 relaxations are compared. 2) Multiple systems with
+same original order are generated and the different statistical properties are studied.
+1) Single system experiment: In this subsection, we consider the experiment where an input
+is applied to a stable randomly generated system of order 10. Noise is then added to the output
+and then applied to the quantizer. The noise values and quantizer setup are as discussed above.
+Given the sensor outputs, the problem is solved via ℓ1 and ℓ0.5 relaxations and the detected
+system orders and outputs are compared.
+Figure 4b plots the original system poles vs those that are associated with the non zero
+coefficients in the vector a and matrix B from the ℓ1 and ℓ0.5 relaxations’ solutions. From the
+figure, it can be concluded that the ℓ0.5 detected a system of order 5 which is less complex than
+the system of order 11 detected by the ℓ1 relaxation. This outlines the out-performance of the
+ℓ0.5 quasi-norm when compared to the ℓ1 convex relaxation.
+In figures 4c and 4d, we plot the sensor input ˆy(k) = y(k) + n(k) and output z(k), vs a finite
+time horizon N for the fourth chunk. The figures show how accurate the considered relaxations,
+whether ℓ1 or ℓ0.5, can represent the sensor inputs and outputs. We define the sensor input
+representation error across a time horizon of length N as, ζin
+x , x ∈ {ℓ1, ℓ0.5} where;
+ζin
+x =
+�
+�
+�
+�
+N−1
+�
+k=0
+(ˆy(k) − ˆyx(k))2,
+(26)
+
+17
+with ˆy(k) as the noisy output from the original system. For the ℓ0.5 relaxation, the representation
+error ζin
+ℓ0.5 was found to be equal 3.3204 which is less than that of the ℓ1 convex relaxation that
+had a value ζin
+ℓ1 = 4.2520. It is important to note that we are not interested in perfectly fitting the
+original system’s output. However, we aim to fit the sensor’s realizations. Hence, we similarly
+define the sensor output representation error ζout
+x , x ∈ {ℓ1, ℓ0.5}, such that,
+ζout
+x
+=
+��
+k∈K4
+(z(k) − zx(k))2,
+(27)
+where, z(k) and zx(k) in (27) are the discrete outputs from the original sensor and the considered
+algorithms while K4 is the set of time indices where the data is available for the fourth chunk.
+From figure 4d, it can be realized that ζout
+ℓ0.5 = ζout
+ℓ1 = 0. In both figures 4c and 4d, the sensor
+levels are indicated by the dotted horizontal lines. The missing instances are marked by ‘x’
+symbol. It can be realized that both algorithms perform a decent job in reconstructing the sensor
+input and output samples at those missing instances.
+B. Multiple system experiment
+Since the systems that we generate to validate our solution method are random, the main idea
+in this part is to study the statistical properties of the derived algorithm solution. We perform an
+experiment where for a given original order, 50 random systems are generated. For each system,
+the same input is applied and the identification problem in (10) is solved, using the ℓ1 norm and
+ℓ0.5 quasi-norm relaxations, given the quantized realizations from the sensor output.
+Figure 5 outlines the different statistical properties from the ℓ1 and ℓ0.5 relaxations. It can
+be realized that for all original system orders, the ℓ0.5 relaxation solution enjoys less mean
+and median values than its counterpart, i.e., ℓ1 relaxation. Moreover, the ℓ0.5 relaxation has a
+maximum value for each original order that is less than that of the ℓ1. It can also be realized
+that in either cases, some systems have a detected order of zero, i.e., the minimum value of the
+whisker is zero, which means that the estimation of the constant r in (6) is enough to describe
+the I/O relationship. Finally, some systems are detected with higher order than the original, this
+because the ℓ0.5 minimization is a non convex problem and hence algorithm 1 converges to a
+local minimum. Moreover, it motivates that the unit circle should be gridded into more points
+to increase precision, i.e., n > 146 mentioned in VI-A, in expense of computational complexity.
+
+18
+(a) Convergence vs iteration number.
+(b) System poles.
+(c) Sensor input
+(d) Sensor output.
+Fig. 4: a) The algorithm convergence. b-d) Single system experiment results. Blue squares, red
+stars and green diamonds are the original, ℓ1 and ℓ0.5 relaxations respectively. Dotted lines in 4c
+and 4d are the used sensor levels. ‘x’ indicates that output data is missing at that instance
+C. System identification using data from a flexible robot arm
+In this part, we consider the identification problem using data collected from the motion of
+a flexible robotic arm. As described in [31], the arm is installed on an electrical motor, where,
+the input represents the reaction torque of the structure to the ground while the output is the
+acceleration of the arm. The data is composed of 1024 samples, which we divide into 20 chunks
+of 50 samples each and hence, we drop the last 24 samples of the data set. We assume a uniform
+2m-levels, m = 2, quantizer that maps I∞ to 22 levels equally spaced on I0.7 with a quantization
+step ∆ = 7/30. Similar to as described in VI-A, we drop 10% of the chunk’s samples, where
+
+0.8
+0.6
+d
+0.4
+0.2
+0
+0
+20
+40
+60
+80
+100
+Iteration numberOriginal poles
+li poles
+lo.5 poles
+Immaginary part
+0.5
+0
+-0.5
+-1
+-1
+-0.5
+0
+0.5
+1
+Real part15
+Original
+6.1
+10
+l0.5
+(M)u+()K
+5
+0
+-5
++
+-10
+0
+10
+20
+30
+40
+50
+k4
+Ori
+inal
+2
+K
+0
+X
+N
+-4
+0
+10
+20
+30
+40
+50
+k19
+Fig. 5: Box plot for the system order statistics. Circles with dots and black squares indicate the
+median and mean values respectively. Bottom/top edges of the boxes are the 25th/75th quantile.
+The whiskers extend from the minimum (downwards) to the maximum (upwards) value.
+the location of the missing samples are chosen at random. For the ℓ0.5 quasi norm algorithm, we
+use the same algorithm initialization as in the previous section while setting ρ to 50 and making
+the algorithm terminates if a budget of 100 iterations is consumed.
+We report the results for the first available data chunk with a threshold value ¯ϵ = 10−3. Figure
+6a plots the detected system order vs ϵ which defines the noise boundaries in the range Iϵ, i.e.
+n(i)(k) ∈ Iϵ. It can be realized from figure 6a that, for both the ℓ1 norm and the ℓ0.5 quasi norm,
+the detected order decreases with the increase of ϵ. This is intuitive because on increasing ϵ,
+the size of the feasibility set increases which enables systems of lower orders to be explored.
+Moreover, a momentarily increase in the system order can happen while increasing ϵ. This is
+because we mainly aim to minimize a relaxed version in (10) instead of the original one in (7)
+and hence, more non zero low value entries can decrease the objective of (10). For all values of ϵ,
+the ℓ0.5 quasi norm algorithm detects a lower order than the ℓ1 convex relaxation. For a chunk of
+size 50 samples, it can be realized that the ℓ1 norm objective recovers systems of orders ∼ 35:40,
+for small values of ϵ. This indicates that the ℓ1 relaxation tends to over fit the data for low values
+of ϵ, while the ℓ0.5 one aims to recover a model which accurately represents it. The norm of
+
+l1
+l0.5
+35
+35
+30
+30
+Identified order
+25
+25
+Identified order
+20
+20
+15
+15
+10
+10
+0
+5
+5
+:
+:
+0
+0
+2
+4
+6
+8
+10
+2
+4
+6
+8
+10
+Original order
+Original order20
+(a) System order vs noise range Iϵ.
+(b) System output error vs noise range Iϵ.
+Fig. 6: Robotic arm experiment results.
+the noiseless system output (quantizer input) error, denoted by ∥y(k) − yx(k)∥2 , x ∈ {ℓ1, ℓ0.5},
+is plotted in figure 6b. With the same justification as in Figure 6a, more systems that might
+have a lower order but higher output error are added to the feasible set when the value of ϵ
+is increased. We are not concerned in exactly fitting the output of the original system, as was
+covered in section VI-A1. However, our goal is to choose the system from the feasibility set
+that fits the realizations of the sensor while having the lowest order.
+VII. CONCLUSION
+In this paper, we presented an approach that aims to find the least order system that is
+compatible with fragmented quantized realizations. This approach allows for the use of a priori
+information on the system and fragmented measurements of the output. The algorithm is based on
+an ADMM approach that aims to solve an ℓp quasi-norm objective by dividing the optimization
+over the variables through iteratively solving simpler sub-problems. The algorithm is tested
+on a synthetic data set, that is randomly generated, and a realistic data set collected through
+the measurement of the movement of a robotic arm. Numerical results presented show that
+the algorithm is very effective in obtaining low complexity explanations of the data collected.
+Further effort is being put into analyzing the convergence of the proposed algorithm, improving
+the numerical performance and its extension to continuous-time systems.
+
+3
+= α
+α = lo.5
+2.5
+l()"
+N
+2
+一
+()
+1.5
+0
+0.1
+0.2
+0.3
+0.4
+Noise limit50
+40
+System order
+30
+20
+10
+0
+0
+0.1
+0.2
+0.3
+0.4
+Noise limit21
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+page_content='1  Parsimonious System Identification from  Fragmented Quantized Measurements  Omar M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='Sleem and Constantino M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Lagoa  Dep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' of Electrical Engineering, Pennsylvania State University, State College, PA 16801, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Email: { oms46@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='edu, cml18@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='edu }  Abstract  Quantization is the process of mapping an input signal from an infinite continuous set to a countable  set with a finite number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' It is a non-linear irreversible process, which makes the traditional  methods of system identification no longer applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In this work, we propose a method for parsimo-  nious linear time invariant system identification when only quantized observations, discerned from noisy  data, are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' More formally, given a priori information on the system, represented by a compact  set containing the poles of the system, and quantized realizations, our algorithm aims at identifying  the least order system that is compatible with the available information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The proposed approach takes  also into account that the available data can be subject to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Our proposed algorithm relies  on an ADMM approach to solve a ℓp, (0 < p < 1), quasi-norm objective problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Numerical results  highlight the performance of the proposed approach when compared to the ℓ1 minimization in terms of  the sparsity of the induced solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Index Terms  System identification, Sparsity, Quantization, ADMM  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' INTRODUCTION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Motivation  Quantization is the division of a quantity into a discrete number of small parts, often assumed  to be integral multiple of a common quantity [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' A classical example of quantization by  rounding off, for the application of estimating densities of histograms, was analyzed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Since  the processing of signals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' speech and image, requires a digital environment, quantization plays  an important role in bridging the analog and digital worlds [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' On one hand, quantization led to  a new research area in control theory called network controlled systems (NCS) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' NCS deals  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='12043v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='SY]  28 Jan 2023  2  with the idea of controlling a process when the input and output signals are transmitted via a  communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' On the other hand, it revealed the incompetence of the classical theory  of system identification in considering quantized measurements [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  When a signal is subject to quantization, the quantization noise can no longer be modeled as  a filtered white (zero mean and independent over time) noise and is signal dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Hence, in  [6], [7] and references therein, the traditional theory of system identification was suggested to  be modified to tackle the fact that the measurements are subject to quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Moreover, from  [8] (section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='1), the classical identification procedures are not suitable for robust identification,  when the signal is subject to quantization, because they identify a set of parameters of a fixed  mathematical structure, where a fixed system order must be assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Inspired by this, various works –which will be discussed in the next section in more detail–  explored the problem of system identification given quantized realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, in this  paper we aim to present a new approach to the problem of (Linear Time Invariant) LTI system  identification from quantized outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This approach allows for the use of a priori information  on the system and fragmented measurements of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In addition, our approach aims to  recover the least order system that is compatible with the data by minimizing an ℓp, (0 < p < 1),  quasi-norm objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  The paper is organized as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' in the remaining of the introduction, we provide a compre-  hensive discussion of the previous related work and our contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Section II introduces the  notations that are used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In section III, we thoroughly describe the system  model used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The parsimonious system identification problem is formally provided in section  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The proposed (Alternating Direction Method of Multipliers) ADMM algorithm based on ℓp  quasi-norm approximation is described in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We validate our approach with an extensive  suite of numerical simulations in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Finally, the paper is concluded in section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Related work  The problem of simple representation of signals using quantization dates back to the 1940’s  and is one of the main threads of information theory [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, rigorous analysis did not  begin until the 1980’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In [10], [11], considering digital feedback control systems, the authors  proposed a way in which one can specify system structures that alleviate the adverse effects of  quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The works in [12], [13] demonstrate that quantization can induce a chaotic behavior  in digital feedback systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The results in [14], [15] are recognized as a quantum leap because  3  the author was able to analyze the behavior of control systems in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The circumstances  under which a discrete unstable LTI system can be stabilized, by choosing feedback control that  depends on the quantized measurements, are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In [16], the authors proposed a control  design methodology, assuming a quantizer with variable sensitivity along with system state, that  stabilizes LTI control systems with quantized measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In [17], the coarsest quantizer that  stabilizes a single input LTI system is shown to be a logarithmic one and can be obtained by  solving a linear quadratic regulator problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Abundant other works investigated the problem of  the stabilization of NCS in different situations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Despite that ample research activity in the stabilization and state estimation, quantization  in system identification problems was still not properly pondered [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In [22], the authors  studied the effect of quantization on I/O data used for system identification in a controlled  plant whose parameters may change during the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' They derived the optimal quantization  scheme and showed that it is coarse near the origin of the signals and dense at a distance  from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This result is opposite to the case of stabilization in [17] and reveals duality between  system identification and stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Similar properties of the optimal quantizer were concluded  in [23], where the author considered a least square error objective function –for parameter  estimation– subject to a constraint on the number of subsections of the quantized signals or the  expectation of the optimal code length for either high or low resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In [24], the problem of  system identification using uniformly quantized realizations was considered, where, the proposed  formulation is a least square minimization of the difference equation errors over all time samples  with the system parameters as optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Regardless of the high accuracy in the  estimation of the unknown information in the I/O data, the proposed method stills suffers the  drawback of high computational complexity and noise neglection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The work in [25] aimed to  solve these drawbacks by exploiting statistical properties instead of deterministic treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In  particular, an identification method for a linear system based on quantized measurements was  derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Using traditional equi-spaced quantizer, an instrumental level identification approach  was proposed to enhance the estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The authors of [26] took this approach a step  further where a variation for the equi-spaced quantizer was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' They showed that using  a generalized noise shaping coder improves the accuracy of the estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Another line of research includes the identification using a general class of quantized observa-  tions that allows the segmentation of the output range into a collection of subsets that may have  unequal, fixed lengths or even design variables such as quantization design in communication  4  systems and NCS [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This serves in favor of understanding the potency of systems with  limited sensor information, which in turns rapports the gap between resource limitations and  identification complexity in sensor and communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In particular, the work in  [28] considered the identification of a gain system by exploiting the information from multiple  thresholds sensor and the convex combination of these thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The results were extended  to the case of a noisy communication channel through which the sensor output information  is transmitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The authors prove that their estimator is asymptotically efficient achieving the  Cramer-Rao lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Furthermore, the results were extended to a finite impulse response  and transfer function models for periodic bounded input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In [29], the authors focused  on relationships between the identification space and time complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' They showed that  the asymptotic efficiency of empirical measure based algorithms yield to a tight bound on  identification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This in turns aids to derive a separation principle of the complexities  (time and space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The gained insights aim to provide a feasible approach for optimal utility  of communication bandwidth resources in magnifying the identification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The role of  dithering noise –adding artificial noise to the observed signal before quantization in order to  mitigate the effects of quantization– at the sensor was studied in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The authors asserted that  tailored dithering noise can considerably simplify the derivation of optimal estimators in the  expense of a decreased signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Contributions  The different methods reviewed in the previous part aim to either stabilize the system, find an  optimal quantization scheme or solve a system identification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In this work, we focus on  the latter problem where, to the best of our knowledge, none of the proposed methods address the  problem of identifying the system of least order that is compatible with collected information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  The problem of identifying systems using collected measurements can also include several  other challenges including;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 1) One can be faced with fragmented data due to the misplacement  of sensors or external disturbances that can possibly make the collected data unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 2) The  ability to handle prior information on the system, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', constraints on the locations of the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  In this paper, we aim to develop an algorithm that tackles the challenges mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  More precisely, we consider a system in which the a priori information can be described  by constraining the locations of the system’s poles to be in a known compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Then, by  exploiting “simple representations” of transfer functions, we develop an efficient algorithm that  5  aims at finding the lowest order system that is compatible with fragmented quantized output  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This algorithm is based on an ADMM approach to the problem of ℓp quasi-norm  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' To validate our analysis, we consider two different numerical examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 1) System  identification with randomly generated data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 2) Identification with actual data collected from  the motion of flexible robotic arm [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The numerical results in both examples show that our  method is competitive against the ℓ1 convex relaxation objective in terms of both the detected  system order and accuracy of the recovered realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  A preliminary version of part of this work was presented in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This journal version includes  a generalized formulation where a non continuous input data stream can be handled (input data  is composed of independent chunks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Moreover, we do not assume a continuous measurement  of the chunks’ outputs from the quantizer, i,e,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' output data is subject to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Unlike  [32], we assume a generalized quantizer whose input is prone to noise and show that it plays  an important role in the sparsity of the induced solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We provide a new experiment to  demonstrate the superior performance of our method in a more practical scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' NOTATIONS  Unless otherwise specified, we denote scalars with non boldface letters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', x, vectors with  lowercase boldface letters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', x, with i-th entry as xi, while matrices are in uppercase, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=',  X, with (i, j)-th entry as xi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Xj,: specifies the j-th row of the matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' R and C are the sets  of real and complex numbers respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For a vector x and matrix X, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='| is an element-wise  absolute value of the applied variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, for a set X, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='| operator stands for the cardinality  of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We use ⪯ for element wise inequality of vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For any constant c > 0, we define  Ic  ∆= [−c, c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For a positive integer n, we let [n]  ∆= {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The p-th norm of a vector x ∈ Rn  is defined such that  ∥x∥p  ∆= (  n  �  i=1  |xi|p)  1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (1)  It is important to note that when 0 < p < 1, the expression in (1) is a quasi-norm satisfying the  same axioms of the norm except the triangular inequality making it a non-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For  a complex number x, we use ¯x to denote the complex conjugate of that number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We let 1 be a  vector of all entries equal to 1, 0 is a vector of zeros and 1X(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=') be the indicator function to the  set X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', it evaluates to zero if its argument belongs to the set X and is +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The  compact set formed by the union of the interior and boundary of a unit circle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' unit desk, on  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 1: System model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  the complex domain centered around the origin is denoted by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Finally, for a matrix X, we let  vec(X) be the vector formed by stacking its rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' SYSTEM DESCRIPTION  We consider the system shown in figure 1, where a discrete time input u(k) on a finite time  horizon is applied to a linear time invariant (LTI) system G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In control systems, the technology  used to sense the process variable (output of the controlled process) often introduces noise,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', noise in electrical signals is due to interference from other electrical sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We let a  measurement noise n(k) ∈ Iϵ be added to the system output y(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The noisy output ˆy(k) is then  measured by the effect of a sensor that quantizes its input to discrete samples z(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In the next  part, we describe each component in figure 1 thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' LTI system G  We consider a stable finite dimensional LTI system G with poles that are contained in the  compact set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The transfer function of the system, in the z-domain, can then represented as  H(z) = r +  �  q∈D  aq  z − q,  (2)  with r ∈ R and aq ∈ C being the coefficient that is associated with pole q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For systems with  repeated poles, an approximation by systems with transfer functions as in (2) can be made with  an arbitrary small precision level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Input data  The system G models the relationship between the input u(k) and the output signal y(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Besides boundness, we impose no constraints on the values of the samples of u(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The stability  of the system G ensures that the output y(k) is bounded as long as u(k) is bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  G7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 2: Input/output data example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The circle indicates that the data is missing at that instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  As mentioned before, and without loss of generality, we assume discrete time data with a  sampling time of 1 unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Moreover, we do not require continuous measurement of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' More  precisely, input data is divided into multiple sets where continuous measurements are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  We refer to these sets as ”chunks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  The upper part of figure 2 provides an input stream example, where T different input data  chunks, with size ni for chunk i ∈ [T], are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For chunk i, the input sample at instance  kj is denoted by u(k(i)  j ) where j ∈ [ni].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Separations between different chunks as well as their  sizes are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Quantizer and output data  We assume a general quantizer, Q, that consists of the set of intervals S = {Si, i ∈ I}, with  the index set I as ordinarily a collection of consecutive integers beginning with 1, together with  a set of quantization levels L = {Li, i ∈ I}, so that the overall quantizer is defined by Q(x) = Li  for x ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The sets Si partition the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' That is, the cells are disjoint and exhaustive [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Without loss of generality, we assume a symmetric quantizer, where L|I| = −L1 as the saturation  level of the quantizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Figure 3 provides an example for a uniform symmetric quantizer with  2m-levels, m = 3, saturation value of 1 and a quantization step ∆ =  1  2m−1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='2857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' A  cosine signal S(t) is applied to the quantizer to produce the discrete signal ¯S(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  In addition, we do not assume that the all the output data stream is available within a chunk,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', the data is subject to fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This arises in cases when intermittent measurements  are collected from the sensor or failure in communication occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The second part of Figure 2  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 3: Sensor operation example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  provides an output example of a uniform 2m levels sensor, with m = 2 and a saturation level of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5, where the output for chunk i ∈ [T] at instance kj is denoted by z(k(i)  j ) with j ∈ [ni].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  As mentioned in the previous section, the input data stream (and correspondingly the output)  chunks’ separations are arbitrary and hence, we assume that the data from different chunks is  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This, along with the time invariance assumption of the system, makes it reasonable  to assume that the data chunks starting instances are the same, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', k(i)  1 = 1 for all i ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For  ease of notation, we drop the subscript and let u(i)(k) and z(i)(k) represent the input and output  samples respectively of chunk i ∈ [T] where k ∈ [ni].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' PROBLEM STATEMENT  Given input/output data u(i)(k) and z(i)(k), we aim to reconstruct the least order system that  is compatible with the input output information and a priori assumptions on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' More  formally, the problem we aim to address can be stated as follows  Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Given  Set D that contains the poles of the LTI system G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Input data chunks u(i)(k), k ∈ [ni], i ∈ [T], which are applied to the system G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  A range Iϵ which includes the measurement noise n(i)(k), k ∈ [ni] and i ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Measurements of the fragmented sensor output realizations z(i)(k) for k ∈ Ki ⊆ [ni].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  S(t)  s(t)  Quantizer  ue  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  signal vali  0  screte  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  1  0  5  10  1  0  1  0  5  10  Time  Continous sianal yalue  Time9  find the most parsimonious system that is compatible with the a priori assumptions and a  posteriori data mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The formulation above assumes only the following a priori information, which are;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  1)the system is stable and 2)the noise is bounded in Iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, any other a priori information  on the system G that can be translated to constraints on the position of the poles (such as  settling time), is compatible with the approach presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Parsimonious identification as a block sparsification problem  From the definition of linear systems, the output at instance k ∈ [ni] within chunk i ∈ [T],  y(i)(k), can be decomposed as,  y(i)(k) = y(i)  zi (k) + y(i)  zs (k),  (3)  where, y(i)  zi (k) is the zero input response at instance k of chunk i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', the response due to  the initial conditions of the system before the input is applied, while y(i)  zs (k) is the zero state  response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' From [33], the zero input response can be written as,  y(i)  zi (k) =  �  q∈D  b(i)  q qk−1,  ∀k ∈ [ni],  ∀i ∈ [T],  (4)  such that, similar to (2), b(i)  q  ∈ C is the coefficient that is associated to pole q for chunk i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  The zero state response is obtained by convolving the input sequence with the system’s impulse  response,  y(i)  zs (k) =  k  �  m=0  u(i)(m)h(k − m),  ∀k ∈ [ni],  ∀i ∈ [T],  (5)  where h(k)  ∆= Z−1  z [H](k) is the system’s impulse response and Z−1  z [H](k) is the inverse z-  transform of H(z) with index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' By taking the inverse z-transform of (2), the impulse response  can be easily found to be  h(k) = δ(k)r +  �  q∈D  aqqk−1step(k − 1),  (6)  where δ(k) is the dirac delta functional and step(·) is the step function defined as,  step(k) =  �  �  �  �  �  1  if k ≥ 0  0  if k < 0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  10  Since system complexity and order are always related with the number of poles used to describe  the system, we aim to reconstruct the system and the associated noise realization n(k) for each  sample, given only the quantized realizations z(k), that can be depicted by the least number  of poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' First, we let Υ : D → CT+1 be the mapping from every pole q to the corresponding  coefficients aq and b(i)  q , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', Υ(q) =  �  aq  b(1)  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  b(T)  q  �⊤  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The problem mentioned earlier can  then be formulated such that, for all k ∈ Ki, i ∈ [T] and q ∈ D, we solve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  min  aq,b(i)  q ,r,n(i)(k)  Cardinality{q ∈ D : Υ(q) ̸= 0},  (7a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  y(i)(k) = y(i)  zi (k) + y(i)  zs (k),  (7b)  y(i)  zi (k) =  �  q∈D  b(i)  q qk−1,  (7c)  y(i)  zs (k) =  k  �  m=0  u(i)(m)h(k − m),  (7d)  h(k) = δ(k)r +  �  q∈D  aqqk−1step(k − 1),  (7e)  ˆy(i)(k) = y(i)(k) + n(i)(k),  (7f)  z(i)(k)=Q(ˆy(i)(k)),  (7g)  n(i)(k) ∈ Iϵ,  (7h)  aq = ¯a¯q,  b(i)  q = ¯b(i)  ¯q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (7i)  Constraint (7i) implies that the coefficients that are associated with complex conjugate poles  have to be complex conjugate as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' PROPOSED SOLUTION  Theoretically, we aim to solve the problem in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, this is not feasible because the  unit circle contains an infinite number of poles which makes the computational complexity of  the problem intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We aim to implement an approximation of the above problem which  is based on using a grid of the unit circle of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The denser the grid, the more accurate  the approximation is to the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, a trade-off could exist as it increases  the problem’s computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' First, we define the vector q⊤= [q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' qn], which is  composed of complex conjugates and real poles resulted from the gridding effect, the vector of  the associated zero state coefficients a⊤=[aq1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' aqn] and the matrix of zero input coefficients  11  B ∈ CT×n, where Bi,:= [b(i)  q1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' b(i)  qn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We also let ni ∈ R|Ki| be the vector of noise realizations  n(i)(k) for k ∈ Ki with chunk i ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Second, we aim to equalize the energy contribution of all the poles and hence, we let the  scaling factor α ∈ Rn be defined as,  αm =  1 − |qm|2  1 − |qm|2N+2  ∀m ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (8)  The scaling factor α aims to make the Hankel matrix formed by the system’s impulse response  has a nuclear norm equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For more information on α and its proper choice, the interested  reader is referred to [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' A good approximation for the problem in (7) can then be defined such  that, for all k ∈ Ki, i ∈ [T] and j ∈ [n], we aim to solve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  min  a,B,r,ni,d ∥d∥0 ,  (9a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  y(i)(k) = y(i)  zi (k) + y(i)  zs (k),  (9b)  y(i)  zi (k) =  �  j∈[n]  αjb(i)  qj qk−1  j  ,  (9c)  y(i)  zs (k) =  k  �  m=0  u(i)(m)h(k − m),  (9d)  h(k) = δ(k)r +  �  q∈D  aqqk−1step(k − 1),  (9e)  ˆy(i)(k) = y(i)(k) + n(i)(k),  (9f)  z(i)(k)=Q(ˆy(i)(k)),  (9g)  n(i)(k) ∈ Iϵ,  (9h)  aqj = ¯a¯qj,  bqj = ¯b¯qj,  (9i)  |a| ⪯ d,  |Bi,:| ⪯ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (9j)  The auxiliary variable d ∈ Rn  + ensures block sparsity of the zero state and zero input coefficients,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', a and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' A proper choice of the vector α, defined in (8), and the use of (9a) and (9j) allow  the identification of the system with the least number of poles, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', least order system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However,  the ℓ0 pseudo-norm is an NP hard problem and hence, using notions of sparsity [35], the objective  function is relaxed using the ℓp(0 < p < 1) quasi-norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', ∥d∥0 in (9a) is replaced with ∥d∥p  p  defined as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  12  For notation simplicity, we define the vector w ∈ C1+n(T+1)+�  i∈[T ] |Ki|, which is the concate-  nation of the variables r, a, vec(B) and ni for i ∈ [T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Let the set D ⊆ C1+n(T+1)+�  i∈[T ] |Ki|×Rn  +  as the set of doubles (w, d) where constraints (9b) to (9j) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Hence, the problem in  (9), after the objective function relaxation, will have the compact representation in the form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  min  w,d  ∥d∥p  p ,  (10a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  w, d ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (10b)  As discussed, we aim to recover the lowest order system and hence, we consider the case  when 0 < p < 1, which lead to a non-convex objective in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In our anaylysis, we consider an  ADMM approach that utilizes the structure of the problem in order to divide the optimization  over the variables via iteratively solving simpler sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Starting with the epi-graph form  of (10) through introducing the auxiliary variable t ∈ Rn, where,  min  w,d,t  1⊤t,  (11)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  ti ≥ |di|p,  i ∈ [n],  w, d ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Let the non-convex set X ⊂ R2 be the epigraph of the scalar function |d|p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', X = {(d, t) ∈  R2 : t ≥ |d|p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Then, (11) can be cast as  min  w,d,t  �  i∈[n]  1X(di, ti) + 1⊤t,  (12)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  w, d ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  In order to write (12) in an ADMM form, we introduce the variables s ∈ C1+n(T+1)+�  i∈[T ] |Ki|,  f and z ∈ Rn, and hence, an equivalent ADMM formulation can be then given by:  min  w,d,t,s,f,z  �  i∈[n]  1X(di, ti) + gD(s, f) + 1⊤z,  (13)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  w = s :  λ1,  d = f :  λ2,  t = z :  θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  13  The dual variables associated with the constraints w = s, d = f and t = z are λ1, λ2 and θ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Hence, the Lagrangian function corresponding to (13) augmented with a quadratic  penalty on the violation of the equality constraints with penalty parameter ρ > 0, is given by:  Lρ(d, t, s, f, w, z, λ1, λ2, θ) =  �  i∈[n]  1X(di, ti) + gD(s, f)+  1⊤z + λ⊤  1(w − s) + λ⊤  2(d − f) + θ⊤(t − z) + ρ  2(∥w − s∥2  2  + ∥d − f∥2  2 + ∥t − z∥2  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (14)  Considering the three block variables Q1 = (d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Q2 = (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' f) and Q3 = (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' ADMM  [36] consists of the following iterations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' where l is the iteration number:  Q{l+1}  1  =argmin  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t  Lρ(Q1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Q{l}  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Q{l}  3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='λ{l}  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='λ{l}  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='θ{l}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (15)  Q{l+1}  2  =argmin  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='f  Lρ(Q{l+1}  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='Q2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='Q{l}  3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='λ{l}  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='λ{l}  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='θ{l}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (16)  Q{l+1}  3  =argmin  w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='z  Lρ(Q{l+1}  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='Q{l+1}  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='Q3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='λ{l}  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='λ{l}  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='θ{l}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (17)  λ{l+1}  1  =λ{l}  1  + ρ(w{l+1} − s{l+1}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (18)  λ{l+1}  2  =λ{l}  2  + ρ(d{l+1} − f {l+1}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (19)  θ{l+1}  1  =θ{l}  1  + ρ(t{l+1} − z{l+1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (20)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' (d, t) update  From the expression of the augmented Lagrangian in (14) and by completing the square, the  update of d and t in (15) can be found by solving the following optimization,  min  d,t  ∥d − (f {l} − λ{l}  2  ρ )∥2  2 + ∥t − (z{l} − θ{l}  ρ )∥2  2,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (di, ti) ∈ X  ∀i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (21)  It can be realized that the problem in (21) enjoys a separable structure and hence is amenable  to decentralization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, it is a non-convex problem due to the nature of the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In [37],  the authors considered a similar problem and it was shown that the element-wise optimization  of (21) boils down to finding the roots, a∗  i , of the scalar 2v polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  a2v  i + u  v  �  a2u  i − ˜tiau  i  �  − ˜xiav  i ,  (22)  where ˜xi = f {l}  i  −  λ{l}  i,2  ρ , ˜ti = z{l}  i  − θ{l}  i  ρ  and u, v ∈ Z+ such that p = u/v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' They showed that, in  proposition 1, the entry-wise solution of (21) is given by (d∗  i , t∗  i ) = (a∗v  i , a∗u  i ) for all i ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  14  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' (s, f) update  By fixing all the remaining variables, the (s, f) update in (16) can be easily shown to be the  solution of the following optimization problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  min  s,f  ∥s−(w{l}+ λ{l}  1  ρ )∥2  2+∥f −(d{l+1} + λ{l}  2  ρ )∥2  2,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (s, f) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (23)  The problem in (23) is clearly a convex optimization one that can be solved by various methods  including sub-gradient projection [38], interior point and ellipsoid methods [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' (w, z) update  From the Lagrangian expression in (14), the w update can be found by solving;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  w{l+1} = argmin  w  ∥w − (s{l+1} − λ{l}  1  ρ )∥2  2  = s{l+1} − λ{l}  1  ρ ,  (24)  while that of z is given by;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  z{l+1} =argmin  z  1⊤z+θ{l}⊤(t{l+1}−z)+ ρ  2∥t{l+1}−z∥2  2  =t{l+1}+ θ{l}−1  ρ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (25)  The steps of the ADMM algorithm described in the previous sections can then be summarized  as in algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' NUMERICAL RESULTS  In this section, we validate the ability of algorithm 1 in solving problem (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For comparison  purposes, we use a convex relaxation of (9), using the ℓ1 norm in the objective, as a baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  We did not include any other solution methods discussed in the literature due to the lack of their  ability to handle the stability of the system when data fragmentation takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Our numerical  results consists mainly of two parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 1) System identification with random data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 2) Identification  with real data from a flexible robot arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In the next parts, we assume that p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  With this selection of p, the algorithm converges more quickly and the polynomial root finding  problem in (22) is easier to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Numerical experiments were carried out for various values  of p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', p ∈ {1  3, 1  4}, however, they were not found to outperform the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Therefore, they  are not included in the numerical results section and still under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  15  Algorithm 1 ADMM algorithm  1: Initialize: w, z, s, f, λ1, λ2, θ, ρ, k = 0, v = 1, u = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  2: repeat  3:  for i ∈ [n] do  4:  solve a2v  i + u  v  �  a2u  i − ˜tiau  i  �  − ˜xiav  i = 0  5:  (d{l+1}  i  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' t{l+1}  i  ) = (a∗v  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' a∗u  i )  6:  ˆd = d{l+1}+ λ{l}  2  ρ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' ˆw = w{l}+ λ{l}  1  ρ  7:  (s{l+1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' f {l+1})=argmin  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='f∈D  ∥s − ˆw∥2  2 + ∥f − ˆd∥2  2  8:  w{l+1} = s{l+1} − λ{l}  1  ρ  9:  z{l+1} = t{l+1} + θ{l}−1  ρ  10:  λ{l+1}  1  = λ{l}  1  + ρ(w{l+1} − s{l+1})  11:  λ{l+1}  2  = λ{l}  2  + ρ(d{l+1} − f {l+1})  12:  θ{l+1}  1  = θ{l}  1  + ρ(t{l+1} − z{l+1})  13:  l = l + 1  14: until convergence  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' System identification with random data  We consider four data chunks, T = 4, with 50 samples per chunk, where the samples of  each chunk are drawn independently from a symmetric uniform distribution on the interval I5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Input chunks are applied to a randomly generated stable LTI system with a known order, where  the initial conditions of the zero input response for each chunk are initialized through samples  of zero mean Gaussian distribution with standard deviation σ = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We assume a uniform  gridding of the unit circle into n = 146 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' As mentioned before in section V, the denser  the grid of the the unit circle is, the better the system is represented but the more complex it  will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' From [41], our choice is a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Realization noise is added to the LTI  system’s output, where samples of the noise, n(k), are drawn independently from a uniform  distribution on the interval I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We assume a symmetric 2m-levels, m = 3, uniform quantizer  that maps the entire domain I∞ to 23 levels equally spaced on the interval I3 with quantization  step ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='8571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 5 samples per chunk, (10%) of the chunk size, are missing from the quantizer  output, where the instances of the missing chunks are random and independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  It is important to highlight that the chunks’ sizes and number of missing samples per chunk  16  could be arbitrary and different among chunks, however, we only assumed that these quantities  are equal among chunks to simplify the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  All the other parameters in step 1 of algorithm 1 are initialized through samples from a  Gaussian distribution of zero mean and 10−1 standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The value of ρ is set to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We  define a threshold ¯ϵ as the value below which a vector entry is considered zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The value of  the threshold ¯ϵ is chosen such that it is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5% of the maximum value of the optimal  vector d, which makes ¯ϵ = 10−3 a good choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The algorithm stops if either ∥d − f∥2 ≤ 10−2  or an iteration budget of 100 iterations is consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This budget value was determined through  a process of trial and error across several repetitions of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In some cases, the  algorithm’s output of ∥d − f∥2 converges to a value that is only slightly greater than 10−2, but  very close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Figure 4a shows the convergence of ∥d−f∥2  ∥f∥2  with respect to the iteration number  for a single run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' It can be realized that a budget of around 80 iterations is enough for the  algorithm to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We perform two different experiments: 1) A single system is considered  and different properties from ℓ1 and ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxations are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 2) Multiple systems with  same original order are generated and the different statistical properties are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  1) Single system experiment: In this subsection, we consider the experiment where an input  is applied to a stable randomly generated system of order 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Noise is then added to the output  and then applied to the quantizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The noise values and quantizer setup are as discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Given the sensor outputs, the problem is solved via ℓ1 and ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxations and the detected  system orders and outputs are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Figure 4b plots the original system poles vs those that are associated with the non zero  coefficients in the vector a and matrix B from the ℓ1 and ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxations’ solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' From the  figure, it can be concluded that the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 detected a system of order 5 which is less complex than  the system of order 11 detected by the ℓ1 relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This outlines the out-performance of the  ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 quasi-norm when compared to the ℓ1 convex relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  In figures 4c and 4d, we plot the sensor input ˆy(k) = y(k) + n(k) and output z(k), vs a finite  time horizon N for the fourth chunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The figures show how accurate the considered relaxations,  whether ℓ1 or ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5, can represent the sensor inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We define the sensor input  representation error across a time horizon of length N as, ζin  x , x ∈ {ℓ1, ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5} where;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  ζin  x =  �  �  �  �  N−1  �  k=0  (ˆy(k) − ˆyx(k))2,  (26)  17  with ˆy(k) as the noisy output from the original system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxation, the representation  error ζin  ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 was found to be equal 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='3204 which is less than that of the ℓ1 convex relaxation that  had a value ζin  ℓ1 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='2520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' It is important to note that we are not interested in perfectly fitting the  original system’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, we aim to fit the sensor’s realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Hence, we similarly  define the sensor output representation error ζout  x , x ∈ {ℓ1, ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5}, such that,  ζout  x  =  ��  k∈K4  (z(k) − zx(k))2,  (27)  where, z(k) and zx(k) in (27) are the discrete outputs from the original sensor and the considered  algorithms while K4 is the set of time indices where the data is available for the fourth chunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  From figure 4d, it can be realized that ζout  ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 = ζout  ℓ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' In both figures 4c and 4d, the sensor  levels are indicated by the dotted horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The missing instances are marked by ‘x’  symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' It can be realized that both algorithms perform a decent job in reconstructing the sensor  input and output samples at those missing instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Multiple system experiment  Since the systems that we generate to validate our solution method are random, the main idea  in this part is to study the statistical properties of the derived algorithm solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We perform an  experiment where for a given original order, 50 random systems are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For each system,  the same input is applied and the identification problem in (10) is solved, using the ℓ1 norm and  ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 quasi-norm relaxations, given the quantized realizations from the sensor output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Figure 5 outlines the different statistical properties from the ℓ1 and ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' It can  be realized that for all original system orders, the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxation solution enjoys less mean  and median values than its counterpart, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', ℓ1 relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Moreover, the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxation has a  maximum value for each original order that is less than that of the ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' It can also be realized  that in either cases, some systems have a detected order of zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', the minimum value of the  whisker is zero, which means that the estimation of the constant r in (6) is enough to describe  the I/O relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Finally, some systems are detected with higher order than the original, this  because the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 minimization is a non convex problem and hence algorithm 1 converges to a  local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Moreover, it motivates that the unit circle should be gridded into more points  to increase precision, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=', n > 146 mentioned in VI-A, in expense of computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  18  (a) Convergence vs iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (b) System poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (c) Sensor input  (d) Sensor output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 4: a) The algorithm convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' b-d) Single system experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Blue squares, red  stars and green diamonds are the original, ℓ1 and ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 relaxations respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Dotted lines in 4c  and 4d are the used sensor levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' ‘x’ indicates that output data is missing at that instance  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' System identification using data from a flexible robot arm  In this part, we consider the identification problem using data collected from the motion of  a flexible robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' As described in [31], the arm is installed on an electrical motor, where,  the input represents the reaction torque of the structure to the ground while the output is the  acceleration of the arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The data is composed of 1024 samples, which we divide into 20 chunks  of 50 samples each and hence, we drop the last 24 samples of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We assume a uniform  2m-levels, m = 2, quantizer that maps I∞ to 22 levels equally spaced on I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='7 with a quantization  step ∆ = 7/30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Similar to as described in VI-A, we drop 10% of the chunk’s samples, where  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='6  d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='2  0  0  20  40  60  80  100  Iteration numberOriginal poles  li poles  lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 poles  Immaginary part  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  1  Real part15  Original  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='1  10  l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  (M)u+()K  5  0  5  +  10  0  10  20  30  40  50  k4  Ori  inal  2  K  0  X  N  4  0  10  20  30  40  50  k19  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 5: Box plot for the system order statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Circles with dots and black squares indicate the  median and mean values respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Bottom/top edges of the boxes are the 25th/75th quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  The whiskers extend from the minimum (downwards) to the maximum (upwards) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  the location of the missing samples are chosen at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 quasi norm algorithm, we  use the same algorithm initialization as in the previous section while setting ρ to 50 and making  the algorithm terminates if a budget of 100 iterations is consumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  We report the results for the first available data chunk with a threshold value ¯ϵ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Figure  6a plots the detected system order vs ϵ which defines the noise boundaries in the range Iϵ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  n(i)(k) ∈ Iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' It can be realized from figure 6a that, for both the ℓ1 norm and the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 quasi norm,  the detected order decreases with the increase of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This is intuitive because on increasing ϵ,  the size of the feasibility set increases which enables systems of lower orders to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Moreover, a momentarily increase in the system order can happen while increasing ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This is  because we mainly aim to minimize a relaxed version in (10) instead of the original one in (7)  and hence, more non zero low value entries can decrease the objective of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For all values of ϵ,  the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 quasi norm algorithm detects a lower order than the ℓ1 convex relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' For a chunk of  size 50 samples, it can be realized that the ℓ1 norm objective recovers systems of orders ∼ 35:40,  for small values of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This indicates that the ℓ1 relaxation tends to over fit the data for low values  of ϵ, while the ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5 one aims to recover a model which accurately represents it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The norm of  l1  l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  35  35  30  30  Identified order  25  25  Identified order  20  20  15  15  10  10  0  5  5  :  :  0  0  2  4  6  8  10  2  4  6  8  10  Original order  Original order20  (a) System order vs noise range Iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  (b) System output error vs noise range Iϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' 6: Robotic arm experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  the noiseless system output (quantizer input) error, denoted by ∥y(k) − yx(k)∥2 , x ∈ {ℓ1, ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5},  is plotted in figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' With the same justification as in Figure 6a, more systems that might  have a lower order but higher output error are added to the feasible set when the value of ϵ  is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' We are not concerned in exactly fitting the output of the original system, as was  covered in section VI-A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' However, our goal is to choose the system from the feasibility set  that fits the realizations of the sensor while having the lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' CONCLUSION  In this paper, we presented an approach that aims to find the least order system that is  compatible with fragmented quantized realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' This approach allows for the use of a priori  information on the system and fragmented measurements of the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The algorithm is based on  an ADMM approach that aims to solve an ℓp quasi-norm objective by dividing the optimization  over the variables through iteratively solving simpler sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' The algorithm is tested  on a synthetic data set, that is randomly generated, and a realistic data set collected through  the measurement of the movement of a robotic arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content=' Numerical results presented show that  the algorithm is very effective in obtaining low complexity explanations of the data collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  Further effort is being put into analyzing the convergence of the proposed algorithm, improving  the numerical performance and its extension to continuous-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='  3  = α  α = lo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  l()"  N  2  一  ()  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='4  Noise limit50  40  System order  30  20  10  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdFLT4oBgHgl3EQfTC8I/content/2301.12043v1.pdf'}
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diff --git a/Z9E1T4oBgHgl3EQfwwUB/content/tmp_files/2301.03413v1.pdf.txt b/Z9E1T4oBgHgl3EQfwwUB/content/tmp_files/2301.03413v1.pdf.txt
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+1
+
+Abstract— This paper presents a transducer network framework
+that supports the amalgamation of multiple transducers into
+single wireless nodes. This approach is aimed at decreasing energy
+consumption by reducing the number of wireless transceivers
+involved in such networks. To make wireless nodes easily
+reconfigurable, a plug and play mechanism is applied to enable
+the clustering of any number of transducers. Furthermore, an
+algorithm is proposed to dynamically detect added and removed
+transducers from a node. Lastly, an XML based protocol is
+devised to allow nodes to communicate a description of their
+layout, measured data and control information. To verify the
+proposed framework, multiple reconfigurable wireless nodes are
+used to monitor the dynamic condition of a multiple rooms during
+a period of 24 hours in order to emulate a smart home scenario.
+Index Terms—Efficient energy consumption, plug and play
+transducers,
+smart
+home,
+transducer
+network,
+wireless
+transducer node.
+I. INTRODUCTION
+ransducer (sensor and actuator) networks measure, process
+and respond to the physical environment and communicate
+their measured information among each other and with
+remote computing nodes. Several works have applied
+transducer networks in various domains [1-4]. This paper
+presents a transducer network framework for smart home
+environments applications.
+In a smart home environment, two kinds of transducer
+networks can be involved, wired and wireless. Each kind has its
+merits and disadvantages. Wired transducer networks are
+restricted by their deployment
+as they require physical
+connections snaked through walls or laid through other less
+esthetically appealing methods; on the other hand, they
+consume less energy compared to wireless transducer networks
+as wired communication is much less energy demanding.
+Wireless transducer networks typically have distinct energy
+consumption patterns that correspond to their phase of
+operation: sensing, processing, and communicating. The
+energy consumption in the first and second phases is negligible
+compared to that of the communicating phase. Pottie et al., for
+example, stated that the required energy for transmitting 1KB
+over 100m distance is about 3 joules and is almost the same as
+processing 1 million operations by a 100 MIPS/W processor [5].
+Therefore, energy consumption is considered as one of the
+major challenges in wireless transducer network design [6, 7].
+B. Hafidh, H. Al Osman, H. Dong and A. El Saddik are with Multimedia
+Computing Research Laboratory (MCRLab), School of Electrical Engineering
+and Computer Science, University of Ottawa, Ottawa, Ontario, Canada (e-mail:
+{bhafi014; halos072; hdong; elsaddik} @uottawa.ca).
+Various mechanisms are proposed by researchers in order to
+minimize or optimize the energy consumption for wirelessly
+communicating sensors. These mechanisms can be divided into
+three ways: software, hardware and communication:
+For software solutions, researchers have implemented
+algorithms in order to minimize the power consumption of
+wireless nodes. For instance, Chou et al. [10] presented a
+compression algorithm that makes use of previous readings in
+order
+to
+minimize
+communication.
+Other
+researchers
+introduced power-aware routing protocols to maximize the
+lifetime of the transducer nodes [11, 12].
+In terms of hardware solutions, researchers have combined
+transducers together in one wireless node to reduce the number
+of nodes in the network and consequently decrease the energy
+consumption of the whole network. Several researchers
+successfully combined multiple transducers in one wireless
+node. Noh et al. [13], for example, presented a design for a
+wireless node that bundles four physically interconnected
+transducers. Similarly, Suh et al. [14] proposed a wireless node
+composed of three transducers, while supporting the option of
+attaching a fourth one through a connecter mechanism.
+Since the communication phase of the wireless node
+consumes most of the energy as mentioned earlier, researchers
+such as Zhang et al., and Surie et al. [8, 9] proposed the use of
+the energy-efficient ZigBee (IEEE 802.15.4) communication
+protocol as opposed to Wi-Fi or Bluetooth in wireless
+transducer networks used for home monitoring purposes.
+Given the related literature presented above, in this work, we
+propose our own hardware solution to minimize the overall
+energy consumption of a transducer network. The existing
+hardware solutions, such as the ones presented by Zhang et al.
+[8] and Surie et al. [9], proposed fixed nodes that are limited by
+the type and number of transducers they support and the way
+the transducers are physically laid.
+Therefore, we present a
+flexible and general-purpose transducer network framework
+that supports the clustering of a large variety of transducers into
+single wireless nodes through a plug and play mechanism.
+Therefore, the resulting transducer network uses a hybrid of
+wired and wireless communication. Wired connections are used
+to
+link
+transducers
+in
+close
+proximity,
+which,
+ensures
+customizability for a wide range of smart home applications as
+it will be demonstrated in this work.
+Our proposed solution can be combined with software and
+communication
+level
+enhancements
+for
+further
+energy
+optimization. In fact, to reduce wireless communication power
+consumption, we have used ZigBee as proposed by [8, 9].
+II. FRAMEWORK DESIGN
+In this section, we provide an overview of our proposed
+A Framework of Reconfigurable Transducer
+Nodes for Smart Home Environments
+Basim Hafidh, Hussein Al Osman, Haiwei Dong, and Abdulmotaleb El Saddik, Fellow, IEEE
+T
+
+2
+framework and discuss its architecture.
+A. Overview
+Our goal is to design a hybrid wired-wireless transducer
+network composed of several plug and play transducers. These
+plug and play transducers share their measured data inside the
+node through wires using a serial communication protocol
+called I2C (inter-integrated circuit) and wirelessly to other
+nodes or central server as shown in Fig.1 through ZigBee links.
+The data produced by the collection of sensors on each node is
+hierarchically organized into an eXtensible Markup Language
+(XML) message that is wirelessly communicated to other nodes.
+Two XML messages are involved: one directed from a wireless
+node to other nodes or a central server which includes the
+node’s overall information, layout description, and periodic
+sensory measurements (see Section 3). The other message, a
+control message, is received by a transducer node to activate its
+vibro-tactile actuators.
+B. Wireless Node Architecture
+Fig. 2 shows the architecture of the wireless node. It is
+composed of the following five main modules:
+1) Transducers module
+The transducers module represents a set of interconnected
+transducer units. Each unit is composed of a microcontroller, an
+analog to digital converter (ADC) or a digital to analog
+converter (DAC). A unique ID is assigned to each transducer
+(see Table I) in order to identify it. These IDs are statically
+pre-assigned and are permanently stored in flash memory.
+2) Communication bus
+The communication bus is responsible for providing the
+necessary communication between the transducers module and
+the data collection module. The onboard communication bus is
+based on I2C.
+3) Data collection module
+This module provides the necessary space to collect all
+measured data that comes from the transducers module, and
+pass them to the interface module. It also collects the upcoming
+signals from the interface module, and forwards them to the
+transducers module in order to activate the running actuators.
+Moreover, this module stores the connectivity status of the
+node’s transducers. This status includes the transducers IDs and
+corresponding statuses (i.e., running or not) which are
+forwarded to the monitoring module.
+4) Monitoring module
+This module is responsible for monitoring the connection
+and disconnection of transducers. When a transducer is
+connected to or disconnected from a node, it triggers an event
+by broadcasting its unique ID through the communication bus.
+This event is detected by the monitoring module which
+recognizes the ID of the transducer. The monitoring module
+records the change regarding the added or removed transducer
+and sends this information to the data collection module in
+order to update the corresponding transducer status.
+5) Interface module
+The interface module provides the necessary communication
+between the transducer node and its external environment. Five
+possible media of communication can be used with the
+proposed transducer node including: USB (RS232), Bluetooth
+technology, Ethernet, Wi-Fi, and ZigBee. However, the ZigBee
+communication technology is considered more preferable in
+designing sensor networks due to its low power consumption
+[15]. This module is responsible for exchanging the messages
+between the data collection module and other nodes or the
+central server.
+III.
+IMPLEMENTATION AND EVALUATION
+As a proof of concept evaluation of our proposed framework,
+we deployed multiple wireless nodes to monitor the status of
+various objects and rooms in home setting. Table I lists the plug
+and play transducers we support in our implementation. The
+main board of the node is supplied by 3.3V rechargeable battery.
+The node’s microcontroller used is Arduino pro-mini which
+runs at 8MHz. The Arduino software runs the data collection
+and
+monitoring
+modules
+mentioned
+above.
+This
+microcontroller shares the data with the wired connected
+transducers.
+Fig. 2. Wireless node architecture.
+Fig. 1. Wireless nodes with reconfigurable transducers.
+
+Transducers
+Monitoring
+Module
+Module
+Communication
+Data Collection
+Interface
+Bus
+Module
+ModuleFSR
+FSR
+FSR
+ACTACT
+ACT
+FSF
+FSR
+ACC
+CO
+Light
+Light
+TMP
+ACC
+Accelerometer
+ACT
+Actuator
+co
+CarbonMonoxide
+FSR
+Force Sensitive Resistor
+TMP
+Temperature
+SR
+SR2
+SR3
+Wirelesstransducernode
+Central server
+Transducer
+Wiredcommunication
+Wirelesscommunication3
+In the current implementation, the transducers cluster
+includes pressure (FSR, force sensitive resistor), ambient light
+(TEMT6000), carbon monoxide gas (MQ-7), triple axes
+accelerometer (ADXL 345), temperature (TMP 102), and flex
+sensors and vibro-tactile actuators. These plug and play
+transducers are implemented on top of Android pro-mini
+microcontrollers. Each transducer has its assigned ID number
+for a unique identification and is able to communicate with
+each
+node’s
+microcontroller
+through
+the
+I2C
+serial
+communication protocol.
+As an example, 6 wireless nodes have been located in
+different places in a house as shown in Fig. 3. Each wireless
+node has been customized depending on the measurement
+environment. For example, Node 1 measures the kitchen’s
+temperature, light intensity and Carbon monoxide (CO) levels.
+Node 2 is placed in the refrigerator and includes pressure
+(placed under the egg tray as an example) and accelerometer
+(attached to the door) sensors. Node 3 has three sensors and
+three actuators mounted on a sofa’s cushions. The sensors can
+be used to monitor sitting patterns and durations and the
+actuators can be used to buzz persons when they remain seated
+for too long. Three nodes are located in the bedroom: Node 4
+with a pressure sensor and actuator mounted on a chair seat,
+Node 5 measures the temperature and light intensity in the
+room, and Node 6 located underneath the pillow with three
+pressure sensors and accelerometer to measure a person’s
+movements while sleeping.
+These nodes and corresponding
+transducers are also presented in the topology of Fig.1. As we
+can see that the same node mainboard hardware and software
+are used with different plug and play transducers depending on
+the type of the physical property to be measured.
+The sampling rate of the sensors used is 1Hz except for that
+of
+the
+accelerometer
+which
+is
+30
+Hz.
+Each
+node’s
+microcontroller sends a measurement XML message including
+the node’s layout description (such as the communication
+protocol, and number and type of the transducers involved) and
+its running sensors’ measurement (such as the timestamp and
+the measured data) to the other nodes or to the central server
+through 1mw ZigBee transceiver (digi XBee S1). The central
+server monitors and stores the received measurement data and
+post-processes this data. The server receives and stores the
+upcoming measured data from the nodes and prepares them for
+visualization to make them meaningful. It also sends control
+signals to the corresponding nodes in order to activate their
+actuators. Fig. 4 shows a visualization of the data measured by
+the 6 wireless nodes in the home setup previously described
+during a period of 24 hours. The white color represents the
+minimum
+measured
+value
+and
+the
+dark represents
+the
+maximum. For example, the first bar represents the data for the
+CO level in the kitchen (Node 1). This level was high from 10
+am to 12 pm and from 8 pm to 9 pm (during which the kitchen
+was used for cooking).
+In order to prove that our proposed framework consumes less
+energy compared to “traditional” wireless transducer networks.
+A traditional wireless network is a set of nodes connected
+through a wireless channel. Each node is composed of a single
+sensor,
+microcontroller,
+and
+wireless
+transceiver.
+Such
+networks have been described in numerous previous works
+including [8, 9, 16, 17, 18]; we have setup a dedicated
+experiment over a period of 24 hours. During the experiment,
+we ran in parallel two transducer networks: the one described
+above using our proposed framework and another traditional
+network composed of the same transducers. To measure the
+energy consumption, the node’s supply voltage and current
+were measured. The current sensor (TI-INA169) was used to
+measure the node’s drawn current. Fig. 5 shows the 24 hour
+energy consumption by each proposed node as compared to
+Fig. 3. A house example with 6 wireless nodes with different reconfigurable
+transducers.
+TABLE I
+PLUG AND PLAY CONFIGURABLE TRANSDUCERS
+ID
+Transducer
+Circuit
+Remarks
+Wireless Node’s
+Core
+(Mainboard)
+Data collection,
+processing and
+communication
+1-20
+Pressure Sensor
+I2C IDs 1-20 are
+reserved for
+pressure sensors
+(FSRs)
+21-40
+Vibrotactile
+Actuator
+I2C IDs 21-40 are
+reserved for
+actuators
+72-75
+Temperature
+Sensor
+I2C IDs 72-75 are
+reserved for
+Temperature
+Sensors
+(TMP102)
+83-84
+Accelerometer
+I2C IDs 83-84 are
+reserved for
+accelerometer
+(ADXL345)
+41,57
+Light Sensor
+I2C IDs 41, 57 are
+reserved for Light
+sensors
+(TSL2561)
+76-78
+CO Gas Sensor
+I2C IDs 76-78 are
+reserved for
+Ambient Light
+sensors
+(TEMT6000)
+85-90
+Flex Sensor
+I2C IDs 85-90 are
+reserved for flex
+sensors 4.5”
+Extension
+
+Node3
+Node 4
+Node5
+Node 6
+Node 1
+Bedroom
+Livingroom
+Node2
+Kitchen
+Bathroom
+Refrigerator1834GND
+ADDR
+INT
+SCL
+OSDA
+UCC4
+traditional transducer nodes that are typically used in wireless
+sensor networks. For example, Node 1 is compared with the
+combination of three traditional transducer nodes. For Node 4
+(the chair node), however, there is no energy saving compared
+to the traditional network of nodes since Node 4 is composed of
+only one pressure sensor and one actuator, and the effect of the
+actuator was neglected as it is run for 30 seconds every 30
+minutes
+to
+buzz
+the
+sitter
+when
+the
+pressure
+sensor
+continuously reads values greater than zero for a period of 30
+minutes. We can see from the figure, the more transducers are
+included in the node, the more energy we save and besides the
+less
+nodes
+can
+be
+involved
+in
+the
+network.
+In
+this
+implementation,
+as
+the
+transducer
+node
+framework
+is
+standardized proposed, the designed wireless transducer was
+efficiently developed.
+IV. CONCLUSION
+In this paper, we proposed a transducer network framework
+composed of wireless nodes that can inter-communicate
+through direct links or a central server. The proposed
+framework was aimed at reducing the energy consumption of
+transducer networks through clustering. A plug and play
+mechanism was conceived to cluster any number of transducers
+together in several possible configurations. The wireless node
+dynamically recognized new added/removed transducers. A
+prototype wireless node core was implemented with several
+plug and play transducers. A proof of concept evaluation was
+done by placing different wireless nodes inside a house. Each
+node was composed of different transducers depending on the
+measured environment or object. A visualization of the data
+collected was shown by the various nodes dispersed throughout
+the home setup.
+We also evaluated the energy consumption of the proposed
+transducer network and compared it to traditional wireless
+transducer networks where each node is dedicated for a single
+transducer. It is shown that our proposed transducer network
+consumes significantly less energy. For our home setup, our
+proposed
+method
+consumes
+46%
+energy
+compared
+to
+traditional approach.
+REFERENCES
+[1]
+M., Mohandes, M. Haleem, M. Deriche, and K. Balakrishnan, "Wireless
+sensor networks for pilgrims tracking," IEEE Embedded Systems Letters,
+vol. 4, no. 4, pp. 106-109, 2012.
+[2]
+I. F. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. Cayirci, “Wireless
+sensor networks: A survey,” Computer Networks, vol. 38, no. 4, pp.
+393–422, 2002.
+[3]
+C. Chong, S. P. Kumar, and S. Member, “Sensor networks : Evolution,
+opportunities, and challenges,” in Proc. IEEE, 2003, vol. 91, no. 8, pp.
+1247–1256.
+[4]
+K. C. Rahman, “A Survey on sensor network,” Int. J. Comput. Inf.
+Technol., vol. 1, no. 1, pp. 76–87, 2010.
+[5]
+G. J. Pottie, and W. J. Kaiser, "Wireless integrated network sensors,"
+Commun. of the ACM, vol. 43, no. 5, pp. 51-58, 2000.
+[6]
+R. Muraleedharan and L. A. Osadciw, “Balancing the performance of a
+sensor network using an ant system,” in Proc. 37th Ann. Conf. on Inform.
+Sci. and Sys., 2003.
+[7]
+L. M. Feeney and M. Nilsson, “Investigating the energy consumption of a
+wireless network interface in an ad hoc networking environment,” in Proc.
+Conf. IEEE Comput. Commun. Soc., 2001, vol. 3, pp. 1548–1557.
+[8]
+J. Zhang, G. Song, H. Wang, and T. Meng, “Design of a wireless sensor
+network based monitoring system for home automation,” in Proc. Int.
+Conf. Futur. Comput. Sci. Appl., 2011, pp. 57–60.
+[9]
+D. Surie, O. Laguionie, and T. Pederson, “Wireless sensor networking of
+everyday objects in a smart home environment,” in Proc. Int. Conf. Intell.
+Sensors, Sens. Networks Inf. Process., 2008, pp. 189–194.
+[10] J. Chou, “A distributed and adaptive sSignal processing approach to
+reducing energy consumption in sensor networks,” in Proc. Conf. of the
+IEEE Commun. Soc., 2003, vol. 2, pp. 1054–1062.
+[11] C.-K. Toh, “Maximum battery life routing to support ubiquitous mobile
+computing in wireless ad hoc networks,” IEEE Commun. Mag., vol. 39,
+no. 6, pp. 138–147, 2001.
+[12] R. C. Shah and J. M. Rabaey, “Energy aware routing for low energy ad
+hoc sensor networks,” in Proc. IEEE Wireless Comm. and Networking
+Conf, 2002, pp. 350–355.
+[13] S.-K. Noh, K.-S. Kim, and Y.-K. Ji, “Design of a room monitoring system
+for wireless sensor networks,” Int. J. Distrib. Sens. Networks, pp. 1–7,
+2013.
+[14] C. Suh and Y.-B. Ko, “Design and implementation of intelligent home
+control systems based on active sensor networks,” IEEE Trans. Consum.
+Electron., vol. 54, no. 3, pp. 1177–1184, 2008.
+[15] J. Lee, Y. Su, and C. Shen, “A Comparative study of wireless protocols :
+Bluetooth, UWB, ZigBee, and Wi-Fi,” in Proc. Annual Conf. IEEE Indus.
+Electron. Soc., 2007, pp. 46–51.
+[16] M. Kusserow, O. Amft, and G. Troster, “Bodyant: Miniature wireless
+sensors for naturalistic monitoring of daily activity,” in Proc. of the 4th
+Conf. on Body Area Networks, 2009.
+[17] C. Park, J. Liu, and P. H. Chou, “Eco: An ultra-compact low-power
+wireless sensor node for real-time motion monitoring,” in Proc. Inform.
+Process. in Sensor Networks, 2005, pp. 398–403.
+[18] A. Flammini, P. Ferrari, D. Marioli, E. Sisinni, “Wired and wireless
+sensor networks for industrial applications,” Microelectro. Jour., vol. 40,
+no. 9, pp. 1322-1336, 2009.
+Fig. 4. Nodes sensors measurement.
+Fig. 5. Energy consumption of each proposed node compared with traditional
+transducer nodes.
+
+High
+cO
+Node 1
+Kitchen
+Light
+Temp.
+Node 2
+Refrig.
+Weight
+L-Pressure
+Node3
+Sofa
+C-Pressure
+R-Pressure
+Node 4
+Pressure
+Chair
+Light
+Node 5
+Bedroom
+Temp.
+L-Pressure
+Node 6
+Pillow
+C-Pressure
+R-Pressure
+Low
+8:00
+10:00
+12:00
+14:00
+16:00
+18:00
+20:00
+22:00
+0:00
+2:00
+4:00
+00:9
+8:00
+Time20
+18
+Proposed muiti-transducernode
+Traditionaltransducernodes
+12
+10
+6
+4
+2
+0
+Kitchen
+Retrigerator
+Sofa
+Chair
+Bedroom
+Pillow
\ No newline at end of file
diff --git a/Z9E1T4oBgHgl3EQfwwUB/content/tmp_files/load_file.txt b/Z9E1T4oBgHgl3EQfwwUB/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7c19aaaefb7f3c264c8cc2d45737726c9d0636d0
--- /dev/null
+++ b/Z9E1T4oBgHgl3EQfwwUB/content/tmp_files/load_file.txt
@@ -0,0 +1,348 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf,len=347
+page_content='1  \uf020  Abstract— This paper presents a transducer network framework  that supports the amalgamation of multiple transducers into  single wireless nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' This approach is aimed at decreasing energy  consumption by reducing the number of wireless transceivers  involved in such networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' To make wireless nodes easily  reconfigurable, a plug and play mechanism is applied to enable  the clustering of any number of transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Furthermore, an  algorithm is proposed to dynamically detect added and removed  transducers from a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Lastly, an XML based protocol is  devised to allow nodes to communicate a description of their  layout, measured data and control information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' To verify the  proposed framework, multiple reconfigurable wireless nodes are  used to monitor the dynamic condition of a multiple rooms during  a period of 24 hours in order to emulate a smart home scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Index Terms—Efficient energy consumption, plug and play  transducers,  smart  home,  transducer  network,  wireless  transducer node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' INTRODUCTION  ransducer (sensor and actuator) networks measure, process  and respond to the physical environment and communicate  their measured information among each other and with  remote computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Several works have applied  transducer networks in various domains [1-4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' This paper  presents a transducer network framework for smart home  environments applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  In a smart home environment, two kinds of transducer  networks can be involved, wired and wireless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Each kind has its  merits and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Wired transducer networks are  restricted by their deployment  as they require physical  connections snaked through walls or laid through other less  esthetically appealing methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' on the other hand, they  consume less energy compared to wireless transducer networks  as wired communication is much less energy demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Wireless transducer networks typically have distinct energy  consumption patterns that correspond to their phase of  operation: sensing, processing, and communicating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The  energy consumption in the first and second phases is negligible  compared to that of the communicating phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Pottie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=', for  example, stated that the required energy for transmitting 1KB  over 100m distance is about 3 joules and is almost the same as  processing 1 million operations by a 100 MIPS/W processor [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Therefore, energy consumption is considered as one of the  major challenges in wireless transducer network design [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Hafidh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Al Osman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Dong and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' El Saddik are with Multimedia  Computing Research Laboratory (MCRLab), School of Electrical Engineering  and Computer Science, University of Ottawa, Ottawa, Ontario, Canada (e-mail:  {bhafi014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' halos072;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' hdong;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' elsaddik} @uottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='ca).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Various mechanisms are proposed by researchers in order to  minimize or optimize the energy consumption for wirelessly  communicating sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' These mechanisms can be divided into  three ways: software, hardware and communication:  For software solutions, researchers have implemented  algorithms in order to minimize the power consumption of  wireless nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' For instance, Chou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' [10] presented a  compression algorithm that makes use of previous readings in  order  to  minimize  communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Other  researchers  introduced power-aware routing protocols to maximize the  lifetime of the transducer nodes [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  In terms of hardware solutions, researchers have combined  transducers together in one wireless node to reduce the number  of nodes in the network and consequently decrease the energy  consumption of the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Several researchers  successfully combined multiple transducers in one wireless  node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Noh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' [13], for example, presented a design for a  wireless node that bundles four physically interconnected  transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Similarly, Suh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' [14] proposed a wireless node  composed of three transducers, while supporting the option of  attaching a fourth one through a connecter mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Since the communication phase of the wireless node  consumes most of the energy as mentioned earlier, researchers  such as Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=', and Surie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' [8, 9] proposed the use of  the energy-efficient ZigBee (IEEE 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='4) communication  protocol as opposed to Wi-Fi or Bluetooth in wireless  transducer networks used for home monitoring purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Given the related literature presented above, in this work, we  propose our own hardware solution to minimize the overall  energy consumption of a transducer network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The existing  hardware solutions, such as the ones presented by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  [8] and Surie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' [9], proposed fixed nodes that are limited by  the type and number of transducers they support and the way  the transducers are physically laid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Therefore, we present a  flexible and general-purpose transducer network framework  that supports the clustering of a large variety of transducers into  single wireless nodes through a plug and play mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Therefore, the resulting transducer network uses a hybrid of  wired and wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Wired connections are used  to  link  transducers  in  close  proximity,  which,  ensures  customizability for a wide range of smart home applications as  it will be demonstrated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Our proposed solution can be combined with software and  communication  level  enhancements  for  further  energy  optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' In fact, to reduce wireless communication power  consumption, we have used ZigBee as proposed by [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' FRAMEWORK DESIGN  In this section, we provide an overview of our proposed  A Framework of Reconfigurable Transducer  Nodes for Smart Home Environments  Basim Hafidh, Hussein Al Osman, Haiwei Dong, and Abdulmotaleb El Saddik, Fellow, IEEE  T  2  framework and discuss its architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Overview  Our goal is to design a hybrid wired-wireless transducer  network composed of several plug and play transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' These  plug and play transducers share their measured data inside the  node through wires using a serial communication protocol  called I2C (inter-integrated circuit) and wirelessly to other  nodes or central server as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='1 through ZigBee links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  The data produced by the collection of sensors on each node is  hierarchically organized into an eXtensible Markup Language  (XML) message that is wirelessly communicated to other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Two XML messages are involved: one directed from a wireless  node to other nodes or a central server which includes the  node’s overall information, layout description, and periodic  sensory measurements (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The other message, a  control message, is received by a transducer node to activate its  vibro-tactile actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Wireless Node Architecture  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 2 shows the architecture of the wireless node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' It is  composed of the following five main modules:  1) Transducers module  The transducers module represents a set of interconnected  transducer units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Each unit is composed of a microcontroller, an  analog to digital converter (ADC) or a digital to analog  converter (DAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' A unique ID is assigned to each transducer  (see Table I) in order to identify it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' These IDs are statically  pre-assigned and are permanently stored in flash memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  2) Communication bus  The communication bus is responsible for providing the  necessary communication between the transducers module and  the data collection module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The onboard communication bus is  based on I2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  3) Data collection module  This module provides the necessary space to collect all  measured data that comes from the transducers module, and  pass them to the interface module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' It also collects the upcoming  signals from the interface module, and forwards them to the  transducers module in order to activate the running actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Moreover, this module stores the connectivity status of the  node’s transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' This status includes the transducers IDs and  corresponding statuses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=', running or not) which are  forwarded to the monitoring module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  4) Monitoring module  This module is responsible for monitoring the connection  and disconnection of transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' When a transducer is  connected to or disconnected from a node, it triggers an event  by broadcasting its unique ID through the communication bus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  This event is detected by the monitoring module which  recognizes the ID of the transducer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The monitoring module  records the change regarding the added or removed transducer  and sends this information to the data collection module in  order to update the corresponding transducer status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  5) Interface module  The interface module provides the necessary communication  between the transducer node and its external environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Five  possible media of communication can be used with the  proposed transducer node including: USB (RS232), Bluetooth  technology, Ethernet, Wi-Fi, and ZigBee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' However, the ZigBee  communication technology is considered more preferable in  designing sensor networks due to its low power consumption  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' This module is responsible for exchanging the messages  between the data collection module and other nodes or the  central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  IMPLEMENTATION AND EVALUATION  As a proof of concept evaluation of our proposed framework,  we deployed multiple wireless nodes to monitor the status of  various objects and rooms in home setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Table I lists the plug  and play transducers we support in our implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The  main board of the node is supplied by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='3V rechargeable battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  The node’s microcontroller used is Arduino pro-mini which  runs at 8MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The Arduino software runs the data collection  and  monitoring  modules  mentioned  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  This  microcontroller shares the data with the wired connected  transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Wireless node architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Wireless nodes with reconfigurable transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Transducers  Monitoring  Module  Module  Communication  Data Collection  Interface  Bus  Module  ModuleFSR  FSR  FSR  ACTACT  ACT  FSF  FSR  ACC  CO  Light  Light  TMP  ACC  Accelerometer  ACT  Actuator  co  CarbonMonoxide  FSR  Force Sensitive Resistor  TMP  Temperature  SR  SR2  SR3  Wirelesstransducernode  Central server  Transducer  Wiredcommunication  Wirelesscommunication3  In the current implementation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' the transducers cluster  includes pressure (FSR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' force sensitive resistor),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' ambient light  (TEMT6000),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' carbon monoxide gas (MQ-7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' triple axes  accelerometer (ADXL 345),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' temperature (TMP 102),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' and flex  sensors and vibro-tactile actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' These plug and play  transducers are implemented on top of Android pro-mini  microcontrollers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Each transducer has its assigned ID number  for a unique identification and is able to communicate with  each  node’s  microcontroller  through  the  I2C  serial  communication protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  As an example, 6 wireless nodes have been located in  different places in a house as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Each wireless  node has been customized depending on the measurement  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' For example, Node 1 measures the kitchen’s  temperature, light intensity and Carbon monoxide (CO) levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Node 2 is placed in the refrigerator and includes pressure  (placed under the egg tray as an example) and accelerometer  (attached to the door) sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Node 3 has three sensors and  three actuators mounted on a sofa’s cushions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The sensors can  be used to monitor sitting patterns and durations and the  actuators can be used to buzz persons when they remain seated  for too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Three nodes are located in the bedroom: Node 4  with a pressure sensor and actuator mounted on a chair seat,  Node 5 measures the temperature and light intensity in the  room, and Node 6 located underneath the pillow with three  pressure sensors and accelerometer to measure a person’s  movements while sleeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  These nodes and corresponding  transducers are also presented in the topology of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' As we  can see that the same node mainboard hardware and software  are used with different plug and play transducers depending on  the type of the physical property to be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  The sampling rate of the sensors used is 1Hz except for that  of  the  accelerometer  which  is  30  Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Each  node’s  microcontroller sends a measurement XML message including  the node’s layout description (such as the communication  protocol, and number and type of the transducers involved) and  its running sensors’ measurement (such as the timestamp and  the measured data) to the other nodes or to the central server  through 1mw ZigBee transceiver (digi XBee S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The central  server monitors and stores the received measurement data and  post-processes this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The server receives and stores the  upcoming measured data from the nodes and prepares them for  visualization to make them meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' It also sends control  signals to the corresponding nodes in order to activate their  actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 4 shows a visualization of the data measured by  the 6 wireless nodes in the home setup previously described  during a period of 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The white color represents the  minimum  measured  value  and  the  dark represents  the  maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' For example, the first bar represents the data for the  CO level in the kitchen (Node 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' This level was high from 10  am to 12 pm and from 8 pm to 9 pm (during which the kitchen  was used for cooking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  In order to prove that our proposed framework consumes less  energy compared to “traditional” wireless transducer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  A traditional wireless network is a set of nodes connected  through a wireless channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Each node is composed of a single  sensor,  microcontroller,  and  wireless  transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Such  networks have been described in numerous previous works  including [8, 9, 16, 17, 18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' we have setup a dedicated  experiment over a period of 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' During the experiment,  we ran in parallel two transducer networks: the one described  above using our proposed framework and another traditional  network composed of the same transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' To measure the  energy consumption, the node’s supply voltage and current  were measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The current sensor (TI-INA169) was used to  measure the node’s drawn current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 5 shows the 24 hour  energy consumption by each proposed node as compared to  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' A house example with 6 wireless nodes with different reconfigurable  transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  TABLE I  PLUG AND PLAY CONFIGURABLE TRANSDUCERS  ID  Transducer  Circuit  Remarks  Wireless Node’s  Core  (Mainboard)  Data collection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  processing and  communication  1-20  Pressure Sensor  I2C IDs 1-20 are  reserved for  pressure sensors  (FSRs)  21-40  Vibrotactile  Actuator  I2C IDs 21-40 are  reserved for  actuators  72-75  Temperature  Sensor  I2C IDs 72-75 are  reserved for  Temperature  Sensors  (TMP102)  83-84  Accelerometer  I2C IDs 83-84 are  reserved for  accelerometer  (ADXL345)  41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='57  Light Sensor  I2C IDs 41,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 57 are  reserved for Light  sensors  (TSL2561)  76-78  CO Gas Sensor  I2C IDs 76-78 are  reserved for  Ambient Light  sensors  (TEMT6000)  85-90  Flex Sensor  I2C IDs 85-90 are  reserved for flex  sensors 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='5”  Extension  Node3  Node 4  Node5  Node 6  Node 1  Bedroom  Livingroom  Node2  Kitchen  Bathroom  Refrigerator1834GND  ADDR  INT  SCL  OSDA  UCC4  traditional transducer nodes that are typically used in wireless  sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' For example, Node 1 is compared with the  combination of three traditional transducer nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' For Node 4  (the chair node), however, there is no energy saving compared  to the traditional network of nodes since Node 4 is composed of  only one pressure sensor and one actuator, and the effect of the  actuator was neglected as it is run for 30 seconds every 30  minutes  to  buzz  the  sitter  when  the  pressure  sensor  continuously reads values greater than zero for a period of 30  minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' We can see from the figure, the more transducers are  included in the node, the more energy we save and besides the  less  nodes  can  be  involved  in  the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  In  this  implementation,  as  the  transducer  node  framework  is  standardized proposed, the designed wireless transducer was  efficiently developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' CONCLUSION  In this paper, we proposed a transducer network framework  composed of wireless nodes that can inter-communicate  through direct links or a central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The proposed  framework was aimed at reducing the energy consumption of  transducer networks through clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' A plug and play  mechanism was conceived to cluster any number of transducers  together in several possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' The wireless node  dynamically recognized new added/removed transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' A  prototype wireless node core was implemented with several  plug and play transducers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' A proof of concept evaluation was  done by placing different wireless nodes inside a house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Each  node was composed of different transducers depending on the  measured environment or object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' A visualization of the data  collected was shown by the various nodes dispersed throughout  the home setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  We also evaluated the energy consumption of the proposed  transducer network and compared it to traditional wireless  transducer networks where each node is dedicated for a single  transducer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' It is shown that our proposed transducer network  consumes significantly less energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' For our home setup, our  proposed  method  consumes  46%  energy  compared  to  traditional approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
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+page_content=' Nodes sensors measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
+page_content=' Energy consumption of each proposed node compared with traditional  transducer nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E1T4oBgHgl3EQfwwUB/content/2301.03413v1.pdf'}
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+Validity in Music Information Research
+Experiments
+Bob L. T. Sturm1 and Arthur Flexer2
+1Division of Speech, Music and Hearing, KTH Stockholm, Sweden,
+bobs@kth.se
+2Institute of Computational Perception, Johannes Kepler
+University Linz, Austria, arthur.flexer@jku.at
+Abstract
+Validity is the truth of an inference made from evidence, such as data
+collected in an experiment, and is central to working scientifically. Given
+the maturity of the domain of music information research (MIR), validity
+in our opinion should be discussed and considered much more than it has
+been so far. Considering validity in one’s work can improve its scientific
+and engineering value. Puzzling MIR phenomena like adversarial attacks
+and performance glass ceilings become less mysterious through the lens
+of validity. In this article, we review the subject of validity in general,
+considering the four major types of validity from a key reference: Shadish
+et al. [2002]. We ground our discussion of these types with a prototypical
+MIR experiment: music classification using machine learning. Through
+this MIR experimentalists can be guided to make valid inferences from
+data collected from their experiments.
+1
+Introduction
+The multi-disciplinary field of Music Information Research (MIR) is focused
+on making music and information about music accessible to a variety of users.
+This ranges from systems for search and retrieval, e.g., Shazam [Wang, 2003]
+or TunePal [Duggan and O’Shea, 2011], to recommendation, e.g., Spotify or
+Pandora, and even to more creative applications like generation. These sys-
+tems consist of many computational components working in domains between
+the acoustic and symbolic where computability is possible [M¨uller, 2015]. The
+effectiveness and reliability of these complex systems and their components are
+of prime importance to the MIR researcher, not to mention other stakeholders.
+The MIR researcher performs experiments to compare approaches for mod-
+elling and retrieving music data. A principal focus is on users, but the cost of
+performing experiments with users is high, and the replicability of such studies is
+difficult. This has motivated the use of computer-based experiments where “test
+1
+arXiv:2301.01578v1  [cs.SD]  4 Jan 2023
+
+collections” serve as proxies for human users, known as the Cranfield Paradigm
+[Cleverdon, 1991]: the researcher predicts the usefulness of an MIR system by
+applying it to a representative sample of the problem domain (e.g., queries and
+relevant documents).
+While such an approach is inexpensive and replicable,
+its relevance and reliability for MIR, and information retrieval in general, have
+been questioned [Voorhees, 2001, Urbano et al., 2013].
+Under the Cranfield Paradigm, state-of-the-art MIR systems perform excep-
+tionally well in reproducing the ground truth of some datasets, e.g., inferring
+rhythm, genre or emotion from audio data. However, slight and irrelevant trans-
+formations of the audio can suddenly render these systems ineffectual [Sturm,
+2014b, Kereliuk et al., 2015, Rodr´ıguez-Algarra et al., 2016, Prinz et al., 2021].
+In one case [Rodr´ıguez-Algarra et al., 2016], a “genre recognition” system re-
+lies on infrasonic signatures, seemingly originating from the data collection, but
+nonetheless imperceptible and irrelevant for human listeners. In another case
+[Sturm et al., 2015], a “rhythm recognition” system seemingly uses tempo to
+infer rhythm, a confusion originating again from the data collection.
+Related are discussions about “glass ceilings” of MIR systems [Aucouturier
+and Pachet, 2004, Pohle et al., 2008], i.e., that the observed barrier to improving
+system performance to perfect or human level is due to the psychophysical and
+cultural factors of music missing from computable features extracted from audio
+recordings [Wiggins, 2009]. MIR researchers striving for great improvement of
+those numbers in established problems and datasets might actually make no
+practical difference for human users [Urbano et al., 2012]. Flexer and Grill [2016]
+identifies one reason contributing to performance glass ceilings: the data used
+to train and test MIR systems can arise from tasks that are not well-posed (e.g.,
+music similarity), and thus render meaningless the pursuit of better numbers
+using that data. The lack of definition in research problems motivates Sturm
+and Collins [2014] to propose MIR researchers solve synthetic yet well-defined
+simplifications of harder, ill-posed problems.
+At the heart of an MIR experiment is the relationship between conclusions
+drawn from its results and their validity, or “truth value” [Shadish et al., 2002].
+Ideally, an experiment will be carefully designed and implemented to answer
+a well-defined hypothesis.1 Its components – units, treatments, design, obser-
+vations, and settings – should be operationalised (translated from theory into
+practice) to maximize quality and minimize cost (money, time, ethics). This is
+the purview of the discipline Design of Experiments: how can one get the best
+evidence for the least cost? A look through the proceedings of MIR’s premier
+conference ISMIR2 and related publications, however, reveals a general lack of
+awareness of validity in the MIR community.
+The first reference explicitly introducing validity to MIR is Urbano [2011],
+discussing it in relation to text information retrieval (IR) literature, and specif-
+ically advocating validity-based meta-evaluation of MIREX results.3
+This is
+1We encourage readers to review Chase [2001] to see the remarkable lengths an experimen-
+talist must go to test even a simple hypothesis.
+2https://ismir.net/conferences/
+3‘Music Information Retrieval Evaluation eXchange’ (MIREX) is an annual evaluation
+2
+
+further developed in Urbano et al. [2013], embracing the validity typology of
+Shadish et al. [2002], and bringing in notions of reliability and efficiency. While
+the review of Schedl et al. [2013] does not explicitly refer to validity or even
+mentions the Cranfield Paradigm, it makes clear how central computer exper-
+iments are to MIR, and criticizes a general lack of consideration of users in
+its conclusions. Sturm [2013] asserts that just reporting figures of merit like
+accuracy is not sufficient to decide whether an MIR system is really recogniz-
+ing “genre” in musical signals, or whether it relies on irrelevant confounding
+factors, later stating that these uncontrolled factors are a danger to the valid-
+ity of conclusions drawn from such experiments [Sturm, 2014b, 2017]. Sturm
+et al. [2014] attempts to define music description (including the “use case”) to
+motivate evaluating music description systems in ways that allow for valid and
+relevant conclusions.
+More recently, Rodr´ıguez-Algarra et al. [2019] propose and test a method for
+detecting the existence of confounding in MIR experiments, thus addressing the
+lack of construct validity of conclusions drawn from them. Liem and Mostert
+[2020] studies construct validity of high-level musical concepts like genre by
+confronting MIR systems with data different from what they have been trained
+on. Flexer et al. [2021] criticizes the lack of external validity of experiments
+on general music similarity and demands focus on specific aspects of music
+similarity and definition of a specific use case. Probably the most recent attempt
+to comprehensively discuss validity and reliability in MIR is an unpublished
+tutorial held at ISMIR 2018 [Urbano and Flexer, 2018].
+The above review of previous work on validity in MIR shows it to be dis-
+persed and fragmented across only relatively few publications. Despite a small
+chorus of calls to address major methodological problems of MIR experiments
+to improve validity in the discipline, e.g., Urbano [2011], Schedl et al. [2013],
+Sturm [2014b], Urbano and Flexer [2018], Liem and Mostert [2020], Flexer et al.
+[2021], there has yet to be published a systematic and critical engagement of
+what validity means in the context of MIR, and how to consider it when design-
+ing, implementing and analyzing experiments.
+In this article, we review the four principal types of validity in Shadish
+et al. [2002], an authoritative resource about validity in causal inference and
+experimental science. Other typologies exist, e.g., Lund [2021], but we focus on
+that of Shadish et al. [2002] because it is an established point of reference, and
+has already been introduced to MIR [Urbano et al., 2013]. For each type, we
+discuss common threats, relate it to the MIR discipline in general, and then more
+concretely in terms of a typical MIR experiment, presented in Sec. 2. We include
+a python jupyter notebook that runs the experiment and computes a variety of
+statistics and results discussed below.4 Section 3 reviews the components of the
+experiment, on which the discussion of validity hinges. Each section 4–7 reviews
+the four types of validity and presents actionable questions which can help MIR
+researchers to scrutinize the conclusions they draw from their experiments.
+campaign for MIR algorithms [Downie, 2006].
+4https://github.com/boblsturm/mirvaliditytutorial
+3
+
+Accuracy
+Precision
+Recall
+f1-score
+LDA
+0.714
+0.711
+0.711
+0.703
+QDA
+0.719
+0.715
+0.723
+0.717
+1NN
+0.662
+0.644
+0.635
+0.638
+3NN
+0.681
+0.673
+0.651
+0.656
+5NN
+0.719
+0.699
+0.687
+0.689
+7NN
+0.695
+0.669
+0.656
+0.659
+9NN
+0.700
+0.681
+0.664
+0.668
+unif
+0.12 ± 0.02
+0.13 ± 0.03
+0.12 ± 0.02
+0.12 ± 0.02
+freq
+0.13 ± 0.02
+0.13 ± 0.03
+0.13 ± 0.02
+0.13 ± 0.02
+maj
+0.16
+0.02
+0.12
+0.03
+Table 1: Accuracy, and macro-averaged precision, recall and f1-score observed
+for several models in a testing partition of BALLROOM [Dixon et al., 2004]. The
+performance of two models randomly selecting labels (with standard deviation)
+are shown in the rows labeled: unif samples labels uniformly; freq samples
+labels according to training data label frequency. The last row maj shows the
+performance of a model choosing the label most frequent in the training data.
+2
+A Typical MIR Experiment
+To ground our general discussion of validity, we present a typical MIR exper-
+iment that exemplifies a considerable amount of MIR research: audio classifi-
+cation using machine learning and a benchmark dataset. Consider the BALL-
+ROOM dataset [Dixon et al., 2004], created around 2004 from downloading
+excerpts of music CDs for sale at a website focused on ballroom dancing. BALL-
+ROOM consists of 698 audio recordings, each about 30 seconds long and labeled
+with one of eight dance styles or music rhythms, e.g., “Waltz”. BALLROOM
+has appeared in dozens of studies [Sturm, 2017], and most often in experiments
+measuring the amount of ground truth labels reproduced by different combina-
+tions of classifiers and features.
+We create a random 70/30 partition, with 488/210 recordings comprising
+the training/testing dataset. From each recording we compute the spectral flux
+onset strength envelope using a hop size of 1024 samples, or 46 ms [McFee
+et al., 2015]. We compute the normalised autocorrelation of the envelope and
+retain the portion relating to lags in [0.23, 4.14] seconds. Finally, we model the
+autocorrelation by a 12th-order autoregressive model, which results in a feature
+of 13 dimensions describing the 30-second audio recording. We normalise each
+dimension of the training features to be zero mean and unit variance.
+We
+normalise the testing data features using the same parameters as for normalising
+the training data features.
+We model features using multivariate Gaussian distributions (linear and
+quadratic discriminant analysis, LDA and QDA), or k-nearest neighbours (KNN).
+We compare the labels inferred by a model to the ground truth and compute the
+accuracy, and macro-averaged precision, recall, and f1-score.5 Table 1 shows the
+5A macro-averaged figure of merit is the mean figure of merit measured across classes,
+regardless of the number of observations of each class.
+4
+
+outcomes of this experiment, which can be reproduced with the supplementary
+python jupyter notebook.6 The last rows show statistics of three other strate-
+gies: unif samples labels according to a uniform prior and freq samples labels
+according to training data label frequency;7 maj selects the most frequent label
+in the training data. These provide comparison points.
+A criticism can be made here that the above experiment is not really repre-
+sentative of MIR experiments today: datasets are now several orders of magni-
+tude larger, and deep learning has in many cases made obsolete the engineering
+of features. However, we will press on with this example for four reasons. First,
+the BALLROOM dataset is analyzed to such an extent that classification exper-
+iments with it illustrate the four types of validity in clear and graspable ways.
+Using a larger dataset can illustrate the same things, but would first require a
+thorough analysis of its composition and provenance. Second, concerns about
+validity are not rendered moot by simply increasing data or shifting the ma-
+chine learning approach. Third, the experiment above is typical in its design: a
+dataset is collected and processed, submitted to a variety of machine learning
+approaches, and measurements are made by applying the trained systems to
+some test collection. Finally, the BALLROOM dataset is unique in that an
+“extended” version of it was created a decade later from the same web resource
+[Marchand and Peeters, 2016]. This allows us to meaningfully test the external
+validity of conclusions about systems trained in BALLROOM, e.g., the concepts
+learned from BALLROOM are relevant to ballroom dance music, and not just
+ballroom dance music in 2004 (see Sec. 7.2).
+3
+Components of an Experiment
+Before discussing the validity of conclusions drawn from the typical MIR exper-
+iment, we must identify its units, treatments, design, observations, and settings.
+Treatments are the things applied to units in order to cause an effect (or not
+in the case of a control), units are the things that are treated, and observations
+are what is measured on a unit. The design specifies which treatment is applied
+to which unit, and settings involve time, place, and condition. To make this
+more concrete, consider a medical experiment in which the effect of a treatment
+on blood pressure is being studied. A number of people are sampled from a
+population, some of whom will receive the treatment while the others receive
+a placebo (control). The design describes which people get the treatment, and
+which do not. The observation is the blood pressure of a person after one month.
+The setting can include particulars of the population (rural or urban), place of
+treatment (hospital or home), and so on. The experimentalist contrasts blood
+pressure observations across groups to conclude if the treatment has an effect.
+Returning to our typical MIR experiment, we wish to determine the effec-
+tiveness of different machine learning (ML) algorithms in predicting the labels of
+6See footnote 4.
+7The mean and standard deviation of these figures of merit can be found in theory, but
+here they are found empirically. See the accompanying code in footnote 4.
+5
+
+a test recording dataset. There are two possibilities here. We can see the treat-
+ments as the ten algorithms, and the units as the entire testing dataset (how
+does the dataset respond to each algorithm?), or we can see the entire testing
+dataset as the one treatment, and the units as the ten algorithms (how does
+each algorithm respond to the dataset?). Since Table 1 reports figures of merit
+(observations) of each algorithm on the entire dataset, the latter interpretation
+motivates conclusions about the effectiveness of particular algorithms. In this
+case, the design is simple: each unit is given the same treatment. The setting
+involves the partitioning of BALLROOM, the features extracted, random seeds,
+software libraries, and so on.
+4
+Statistical Conclusion Validity
+Statistical conclusion validity is “the validity of inferences about covariation
+between two variables” [Shadish et al., 2002]. This includes concluding that a
+covariation exists, and perhaps its strength as well. This is the level at which
+one is concerned with statistical significance, i.e., that an observed covariation
+between treatment and effect is not likely to arise by chance. Shadish et al.
+[2002] (p. 45) includes a table of nine different threats to statistical conclusion
+validity.
+Four threats relevant to computer-based experiments are: violated
+assumptions about the statistics underlying the responses (and the use of the
+wrong statistical test, a type III error [Kimball, 1957]); a sample size too small to
+reliably detect covariation (lack of power); the purposeful search for significant
+results by trying multiple analyses and data selections (“p-hacking”, Head et al.
+[2015]); and increased variance in observations due to the heterogeneity of units.
+4.1
+MIR concerns about statistical conclusion validity
+Are my results statistically significant? Null hypothesis statistical testing (NHST)
+quantifies whether the observed effects of the treatments on the responses arise
+by mere chance, as well as the direction of effect and its size. This answers the
+question: are the results statistically significant? Fundamentals about statistical
+testing in MIR are discussed by Flexer [2006], for Artificial Intelligence in gen-
+eral by Cohen [1995], and for machine learning by Japkowicz and Shah [2011].
+One must take care in selecting a statistical test to use; each one makes strong
+assumptions that could be violated. NHST is most straightforwardly applicable
+to completely randomized experimental designs [Bailey, 2008], thereby reducing
+the possibility of structure in units and treatments interfering with the responses
+(which results in confounding, discussed in the next section). Most MIR exper-
+iments cannot use complete randomisation because the target population from
+which samples come is unclear (what is a random sample of “sad” music, with
+the term “sad” being quite ill-defined?), and so the kinds of conclusions that
+can be made with NHST in MIR are limited.8
+8Experimental designs that cannot be completely randomised are called quasi-experimental
+designs, which is another major topic of Shadish et al. [2002].
+6
+
+Is the observed statistical significance relevant for a user? In MIR, even if one
+finds statistical significance, this may not generalise to a perceivable difference
+for actual users interacting with the “improved” MIR system. As an example
+from MIR, a crowd-sourced user evaluation [Urbano et al., 2012] demonstrates
+that there is an upper bound of user satisfaction with music recommendation
+systems of about 80%, since this was the highest percentage of users agreeing
+that two systems “are equally good.”
+In addition, for the MIREX task of
+Audio Music Similarity and Retrieval Urbano et al. [2012] demonstrate that
+statistically significant differences between algorithms can be so small that they
+make no practical difference for users.
+4.2
+Statistical conclusion validity in the typical MIR ex-
+periment
+Let us now consider the typical MIR experiment in Sec. 2 and reason about
+what conclusions we can draw from it that have statistical conclusion validity.
+Table 1 clearly shows that each response of machine learning (ML) to the dataset
+is greater than the random approaches unif, freq and maj. How likely is it that
+any of the responses of ML is due to chance, i.e., that any of the ML approaches
+is actually no better than one of the random approaches? Since we have the
+empirical distributions for unif and freq, we can estimate the probability of
+either of them resulting in, e.g., a macro-average recall at least as large as
+0.6: p < e−200.9
+Hence, a valid statistical conclusion is that we observe a
+significant covariation between the use of ML with these particular features and
+the responses measured on a specific partition of BALLROOM. An equivalent
+conclusion is that each response of these trained ML systems treated by this
+partition of BALLROOM are inconsistent with choosing randomly.
+Whereas the conclusions above relate to the use of ML, one might consider
+statistical conclusions relating to the type of ML, i.e., Gaussian modelling (LDA
+and QDA) vs. nearest neighbour modelling (KNN), or LDA vs. QDA. One may
+want to conclude that Gaussian modelling is better than nearest neighbour
+modelling with these features on BALLROOM, or even more precisely, QDA
+performs the best with these features on BALLROOM. This involves looking
+at the differences between responses, called contrasts. While there is not much
+reason to suspect that the particular 70/30 partition of BALLROOM used is
+unusually beneficial to one kind of ML model over the other, what we do not
+know from our experiment is the distribution of any response due to ML, and
+so of any difference of responses due to ML, related to the random partitioning
+of BALLROOM. We have two measurements of Gaussian models, and five of
+nearest neighbour models, but we cannot simply compute and compare statistics
+9This is the probability of observing a macro-averaged recall at least as big as 0.6 from the
+models selecting labels randomly. Specifically, it is the probability of sampling a number from
+a random variable distributed Gaussian with mean 0.125 and standard deviation 0.02 in the
+domain [0.6, ∞). See the accompanying jupyter notebook. Also note that a macro-averaged
+recall of 0.6 is a pessimistic choice. Using the higher numbers in Table 1 results in even lower
+probability.
+7
+
+of these without involving auxiliary information not present in the experiment,
+e.g., how the decision boundaries of an ML model vary according to variation
+in the training data.
+In fact, if we conclude from Table 1 that Gaussian modelling performs bet-
+ter than nearest neighbour modelling with these features on 70/30 partitions of
+BALLROOM, we would be wrong. This is a “type I error”, which is concluding
+there to be a significant difference when in fact there is none. When we perform
+this experiment 1000 times with random 70/30 partitions we observe that the
+difference between the best response of a Gaussian model and the best response
+of a nearest neighbour model is distributed Gaussian, and that the probability
+of observing zero difference or less is p > 0.41 for any of the figures of merit
+(see the accompanying jupyter notebook). That is, the more statistically valid
+conclusion is that three out of five times randomly partitioning BALLROOM
+the best Gaussian model performs better than the best nearest neighbor model.
+With this new data we see that the results in Table 1 do not merit a valid sta-
+tistical conclusion that one of the modelling approaches is significantly better or
+worse than any other in BALLROOM using these features for any significance
+level α < 0.4. On the other hand, if we were to conclude that QDA performs
+better than nearest neighbour modelling with these features on 70/30 random
+partitions of BALLROOM, we would be correct – but only for macro-averaged
+recall. Performing 1000 repetitions of our experiment shows that the distribu-
+tion of the difference between the response from QDA and the best response of
+all others is distributed such that the probability of observing a negative differ-
+ence is p < 0.046 only for recall (see the accompanying jupyter notebook). In
+other words, less than one out of 20 times do we see QDA perform worse than
+the other models in terms of recall.
+In summary, the most general statistical conclusion we can make from Table
+1 is that the responses we observe from ML are highly inconsistent with the
+responses of models choosing randomly. Each ML model knows something about
+BALLROOM linking the features computed from a music recording with its
+label. Because we do not know the amount of variation in any response due to
+partitioning in Table 1, we cannot make any valid statistical conclusion about
+which type of ML model is the best, or which particular realisation is the best,
+for this particular dataset. In order to go further, we had to run the experiment
+multiple times to obtain distributions of the contrasts. Even then, however, we
+cannot say anything about the cause of significant differences yet. This is where
+the notion of internal validity becomes relevant.
+5
+Internal Validity
+Internal validity is “the validity of inferences about whether the observed covari-
+ation between two variables is causal” [Shadish et al., 2002]. While statistical
+conclusion validity is concerned only with the strength of covariation between
+treatment and responses, internal validity is focused on the cause of a particular
+response to the treatment. Shadish et al. [2002] (p. 55) includes a table of nine
+8
+
+different threats to causal conclusions. Several of these involve confounding,
+which is the confusion of the treatment with other factors arising from poor
+operationalisation in an experiment.
+As a concrete example, consider an experiment measuring the effects of two
+different medicines on lowering blood pressure, but where one medicine is given
+to young patients and the other is given to elderly patients. This experimental
+design confounds the two medicines and patient age, and so the effects caused by
+the two factors cannot be disambiguated. Any conclusion from this experiment
+about the effects of the medicines lacks internal validity.
+5.1
+MIR concerns about internal validity
+Does my data collection introduce confounds? One’s methodology for collect-
+ing music data might unintentionally introduce structure. For instance, Sturm
+[2017] discusses how the BALLROOM dataset was assembled by downloading
+excerpts of music CDs sold at a website selling music for ballroom dance compe-
+titions. Ballroom dance competitions are regulated by organisations, e.g., World
+DanceSport Federation (WDSF),10 to ensure uniformity of events for competi-
+tors around the world. These organisations set strict requirements of tempo of
+each dance such that high skill is required of the dancers. Hence, the labels of
+the BALLROOM dataset can mean: 1) the rhythm of the music; 2) the type of
+dance; 3) the strict tempo requirements of the dance.
+Does my data partitioning introduce confounds? Dataset partitioning can
+also introduce confounds, e.g., “bleeding ground truth.” An example is to first
+segment recordings into short (e.g., 40ms) time frames and then partition these
+frames into training and testing sets, thus spreading highly correlated features
+across these sets. In the context of audio-based genre classification, the presence
+of songs from the same artists or albums in both training and test data has
+been shown to artificially inflate performance [Pampalk et al., 2005, Flexer and
+Schnitzer, 2010]. Flexer [2007] shows that audio-based genre classification using
+very direct representations of spectral content degrade more when employing
+artist/album filters than classification based on more abstract kind of features
+like rhythmic content (fluctuation patterns). This insight that problems of data
+partitioning can affect MIR systems in quite different ways and hence change
+performance rankings has been confirmed in another meta-study [Sturm, 2014a].
+Research on explainable and interpretable MIR might help to identify con-
+founds, see e.g., [Mishra et al., 2018], but latest results have shown that many
+popular explainers struggle when being probed with deliberate confounding
+transformations [Praher et al., 2021, Hoedt et al., 2022b].
+A general frame-
+work for detecting and characterising effects of confounding in MIR experi-
+ments through interventions is proposed by Sturm [2014b] and further refined
+in Rodr´ıguez-Algarra et al. [2019]. More work has yet to be done in the do-
+main of causal inference in MIR, but is difficult because its core concepts, like
+“similarity”, need to be operationalised first. Otherwise disentangling variables
+10https://www.worlddancesport.org/
+9
+
+potentially influencing measurements is not possible.
+5.2
+Internal validity in the typical MIR experiment
+Of interest is what it is in our trained ML models of Sec. 2 causing their response
+to be inconsistent with random selection. Knowing how Gaussian models used
+in LDA and QDA are built – mean and covariance parameters are estimated
+from training data – an internally valid conclusion is that these models work
+well in BALLROOM because likelihood distributions estimated from the train-
+ing data also fit the testing data well. Similarly, knowing how nearest neighbour
+classifiers work, an internally valid conclusion is that in BALLROOM the neigh-
+bourhoods of the labeled features in the training data include to a large extent
+testing data features with the same labels. Another internally valid conclusion
+is that the high performances of these ML models in BALLROOM are caused by
+the features together with the expressivity of the models capturing information
+related to the labels in BALLROOM. Our ML models have learned something
+about BALLROOM, which causes their performance to be significantly better
+than random selection.
+With reference to the aims of MIR research, we want to conclude something
+more specific, e.g., our ML models have learned to recognize the rhythms in
+BALLROOM. This is certainly one explanation consistent with the performance
+of our systems, but is it the only one? The internal validity of this conclusion
+relies on a key assumption: inferring the correct labels of BALLROOM can only
+be the result of learning to discriminate between and identify the rhythms in
+BALLROOM. In other words, we must assume that there is no other way to
+accurately infer labels in BALLROOM than by perceiving rhythm.
+It is not difficult to imagine other ways to infer the labels in BALLROOM
+by considering the multitude of variables present in the music recordings. One
+can hear instrumentation unique to each class, e.g., many recordings labeled
+“Tango” feature strings, piano and accordion. Many recordings labeled “Waltz”
+and “Viennese Waltz” feature strings. Many recordings labeled “Quickstep”
+and “Jive” feature brass, piano, vocals, and drums. So perceiving instrumenta-
+tion is one way to infer labels in BALLROOM with more success than random
+selection. We can reject such an explanation knowing that the time-domain
+nature of the features we are using are not sensitive to timbre, and so cannot
+clearly express the sound characteristic of an instrument. Then, are there time-
+domain variables other than rhythm that are closely associated with the labels
+in BALLROOM?
+One possible variable is tempo. If tempo is correlated with rhythm in BALL-
+ROOM then tempo estimation is another way a ML model can reproduce the
+ground truth labels. From listening to BALLROOM recordings labeled “Waltz”,
+“Quickstep” or “Cha cha cha”, one can hear they feature different rhythms and
+different tempi, but discriminating between them can be done on the basis of
+tempo alone. Tempo and rhythm are related musical characteristics, but they
+are not one and the same thing [Sethares, 2007].
+Let us perform an experiment to test the sensitivity of our trained ML models
+10
+
+Figure 1: Accuracies of the ML models in Table 1 with test recordings under-
+going pitch-preserving time dilation. Horizontal dashed line in grey is the mean
+accuracy of the best random system plus twice its standard deviation.
+to tempo. We perform an intervention where we alter all test recordings by
+some amount of pitch-preserving time dilation, and then measure the responses
+of the models to these new “treatments”. Dilating a recording by an amount
+1.1 increases its duration by 10% – or equivalently, makes the music in the
+recording have a tempo that is 10% slower. Dilating a recording by an amount
+0.9 decreases its duration by 10% – or equivalently, makes the music in the
+recording have a tempo that is 10% faster. Figure 1 shows how the accuracy
+of all ML models from Table 1 covaries with the amount of dilation.
+Our
+intervention has clearly revealed the extent to which the ML models rely on the
+tempi in the test data – which is no surprise given the background information
+of BALLROOM in the previous subsection.
+The validity of the conclusion that the responses of our ML models are
+caused by their recognition of the rhythms in BALLROOM relies on too strong
+of an assumption about BALLROOM, i.e., that rhythm recognition is the only
+way to infer labels in BALLROOM. The experimental design does not account
+for the structure present in the dataset; we do not control for other ways of
+inferring the labels of BALLROOM, which are guaranteed to exist by its very
+construction. From Table 1 and our experimental design, we thus cannot be any
+more specific in our causal inference than this: the responses of our ML models
+are caused by their having learned something about BALLROOM. At best, they
+are relying on at least tempo and rhythm. This then calls into question how
+comparing predictions with ground truth in BALLROOM relates to the ability
+we might actually want to measure, that is the recognition of rhythm. This is
+where the notion of construct validity becomes relevant.
+6
+Construct Validity
+Construct validity is “the validity of inferences about the higher order constructs
+that represent sampling particulars” [Shadish et al., 2002]. This involves the re-
+11
+
+0.8
+LDA
+QDA
+0.6
+INN
+Accuracy
+3NN
+5NN
+0.4
+7NN
+NN6
+0.2
+0.85
+0.90
+0.95
+1.00
+1.05
+1.10
+1.15
+DilationAmountlationship between what is meant to be inferred by the experimentalist from
+an experiment and what is actually measured, i.e., the operationalisation of the
+experimentalist’s intention. For instance, directly measuring the blood pressure
+of a person involves an invasive procedure inserting a measuring device in their
+veins. Blood pressure can be measured less invasively but indirectly by exter-
+nally applying known pressure to a vein and listening for when blood flow ceases.
+Knowledge about the incompressibility of liquids in closed systems makes the
+measurement of pressure in the balloon a relevant measure of blood pressure.
+Shadish et al. [2002] (p. 73) includes a table of fourteen different threats to
+construct validity, but several of these are irrelevant to computer-based exper-
+iments. The main threat is a questionable relationship between what is being
+measured and what is intended to be measured. Selecting a measure by conve-
+nience but not relevance, sampling from convenient populations, and a lack of
+definition of what is intended to be measured, are threats to construct validity.
+Construct validity involves more than just how something is measured; it also
+involves what is measured and in what settings.
+6.1
+MIR concerns about construct validity
+How is classification accuracy, or any figure of merit, in a labeled music dataset
+related to X? Two examples in MIR are the use of “genre” classification accu-
+racy as an indirect measure of music similarity [Pohle et al., 2008], or IR user
+satisfaction (see, e.g., [Schedl et al., 2013] for a discussion). The relationship
+between these is very tenuous, especially so considering that accuracy itself is
+an unreliable measure of whether or not a system has learned anything relevant
+to music [Sturm, 2013, 2014b]. A key reference in this respect is that of Pfungst
+[1911] describing a series of experiments in trying to reliably measure the arith-
+metic acumen of a horse that was only able to tap out answers. Counting the
+number of correct answers tapped out by the horse, no matter how many ques-
+tions are asked, is irrelevant without considering how each question is posed (the
+setting). The key to Pfungst discovering the cause of the horse’s apparent arith-
+metic acumen involved changing the setting: the questions remained the same,
+and accuracy of correct response was measured, but how the questions were
+posed was changed in order to control for different factors of the experiment.
+The same is true for MIR.
+What is the “use case” of the system to be tested? To counter threats to
+construct validity the MIR experimentalist must operationalise as much as pos-
+sible the use case of the system to be built and tested. An attempt to do so for
+music description is in Sturm et al. [2014], which emphasises the need to define
+success criteria. The experimentalist must determine how their method of mea-
+surement relates to the success criteria, e.g., relating classification accuracy in
+BALLROOM to the satisfaction of a specific user.
+How can we test the construct validity of a conclusion? One possibility is to
+assess the outcomes of different experiments which are supposed to measure the
+same higher order constructs. An example in MIR is to study correlations of
+different genre classifiers when given identical inputs [Liem and Mostert, 2020].
+12
+
+Low correlations between classifiers point to problems of construct validity. A
+related topic is that of adversarial examples, which casts doubt on the con-
+clusion that the high accuracy of an MIR system in some dataset reflects its
+“perception” of the music in the waveform.
+Adversarial examples have first
+been described in image analysis [Szegedy et al., 2014], where imperceptible
+perturbations of input data significantly degraded classification accuracy.
+For music genre classification systems, irrelevant audio filtering transforma-
+tions of music signals are used in Sturm [2014b] to both deflate and inflate
+classification accuracy to be no better than chance level or perfect 100%. The
+irrelevance is ascertained with listening tests and real human subjects, with the
+transformation being audible but not changing the clear impression of a certain
+musical genre. The same problematic behavior has been documented concerning
+music emotion classification [Sturm, 2014b]. For deep music embedding a test
+based on imperceptible audio transformations has been proposed [Kim et al.,
+2019], essentially verifying distance consistency both in the input audio space
+and corresponding latent deep space.
+Following these so-called untargeted attacks which try to change a predic-
+tion to an arbitrary target, targeted attacks aiming at changing predictions to
+specific classes have been explored. A targeted attack on genre recognition has
+been reported [Kereliuk et al., 2015], where magnitude spectral frames com-
+puted from audio are treated as images and attacked using approaches from
+image object recognition. For music instrument classification a targeted attack
+allowing to add perturbations directly to audio waveforms instead of spectro-
+grams has also been presented [Prinz et al., 2021]. Quite similar to the results
+by [Kereliuk et al., 2015], the attacks were able to reduce the accuracy close
+to a random baseline and produce misclassifications to any desired instrument.
+Signal perturbations were almost imperceptible apart from some high-frequency
+deviations. The authors also artificially boosted playcounts via an attack on a
+real-world music recommender, thereby demonstrating that such attacks can be
+a security issue in MIR. Follow-up work presented lines of defence against such
+malicious attacks [Hoedt et al., 2022a].
+6.2
+Construct validity in the typical MIR experiment
+When it comes to our typical MIR experiment in Sec.
+2, we are interested
+in making construct inferences around the latent ability of rhythm recognition
+we are supposedly measuring.
+For instance, one construct inference is that
+our features measure relevant aspects of rhythm in recorded music. In some
+sense, by their definition from basic signal processing components, our features
+come from temporal aspects that are certainly relevant to rhythm. Our features
+are also reliant on acoustic information, and in particular there being high-
+contrast differences in onsets captured by spectral flux – hence limiting their
+relationship to rhythms played by particular kinds of instruments with sharp
+attacks. However, we have seen above that the features are also indicative of
+tempo, which is not rhythm [Sethares, 2007], and that tempo is another path an
+ML algorithm can use to get to the rhythm label. Hence we are left to question
+13
+
+the relationship of our features to the concept we are trying to operationalise,
+i.e., rhythm.
+The construction of the BALLROOM dataset, intended to reflect different
+ballroom dance rhythms, is closely related to the validity of construct inferences
+derived from its use.
+How do the “Waltz” excerpts exemplify the “Waltz”
+rhythm? Is there one “Waltz” rhythm? In BALLROOM, there are actually
+two different labels for waltz: “Waltz” and “Viennese Waltz.” The distinction
+between them is based in part on tempo, according to the World Sport Dance
+Federation, a Viennese waltz is to be performed at a tempo between 174-180
+BPM Federation [2014].
+Having a system label any partition of the BALLROOM dataset provides no
+reliable measure of a system’s ability to recognise rhythm without changing the
+setting to control for other factors. It is not as simple as choosing a different
+feature, measure, cross-validation method, or using a particular statistical test.
+One must change the experiment itself such that rhythm recognition is what
+is actually being measured. This means that BALLROOM can still be useful
+to measuring the rhythm recognition of a ML system. Indeed, in the previ-
+ous section we used it to disprove the causal claim that the good performance
+of the ML systems of Table 1 is caused by their ability to recognize rhythm.
+Might performance in BALLROOM also be an indication of performance in
+other datasets focused on rhythm? This is where the notion of external validity
+becomes relevant.
+7
+External Validity
+External validity is “the validity of inferences about the extent to which a causal
+relationship holds over variations in experimental units, settings, treatment
+variables and measurement variables” [Shadish et al., 2002]. More generally,
+external validity is the truth of a generalised causal inference drawn from an ex-
+periment. An example is inferring that medicine found to lower blood pressure
+in patients living in Germany will also lower blood pressure in people living in
+Mexico – a conclusion that can lack validity due to differences in diet, living
+and working conditions, and so on. Another example is that increasing the dose
+of the medicine will cause blood pressure to lower further in the studied popu-
+lation. If a causal inference we draw from an experiment lacks internal validity,
+then generalising that inference to include variations not tested will not have
+external validity. Shadish et al. [2002] (p. 87) includes a table of five different
+threats to external validity, which are in addition to the threats to internal va-
+lidity. The main threat is that variation of the components of the experiment
+might destroy the causal inference that holds in the experiment. For instance, a
+medication may work for the type of illness tested, but that type of illness may
+not be generalisable to other closely related illnesses.
+14
+
+7.1
+MIR concerns about external validity
+Does my model generalize to out-of-sample data? The standard approach in
+evaluating MIR classification systems is to use separate train and test sets in
+cross-validation experiments to obtain seemingly unbiased estimates of perfor-
+mance. However, if such MIR systems are exposed to independent out-of-sample
+data often severe loss of performance is observed. One example are experiments
+on genre recognition where accuracy results do not hold when evaluated across
+different collections that are not part of the training sets [Bogdanov et al., 2016,
+2019]. The results do not generalize to supposedly identical genre labels in dif-
+ferent collections, which reflects a lack of external validity. Genre labels like
+e.g. ’Rock’ will be used differently by different annotators working on these
+collections – which is a threat to construct validity. Another example are how
+different audio encodings affect subsequent computation of descriptors and clas-
+sification results [Urbano et al., 2014], or how in general differences in software
+implementations diminish replicability [McFee et al., 2018].
+Do different raters agree on a ground truth? Human perception of music
+is highly subjective resulting in possible low inter-rater agreement. Therefore
+only a certain amount of agreement can be expected if several human subjects
+are asked to rate the same song pairs according to their perceived similarity,
+depending on a number of subjective factors [Schedl et al., 2013, Flexer and Grill,
+2016] like personal taste, listening history, familiarity with the music, current
+mood, etc. Concerning annotation of music, Seyerlehner et al. [2010] shows that
+the performance of humans classifying songs into 19 genres ranges from modest
+26% to 71%. Audio-based grounding of everyday musical terms shows the same
+problematic results [Aucouturier, 2009]. It has even been argued [Wiggins, 2009]
+that no such thing as an immovable ‘ground’ exists in the context of music,
+because music itself is subjective, highly context-dependent and dynamic.
+The lack of inter-rater agreement presents a problem of external validity be-
+cause inferences from the experiment do not generalize from users or annotators
+in the experiment to the intended target population of arbitrary users/annotators.
+It is also a problem of reliability, since different groups of users or annotators
+with their differing subjective opinions will impede repeatability of experimen-
+tal results. This lack of inter-rater agreement presents an upper bound for MIR
+approaches, since it is not meaningful to have computational models going be-
+yond the level of human agreement. Such upper bounds have been reported
+[Jones et al., 2007, Schedl et al., 2013, Flexer and Grill, 2016] for the MIREX
+tasks of ‘Audio Music Similarity and Retrieval’ (AMS) and ‘Music Structural
+Segmentation’ (MSS). For AMS the upper bound has already been reached in
+2009, while for MSS the upper bound is within reach for at least some genres
+of music. Comparable results exist concerning music structure analysis [Nieto
+et al., 2014] and chord estimation [Ni et al., 2013, Koops et al., 2019].
+Do raters agree with themselves at different points in time? Going beyond
+the question of whether different annotators agree on a ground truth one can also
+access what the level of agreement within one person is when faced with identical
+annotation tasks at different points in time. A high intra-rater agreement would
+15
+
+help to overcome the problem of upper bounds in MIR systems since it would
+make personalization of models meaningful, i.e. to have separate models for
+individual persons. However, at least for the task of general music similarity
+it has been shown that intra-rater agreement is only slightly higher than inter-
+rater agreement [Flexer et al., 2021], with the absolute level also depending on
+music material and mood of raters at test time. An approach to personalize
+chord labels for individual annotators via deep learning was more successful
+[Koops et al., 2020].
+Returning to the impact of irrelevant transformations [Sturm, 2014b, Kere-
+liuk et al., 2015, Rodr´ıguez-Algarra et al., 2016] and the existence of adversarial
+examples [Prinz et al., 2021], one can ask, does my model generalize to these
+kinds of attacks? More constructively, one can seek ways to lessen the impact of
+these attacks, thus possibly increasing the generalization of the models [Hoedt
+et al., 2022a].
+7.2
+External validity in the typical MIR experiment
+Considering the typical MIR experiment in Sec. 2, we cannot validly conclude
+that any of our models is recognizing rhythm in general because we do not know
+if they are recognizing rhythm in BALLROOM. Our dilation intervention ex-
+periment in Sec. 5.2 reveals that all of the models lose their supposed ability to
+recognize rhythm in BALLROOM, so there is no reason to infer they will rec-
+ognize rhythm elsewhere. One causal conclusion we might generalise is that all
+our models perform well in BALLROOM because they have learned something
+about BALLROOM — a curated set of recordings downloaded from a specific
+website in 2004. Might our models have also learned something about other
+recordings from that same website but collected many years later?
+The extended BALLROOM dataset (X-BALLROOM) [Marchand and Peeters,
+2016] consists of 3,484 audio recordings in the same eight dance styles or mu-
+sic rhythms as BALLROOM, but downloaded from the same website over a
+decade later.
+This gives us a chance to test our hypothesis.
+The figures of
+merit measured from our models trained in BALLROOM but applied to all
+of X-BALLROOM are shown in Table 2. We still see significant covariation
+between response and the use of ML with our features. By and large, what-
+ever concepts our ML models have learned about BALLROOM carry over to
+X-BALLROOM – but we still do not know whether or not those concepts have
+to do with rhythm.
+We can take this opportunity to test the generalizability of a conclusion
+about Gaussian modelling outperforming nearest neighbor modelling from Table
+1. Training and testing the same ML models with a 70/30 random partition of
+X-BALLROOM produces the results in Table 3. We still see QDA performing
+the best, but the nearest neighbor models show large performance increases.
+The cause of this change in performance should be investigated, and related
+to the confounding known to exist in BALLROOM. Does the confounding also
+exist in X-BALLROOM, suggested by the results in Table 2? This shows how
+the observations made in the typical MIR experiment represent the beginning
+16
+
+Accuracy
+Precision
+Recall
+f1-score
+LDA
+0.659
+0.647
+0.643
+0.643
+QDA
+0.682
+0.678
+0.672
+0.673
+1NN
+0.622
+0.616
+0.602
+0.604
+3NN
+0.636
+0.629
+0.610
+0.613
+5NN
+0.644
+0.643
+0.617
+0.619
+7NN
+0.647
+0.646
+0.619
+0.621
+9NN
+0.645
+0.643
+0.615
+0.618
+unif
+0.12 ± 0.01
+0.13 ± 0.01
+0.12 ± 0.01
+0.12 ± 0.01
+freq
+0.13 ± 0.01
+0.13 ± 0.01
+0.12 ± 0.01
+0.12 ± 0.01
+maj
+0.13
+0.02
+0.12
+0.03
+Table 2: As in Table 1, models trained in BALLROOM and tested in all of
+X-BALLROOM [Marchand and Peeters, 2016].
+Mod
+Accuracy
+Precision
+Recall
+f1-score
+LDA
+0.726
+0.721
+0.722
+0.717
+QDA
+0.763
+0.764
+0.773
+0.758
+1NN
+0.702
+0.691
+0.693
+0.690
+3NN
+0.741
+0.731
+0.727
+0.727
+5NN
+0.750
+0.744
+0.736
+0.738
+7NN
+0.753
+0.746
+0.736
+0.738
+9NN
+0.751
+0.746
+0.734
+0.737
+unif
+0.12 ± 0.01
+0.13 ± 0.01
+0.12 ± 0.01
+0.12 ± 0.01
+freq
+0.13 ± 0.01
+0.13 ± 0.01
+0.13 ± 0.01
+0.12 ± 0.01
+maj
+0.13
+0.02
+0.12
+0.03
+Table 3: As in Table 1, but models trained and tested in X-BALLROOM [Marc-
+hand and Peeters, 2016].
+of avenues for exploration, sources of hypotheses, and not the final result.
+8
+Conclusion
+This article provides a review of the notion of validity based on the typology
+given in Shadish et al. [2002]. It brings together the few sources in MIR that
+mention anything to do with validity, and several sources that do not but are
+related.
+This article does not aim to prescribe how to design and perform
+experiments such that valid conclusions can be drawn from them. Instead it
+aims to bring within the realm of MIR what validity means, why it is important,
+and how it can be threatened.
+In MIR the predominant experimental methodology is given by the Cran-
+field paradigm: train a model on a partition of a dataset and count the number
+of correct answers on another partition. This kind of experiment is inexpen-
+sive to do with the data conveniently at hand, and provides numbers that can
+be compared in ways that convince peer reviewers that progress has been ac-
+complished [Hand, 2006]. Despite various appeals [Peeters et al., 2012, Schedl
+et al., 2013] and beseechings [Urbano et al., 2012, 2013, Sturm, 2013, 2014b,
+Flexer and Grill, 2016, Flexer et al., 2021], such an experimental approach is
+17
+
+still standard in the field and its serious flaws are ignored. Any inference from
+this experiment that is more general than “the system has learned something
+about this dataset” lacks internal, construct and external validity. This does
+not mean that all such inferences are false, just that they cannot follow from
+the experiment as designed and implemented. Reproducing the ground truth
+of a dataset represents a beginning and must be followed by a search for the
+causes of the observed behavior. One must resist the urge to conclude that a
+system must be doing whatever is hoped for.
+Shadish et al. [2002] provides an established starting point for MIR, but there
+exist other types of validity. For instance, Lund [2021] revises the typology of
+Shadish et al. [2002] to address ambiguities between causes and treatments, to
+better define aspects of settings, and to establish a hierarchical ordering of five
+types of validity: statistical conclusion, causal, construct, generalization and
+theoretical.
+An important distinction in this typology from that of Shadish
+et al. [2002] is its emphasis on a major aim of basic research: to contribute
+theory.
+Other kinds of validity include ecological, convergent, and criterion
+[Urbano, 2011]; but these still deal with the kind of conclusion one is drawing
+from evidence collected in some way.
+As a final note, a frustration when encountering Shadish et al. [2002] as an
+engineer is that of its 623 pages there are only five pages with at least one equa-
+tion on them. Instead, Shadish et al. [2002] describe experiments and how each
+type of validity manifests in the conclusions drawn, with specific threats to the
+reasoning of those conclusions. Experiments, not to mention experimentalists,
+are such complex assemblages that expressing them in formal ways that appear
+to permit the computation of numbers that relate to each type of validity would
+probably have very limited applicability, and then only be understood by a lim-
+ited audience. The language of validity is reason, and we hope this manuscript
+will inspire MIR researchers to think creatively about the phenomena they ob-
+serve to discover their causes.
+Acknowledgments
+We thank Juli´an Urbano and Hugo Maruri-Aguilar for helpful discussions during
+the drafting of this manuscript, as well as the constructive criticisms of reviewers
+from the Transactions of the Society for Music Information Retrieval. The con-
+tribution of Sturm is supported by a project that has received funding from the
+European Research Council (ERC) under the European Union’s Horizon 2020
+research and innovation programme (Grant agreement No. 864189 MUSAiC:
+Music at the Frontiers of Artificial Creativity and Criticism). The contribu-
+tion of Flexer is supported by funding from the Austrian Science Fund (FWF,
+project number P 31988). For the purpose of open access, the authors have
+applied a CC BY public copyright license to any author accepted manuscript
+version arising from this submission.
+18
+
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+K. Seyerlehner, G. Widmer, and P. Knees. A comparison of human, automatic
+and collaborative music genre classification and user centric evaluation of
+genre classification systems. In International Workshop on Adaptive Multi-
+media Retrieval, pages 118–131. Springer, 2010.
+W. R. Shadish, T. D. Cook, and D. T. Campbell.
+Experimental and quasi-
+experimental designs for generalized causal inference. Boston: Houghton Mif-
+flin, 2002.
+B. L. Sturm. Classification accuracy is not enough. Journal of Intelligent In-
+formation Systems, 41(3):371–406, 2013.
+B. L. Sturm. The state of the art ten years after a state of the art: Future
+research in music information retrieval. Journal of New Music Research, 43
+(2):147–172, 2014a.
+B. L. Sturm. A simple method to determine if a music information retrieval
+system is a “horse”.
+IEEE Transactions on Multimedia, 16(6):1636–1644,
+2014b.
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+B. L. Sturm. The “horse” inside: seeking causes behind the behaviors of music
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+Computers in Entertainment (CIE), 14(2):1–32,
+2017.
+B. L. Sturm and N. Collins. The kiki-bouba challenge: Algorithmic compo-
+sition for content-based mir research & development. In Proceedings of the
+15th Conference of the International Society for Music Information Retrieval,
+pages 21–26, 2014.
+B. L. Sturm, R. Bardeli, T. Langlois, and V. Emiya. Formalizing the problem of
+music description. In Proceedings of the 15th Conference of the International
+Society for Music Information Retrieval, pages 89–94, 2014.
+B. L. Sturm, C. Kereliuk, and J. Larsen.
+¿El Caballo Viejo?
+Latin genre
+recognition with deep learning and spectral periodicity. In Proceedings of the
+International Conference on Mathematics and Computation in Music, pages
+335–346, 2015.
+C. Szegedy, W. Zaremba, I. Sutskever, J. Bruna, D. Erhan, I. J. Goodfellow,
+and R. Fergus. Intriguing properties of neural networks. In Proceedings of the
+2nd International Conference on Learning Representations, ICLR, 2014.
+J. Urbano. Information retrieval meta-evaluation: Challenges and opportunities
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+J. Urbano and A. Flexer. Statistical analysis of results in music information
+retrieval: why and how (abstract). In Proceedings of the International Society
+for Music Information Retrieval Conference, pages xli–xlii, 2018.
+J. Urbano, J. S. Downie, B. Mcfee, and M. Schedl. How significant is statistically
+significant? the case of audio music similarity and retrieval. In Proceedings
+of the 13th Conference of the International Society for Music Information
+Retrieval, pages 181–186, 2012.
+J. Urbano, M. Schedl, and X. Serra. Evaluation in music information retrieval.
+Journal of Intelligent Information Systems, 41(3):345–369, 2013.
+J. Urbano, D. Bogdanov, H. Boyer, E. G´omez Guti´errez, X. Serra, et al. What
+is the effect of audio quality on the robustness of mfccs and chroma features?
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+E. M. Voorhees. The philosophy of information retrieval evaluation. In Proceed-
+ings of the Cross-Language Evaluation Forum, 2001.
+A. Wang. An industrial strength audio search algorithm. In Proceedings of the
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+pages 7–13, Oct. 2003.
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+24
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf,len=1103
+page_content='Validity in Music Information Research  Experiments  Bob L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Sturm1 and Arthur Flexer2  1Division of Speech, Music and Hearing, KTH Stockholm, Sweden,  bobs@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='se  2Institute of Computational Perception, Johannes Kepler  University Linz, Austria, arthur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='flexer@jku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='at  Abstract  Validity is the truth of an inference made from evidence, such as data  collected in an experiment, and is central to working scientifically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Given  the maturity of the domain of music information research (MIR), validity  in our opinion should be discussed and considered much more than it has  been so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Considering validity in one’s work can improve its scientific  and engineering value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Puzzling MIR phenomena like adversarial attacks  and performance glass ceilings become less mysterious through the lens  of validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In this article, we review the subject of validity in general,  considering the four major types of validity from a key reference: Shadish  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We ground our discussion of these types with a prototypical  MIR experiment: music classification using machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Through  this MIR experimentalists can be guided to make valid inferences from  data collected from their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  1  Introduction  The multi-disciplinary field of Music Information Research (MIR) is focused  on making music and information about music accessible to a variety of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  This ranges from systems for search and retrieval, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', Shazam [Wang, 2003]  or TunePal [Duggan and O’Shea, 2011], to recommendation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', Spotify or  Pandora, and even to more creative applications like generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' These sys-  tems consist of many computational components working in domains between  the acoustic and symbolic where computability is possible [M¨uller, 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The  effectiveness and reliability of these complex systems and their components are  of prime importance to the MIR researcher, not to mention other stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The MIR researcher performs experiments to compare approaches for mod-  elling and retrieving music data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A principal focus is on users, but the cost of  performing experiments with users is high, and the replicability of such studies is  difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This has motivated the use of computer-based experiments where “test  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='01578v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='SD]  4 Jan 2023  collections” serve as proxies for human users, known as the Cranfield Paradigm  [Cleverdon, 1991]: the researcher predicts the usefulness of an MIR system by  applying it to a representative sample of the problem domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', queries and  relevant documents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  While such an approach is inexpensive and replicable,  its relevance and reliability for MIR, and information retrieval in general, have  been questioned [Voorhees, 2001, Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Under the Cranfield Paradigm, state-of-the-art MIR systems perform excep-  tionally well in reproducing the ground truth of some datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', inferring  rhythm, genre or emotion from audio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' However, slight and irrelevant trans-  formations of the audio can suddenly render these systems ineffectual [Sturm,  2014b, Kereliuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2015, Rodr´ıguez-Algarra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2016, Prinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  In one case [Rodr´ıguez-Algarra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2016], a “genre recognition” system re-  lies on infrasonic signatures, seemingly originating from the data collection, but  nonetheless imperceptible and irrelevant for human listeners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In another case  [Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2015], a “rhythm recognition” system seemingly uses tempo to  infer rhythm, a confusion originating again from the data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Related are discussions about “glass ceilings” of MIR systems [Aucouturier  and Pachet, 2004, Pohle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2008], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', that the observed barrier to improving  system performance to perfect or human level is due to the psychophysical and  cultural factors of music missing from computable features extracted from audio  recordings [Wiggins, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' MIR researchers striving for great improvement of  those numbers in established problems and datasets might actually make no  practical difference for human users [Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Flexer and Grill [2016]  identifies one reason contributing to performance glass ceilings: the data used  to train and test MIR systems can arise from tasks that are not well-posed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=',  music similarity), and thus render meaningless the pursuit of better numbers  using that data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The lack of definition in research problems motivates Sturm  and Collins [2014] to propose MIR researchers solve synthetic yet well-defined  simplifications of harder, ill-posed problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  At the heart of an MIR experiment is the relationship between conclusions  drawn from its results and their validity, or “truth value” [Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Ideally, an experiment will be carefully designed and implemented to answer  a well-defined hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='1 Its components – units, treatments, design, obser-  vations, and settings – should be operationalised (translated from theory into  practice) to maximize quality and minimize cost (money, time, ethics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This is  the purview of the discipline Design of Experiments: how can one get the best  evidence for the least cost?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A look through the proceedings of MIR’s premier  conference ISMIR2 and related publications, however, reveals a general lack of  awareness of validity in the MIR community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The first reference explicitly introducing validity to MIR is Urbano [2011],  discussing it in relation to text information retrieval (IR) literature, and specif-  ically advocating validity-based meta-evaluation of MIREX results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='3  This is  1We encourage readers to review Chase [2001] to see the remarkable lengths an experimen-  talist must go to test even a simple hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  2https://ismir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='net/conferences/  3‘Music Information Retrieval Evaluation eXchange’ (MIREX) is an annual evaluation  2  further developed in Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2013], embracing the validity typology of  Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002], and bringing in notions of reliability and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' While  the review of Schedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2013] does not explicitly refer to validity or even  mentions the Cranfield Paradigm, it makes clear how central computer exper-  iments are to MIR, and criticizes a general lack of consideration of users in  its conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Sturm [2013] asserts that just reporting figures of merit like  accuracy is not sufficient to decide whether an MIR system is really recogniz-  ing “genre” in musical signals, or whether it relies on irrelevant confounding  factors, later stating that these uncontrolled factors are a danger to the valid-  ity of conclusions drawn from such experiments [Sturm, 2014b, 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Sturm  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2014] attempts to define music description (including the “use case”) to  motivate evaluating music description systems in ways that allow for valid and  relevant conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  More recently, Rodr´ıguez-Algarra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2019] propose and test a method for  detecting the existence of confounding in MIR experiments, thus addressing the  lack of construct validity of conclusions drawn from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Liem and Mostert  [2020] studies construct validity of high-level musical concepts like genre by  confronting MIR systems with data different from what they have been trained  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Flexer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2021] criticizes the lack of external validity of experiments  on general music similarity and demands focus on specific aspects of music  similarity and definition of a specific use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Probably the most recent attempt  to comprehensively discuss validity and reliability in MIR is an unpublished  tutorial held at ISMIR 2018 [Urbano and Flexer, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The above review of previous work on validity in MIR shows it to be dis-  persed and fragmented across only relatively few publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Despite a small  chorus of calls to address major methodological problems of MIR experiments  to improve validity in the discipline, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', Urbano [2011], Schedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2013],  Sturm [2014b], Urbano and Flexer [2018], Liem and Mostert [2020], Flexer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  [2021], there has yet to be published a systematic and critical engagement of  what validity means in the context of MIR, and how to consider it when design-  ing, implementing and analyzing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  In this article, we review the four principal types of validity in Shadish  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002], an authoritative resource about validity in causal inference and  experimental science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Other typologies exist, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', Lund [2021], but we focus on  that of Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] because it is an established point of reference, and  has already been introduced to MIR [Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For each type, we  discuss common threats, relate it to the MIR discipline in general, and then more  concretely in terms of a typical MIR experiment, presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We include  a python jupyter notebook that runs the experiment and computes a variety of  statistics and results discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='4 Section 3 reviews the components of the  experiment, on which the discussion of validity hinges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Each section 4–7 reviews  the four types of validity and presents actionable questions which can help MIR  researchers to scrutinize the conclusions they draw from their experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  campaign for MIR algorithms [Downie, 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='com/boblsturm/mirvaliditytutorial  3  Accuracy  Precision  Recall  f1-score  LDA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content='668  unif  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='02  freq  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='02  maj  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content='03  Table 1: Accuracy, and macro-averaged precision, recall and f1-score observed  for several models in a testing partition of BALLROOM [Dixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The  performance of two models randomly selecting labels (with standard deviation)  are shown in the rows labeled: unif samples labels uniformly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' freq samples  labels according to training data label frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The last row maj shows the  performance of a model choosing the label most frequent in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  2  A Typical MIR Experiment  To ground our general discussion of validity, we present a typical MIR exper-  iment that exemplifies a considerable amount of MIR research: audio classifi-  cation using machine learning and a benchmark dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Consider the BALL-  ROOM dataset [Dixon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2004], created around 2004 from downloading  excerpts of music CDs for sale at a website focused on ballroom dancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' BALL-  ROOM consists of 698 audio recordings, each about 30 seconds long and labeled  with one of eight dance styles or music rhythms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', “Waltz”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' BALLROOM  has appeared in dozens of studies [Sturm, 2017], and most often in experiments  measuring the amount of ground truth labels reproduced by different combina-  tions of classifiers and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  We create a random 70/30 partition, with 488/210 recordings comprising  the training/testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' From each recording we compute the spectral flux  onset strength envelope using a hop size of 1024 samples, or 46 ms [McFee  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We compute the normalised autocorrelation of the envelope and  retain the portion relating to lags in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='23, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='14] seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Finally, we model the  autocorrelation by a 12th-order autoregressive model, which results in a feature  of 13 dimensions describing the 30-second audio recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We normalise each  dimension of the training features to be zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  We  normalise the testing data features using the same parameters as for normalising  the training data features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  We model features using multivariate Gaussian distributions (linear and  quadratic discriminant analysis, LDA and QDA), or k-nearest neighbours (KNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  We compare the labels inferred by a model to the ground truth and compute the  accuracy, and macro-averaged precision, recall, and f1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='5 Table 1 shows the  5A macro-averaged figure of merit is the mean figure of merit measured across classes,  regardless of the number of observations of each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  4  outcomes of this experiment, which can be reproduced with the supplementary  python jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='6 The last rows show statistics of three other strate-  gies: unif samples labels according to a uniform prior and freq samples labels  according to training data label frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='7 maj selects the most frequent label  in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' These provide comparison points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  A criticism can be made here that the above experiment is not really repre-  sentative of MIR experiments today: datasets are now several orders of magni-  tude larger, and deep learning has in many cases made obsolete the engineering  of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' However, we will press on with this example for four reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' First,  the BALLROOM dataset is analyzed to such an extent that classification exper-  iments with it illustrate the four types of validity in clear and graspable ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Using a larger dataset can illustrate the same things, but would first require a  thorough analysis of its composition and provenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Second, concerns about  validity are not rendered moot by simply increasing data or shifting the ma-  chine learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Third, the experiment above is typical in its design: a  dataset is collected and processed, submitted to a variety of machine learning  approaches, and measurements are made by applying the trained systems to  some test collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Finally, the BALLROOM dataset is unique in that an  “extended” version of it was created a decade later from the same web resource  [Marchand and Peeters, 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This allows us to meaningfully test the external  validity of conclusions about systems trained in BALLROOM, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', the concepts  learned from BALLROOM are relevant to ballroom dance music, and not just  ballroom dance music in 2004 (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  3  Components of an Experiment  Before discussing the validity of conclusions drawn from the typical MIR exper-  iment, we must identify its units, treatments, design, observations, and settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Treatments are the things applied to units in order to cause an effect (or not  in the case of a control), units are the things that are treated, and observations  are what is measured on a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The design specifies which treatment is applied  to which unit, and settings involve time, place, and condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' To make this  more concrete, consider a medical experiment in which the effect of a treatment  on blood pressure is being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A number of people are sampled from a  population, some of whom will receive the treatment while the others receive  a placebo (control).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The design describes which people get the treatment, and  which do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The observation is the blood pressure of a person after one month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The setting can include particulars of the population (rural or urban), place of  treatment (hospital or home), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The experimentalist contrasts blood  pressure observations across groups to conclude if the treatment has an effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Returning to our typical MIR experiment, we wish to determine the effec-  tiveness of different machine learning (ML) algorithms in predicting the labels of  6See footnote 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  7The mean and standard deviation of these figures of merit can be found in theory, but  here they are found empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' See the accompanying code in footnote 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  5  a test recording dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' There are two possibilities here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We can see the treat-  ments as the ten algorithms, and the units as the entire testing dataset (how  does the dataset respond to each algorithm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' ), or we can see the entire testing  dataset as the one treatment, and the units as the ten algorithms (how does  each algorithm respond to the dataset?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Since Table 1 reports figures of merit  (observations) of each algorithm on the entire dataset, the latter interpretation  motivates conclusions about the effectiveness of particular algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In this  case, the design is simple: each unit is given the same treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The setting  involves the partitioning of BALLROOM, the features extracted, random seeds,  software libraries, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  4  Statistical Conclusion Validity  Statistical conclusion validity is “the validity of inferences about covariation  between two variables” [Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This includes concluding that a  covariation exists, and perhaps its strength as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This is the level at which  one is concerned with statistical significance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', that an observed covariation  between treatment and effect is not likely to arise by chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  [2002] (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 45) includes a table of nine different threats to statistical conclusion  validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Four threats relevant to computer-based experiments are: violated  assumptions about the statistics underlying the responses (and the use of the  wrong statistical test, a type III error [Kimball, 1957]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' a sample size too small to  reliably detect covariation (lack of power);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' the purposeful search for significant  results by trying multiple analyses and data selections (“p-hacking”, Head et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  [2015]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' and increased variance in observations due to the heterogeneity of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='1  MIR concerns about statistical conclusion validity  Are my results statistically significant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Null hypothesis statistical testing (NHST)  quantifies whether the observed effects of the treatments on the responses arise  by mere chance, as well as the direction of effect and its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This answers the  question: are the results statistically significant?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Fundamentals about statistical  testing in MIR are discussed by Flexer [2006], for Artificial Intelligence in gen-  eral by Cohen [1995], and for machine learning by Japkowicz and Shah [2011].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  One must take care in selecting a statistical test to use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' each one makes strong  assumptions that could be violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' NHST is most straightforwardly applicable  to completely randomized experimental designs [Bailey, 2008], thereby reducing  the possibility of structure in units and treatments interfering with the responses  (which results in confounding, discussed in the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Most MIR exper-  iments cannot use complete randomisation because the target population from  which samples come is unclear (what is a random sample of “sad” music, with  the term “sad” being quite ill-defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' ), and so the kinds of conclusions that  can be made with NHST in MIR are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='8  8Experimental designs that cannot be completely randomised are called quasi-experimental  designs, which is another major topic of Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  6  Is the observed statistical significance relevant for a user?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In MIR, even if one  finds statistical significance, this may not generalise to a perceivable difference  for actual users interacting with the “improved” MIR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' As an example  from MIR, a crowd-sourced user evaluation [Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2012] demonstrates  that there is an upper bound of user satisfaction with music recommendation  systems of about 80%, since this was the highest percentage of users agreeing  that two systems “are equally good.”  In addition, for the MIREX task of  Audio Music Similarity and Retrieval Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2012] demonstrate that  statistically significant differences between algorithms can be so small that they  make no practical difference for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='2  Statistical conclusion validity in the typical MIR ex-  periment  Let us now consider the typical MIR experiment in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 2 and reason about  what conclusions we can draw from it that have statistical conclusion validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Table 1 clearly shows that each response of machine learning (ML) to the dataset  is greater than the random approaches unif, freq and maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' How likely is it that  any of the responses of ML is due to chance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', that any of the ML approaches  is actually no better than one of the random approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Since we have the  empirical distributions for unif and freq, we can estimate the probability of  either of them resulting in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', a macro-average recall at least as large as  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='6: p < e−200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='9  Hence, a valid statistical conclusion is that we observe a  significant covariation between the use of ML with these particular features and  the responses measured on a specific partition of BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' An equivalent  conclusion is that each response of these trained ML systems treated by this  partition of BALLROOM are inconsistent with choosing randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Whereas the conclusions above relate to the use of ML, one might consider  statistical conclusions relating to the type of ML, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', Gaussian modelling (LDA  and QDA) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' nearest neighbour modelling (KNN), or LDA vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' QDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' One may  want to conclude that Gaussian modelling is better than nearest neighbour  modelling with these features on BALLROOM, or even more precisely, QDA  performs the best with these features on BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This involves looking  at the differences between responses, called contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' While there is not much  reason to suspect that the particular 70/30 partition of BALLROOM used is  unusually beneficial to one kind of ML model over the other, what we do not  know from our experiment is the distribution of any response due to ML, and  so of any difference of responses due to ML, related to the random partitioning  of BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We have two measurements of Gaussian models, and five of  nearest neighbour models, but we cannot simply compute and compare statistics  9This is the probability of observing a macro-averaged recall at least as big as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='6 from the  models selecting labels randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Specifically, it is the probability of sampling a number from  a random variable distributed Gaussian with mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='125 and standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='02 in the  domain [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='6, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' See the accompanying jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Also note that a macro-averaged  recall of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='6 is a pessimistic choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Using the higher numbers in Table 1 results in even lower  probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  7  of these without involving auxiliary information not present in the experiment,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', how the decision boundaries of an ML model vary according to variation  in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  In fact, if we conclude from Table 1 that Gaussian modelling performs bet-  ter than nearest neighbour modelling with these features on 70/30 partitions of  BALLROOM, we would be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This is a “type I error”, which is concluding  there to be a significant difference when in fact there is none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' When we perform  this experiment 1000 times with random 70/30 partitions we observe that the  difference between the best response of a Gaussian model and the best response  of a nearest neighbour model is distributed Gaussian, and that the probability  of observing zero difference or less is p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='41 for any of the figures of merit  (see the accompanying jupyter notebook).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' That is, the more statistically valid  conclusion is that three out of five times randomly partitioning BALLROOM  the best Gaussian model performs better than the best nearest neighbor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  With this new data we see that the results in Table 1 do not merit a valid sta-  tistical conclusion that one of the modelling approaches is significantly better or  worse than any other in BALLROOM using these features for any significance  level α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' On the other hand, if we were to conclude that QDA performs  better than nearest neighbour modelling with these features on 70/30 random  partitions of BALLROOM, we would be correct – but only for macro-averaged  recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Performing 1000 repetitions of our experiment shows that the distribu-  tion of the difference between the response from QDA and the best response of  all others is distributed such that the probability of observing a negative differ-  ence is p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='046 only for recall (see the accompanying jupyter notebook).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In  other words, less than one out of 20 times do we see QDA perform worse than  the other models in terms of recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  In summary, the most general statistical conclusion we can make from Table  1 is that the responses we observe from ML are highly inconsistent with the  responses of models choosing randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Each ML model knows something about  BALLROOM linking the features computed from a music recording with its  label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Because we do not know the amount of variation in any response due to  partitioning in Table 1, we cannot make any valid statistical conclusion about  which type of ML model is the best, or which particular realisation is the best,  for this particular dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In order to go further, we had to run the experiment  multiple times to obtain distributions of the contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Even then, however, we  cannot say anything about the cause of significant differences yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This is where  the notion of internal validity becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  5  Internal Validity  Internal validity is “the validity of inferences about whether the observed covari-  ation between two variables is causal” [Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' While statistical  conclusion validity is concerned only with the strength of covariation between  treatment and responses, internal validity is focused on the cause of a particular  response to the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 55) includes a table of nine  8  different threats to causal conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Several of these involve confounding,  which is the confusion of the treatment with other factors arising from poor  operationalisation in an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  As a concrete example, consider an experiment measuring the effects of two  different medicines on lowering blood pressure, but where one medicine is given  to young patients and the other is given to elderly patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This experimental  design confounds the two medicines and patient age, and so the effects caused by  the two factors cannot be disambiguated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Any conclusion from this experiment  about the effects of the medicines lacks internal validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='1  MIR concerns about internal validity  Does my data collection introduce confounds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' One’s methodology for collect-  ing music data might unintentionally introduce structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For instance, Sturm  [2017] discusses how the BALLROOM dataset was assembled by downloading  excerpts of music CDs sold at a website selling music for ballroom dance compe-  titions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Ballroom dance competitions are regulated by organisations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', World  DanceSport Federation (WDSF),10 to ensure uniformity of events for competi-  tors around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' These organisations set strict requirements of tempo of  each dance such that high skill is required of the dancers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Hence, the labels of  the BALLROOM dataset can mean: 1) the rhythm of the music;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 2) the type of  dance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 3) the strict tempo requirements of the dance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Does my data partitioning introduce confounds?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Dataset partitioning can  also introduce confounds, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', “bleeding ground truth.” An example is to first  segment recordings into short (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 40ms) time frames and then partition these  frames into training and testing sets, thus spreading highly correlated features  across these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In the context of audio-based genre classification, the presence  of songs from the same artists or albums in both training and test data has  been shown to artificially inflate performance [Pampalk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2005, Flexer and  Schnitzer, 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Flexer [2007] shows that audio-based genre classification using  very direct representations of spectral content degrade more when employing  artist/album filters than classification based on more abstract kind of features  like rhythmic content (fluctuation patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This insight that problems of data  partitioning can affect MIR systems in quite different ways and hence change  performance rankings has been confirmed in another meta-study [Sturm, 2014a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Research on explainable and interpretable MIR might help to identify con-  founds, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', [Mishra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2018], but latest results have shown that many  popular explainers struggle when being probed with deliberate confounding  transformations [Praher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2021, Hoedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2022b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  A general frame-  work for detecting and characterising effects of confounding in MIR experi-  ments through interventions is proposed by Sturm [2014b] and further refined  in Rodr´ıguez-Algarra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' More work has yet to be done in the do-  main of causal inference in MIR, but is difficult because its core concepts, like  “similarity”, need to be operationalised first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Otherwise disentangling variables  10https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='worlddancesport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='org/  9  potentially influencing measurements is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='2  Internal validity in the typical MIR experiment  Of interest is what it is in our trained ML models of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 2 causing their response  to be inconsistent with random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Knowing how Gaussian models used  in LDA and QDA are built – mean and covariance parameters are estimated  from training data – an internally valid conclusion is that these models work  well in BALLROOM because likelihood distributions estimated from the train-  ing data also fit the testing data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Similarly, knowing how nearest neighbour  classifiers work, an internally valid conclusion is that in BALLROOM the neigh-  bourhoods of the labeled features in the training data include to a large extent  testing data features with the same labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Another internally valid conclusion  is that the high performances of these ML models in BALLROOM are caused by  the features together with the expressivity of the models capturing information  related to the labels in BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Our ML models have learned something  about BALLROOM, which causes their performance to be significantly better  than random selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  With reference to the aims of MIR research, we want to conclude something  more specific, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', our ML models have learned to recognize the rhythms in  BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This is certainly one explanation consistent with the performance  of our systems, but is it the only one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The internal validity of this conclusion  relies on a key assumption: inferring the correct labels of BALLROOM can only  be the result of learning to discriminate between and identify the rhythms in  BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In other words, we must assume that there is no other way to  accurately infer labels in BALLROOM than by perceiving rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  It is not difficult to imagine other ways to infer the labels in BALLROOM  by considering the multitude of variables present in the music recordings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' One  can hear instrumentation unique to each class, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', many recordings labeled  “Tango” feature strings, piano and accordion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Many recordings labeled “Waltz”  and “Viennese Waltz” feature strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Many recordings labeled “Quickstep”  and “Jive” feature brass, piano, vocals, and drums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' So perceiving instrumenta-  tion is one way to infer labels in BALLROOM with more success than random  selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We can reject such an explanation knowing that the time-domain  nature of the features we are using are not sensitive to timbre, and so cannot  clearly express the sound characteristic of an instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Then, are there time-  domain variables other than rhythm that are closely associated with the labels  in BALLROOM?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  One possible variable is tempo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' If tempo is correlated with rhythm in BALL-  ROOM then tempo estimation is another way a ML model can reproduce the  ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' From listening to BALLROOM recordings labeled “Waltz”,  “Quickstep” or “Cha cha cha”, one can hear they feature different rhythms and  different tempi, but discriminating between them can be done on the basis of  tempo alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Tempo and rhythm are related musical characteristics, but they  are not one and the same thing [Sethares, 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Let us perform an experiment to test the sensitivity of our trained ML models  10  Figure 1: Accuracies of the ML models in Table 1 with test recordings under-  going pitch-preserving time dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Horizontal dashed line in grey is the mean  accuracy of the best random system plus twice its standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  to tempo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We perform an intervention where we alter all test recordings by  some amount of pitch-preserving time dilation, and then measure the responses  of the models to these new “treatments”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Dilating a recording by an amount  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='1 increases its duration by 10% – or equivalently, makes the music in the  recording have a tempo that is 10% slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Dilating a recording by an amount  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='9 decreases its duration by 10% – or equivalently, makes the music in the  recording have a tempo that is 10% faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Figure 1 shows how the accuracy  of all ML models from Table 1 covaries with the amount of dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Our  intervention has clearly revealed the extent to which the ML models rely on the  tempi in the test data – which is no surprise given the background information  of BALLROOM in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The validity of the conclusion that the responses of our ML models are  caused by their recognition of the rhythms in BALLROOM relies on too strong  of an assumption about BALLROOM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', that rhythm recognition is the only  way to infer labels in BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The experimental design does not account  for the structure present in the dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' we do not control for other ways of  inferring the labels of BALLROOM, which are guaranteed to exist by its very  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' From Table 1 and our experimental design, we thus cannot be any  more specific in our causal inference than this: the responses of our ML models  are caused by their having learned something about BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' At best, they  are relying on at least tempo and rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This then calls into question how  comparing predictions with ground truth in BALLROOM relates to the ability  we might actually want to measure, that is the recognition of rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This is  where the notion of construct validity becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  6  Construct Validity  Construct validity is “the validity of inferences about the higher order constructs  that represent sampling particulars” [Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This involves the re-  11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='8  LDA  QDA  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='6  INN  Accuracy  3NN  5NN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='4  7NN  NN6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='15  DilationAmountlationship between what is meant to be inferred by the experimentalist from  an experiment and what is actually measured, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', the operationalisation of the  experimentalist’s intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For instance, directly measuring the blood pressure  of a person involves an invasive procedure inserting a measuring device in their  veins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Blood pressure can be measured less invasively but indirectly by exter-  nally applying known pressure to a vein and listening for when blood flow ceases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Knowledge about the incompressibility of liquids in closed systems makes the  measurement of pressure in the balloon a relevant measure of blood pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 73) includes a table of fourteen different threats to  construct validity, but several of these are irrelevant to computer-based exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The main threat is a questionable relationship between what is being  measured and what is intended to be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Selecting a measure by conve-  nience but not relevance, sampling from convenient populations, and a lack of  definition of what is intended to be measured, are threats to construct validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Construct validity involves more than just how something is measured;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' it also  involves what is measured and in what settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='1  MIR concerns about construct validity  How is classification accuracy, or any figure of merit, in a labeled music dataset  related to X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Two examples in MIR are the use of “genre” classification accu-  racy as an indirect measure of music similarity [Pohle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2008], or IR user  satisfaction (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', [Schedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2013] for a discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The relationship  between these is very tenuous, especially so considering that accuracy itself is  an unreliable measure of whether or not a system has learned anything relevant  to music [Sturm, 2013, 2014b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A key reference in this respect is that of Pfungst  [1911] describing a series of experiments in trying to reliably measure the arith-  metic acumen of a horse that was only able to tap out answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Counting the  number of correct answers tapped out by the horse, no matter how many ques-  tions are asked, is irrelevant without considering how each question is posed (the  setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The key to Pfungst discovering the cause of the horse’s apparent arith-  metic acumen involved changing the setting: the questions remained the same,  and accuracy of correct response was measured, but how the questions were  posed was changed in order to control for different factors of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The same is true for MIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  What is the “use case” of the system to be tested?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' To counter threats to  construct validity the MIR experimentalist must operationalise as much as pos-  sible the use case of the system to be built and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' An attempt to do so for  music description is in Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2014], which emphasises the need to define  success criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The experimentalist must determine how their method of mea-  surement relates to the success criteria, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', relating classification accuracy in  BALLROOM to the satisfaction of a specific user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  How can we test the construct validity of a conclusion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' One possibility is to  assess the outcomes of different experiments which are supposed to measure the  same higher order constructs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' An example in MIR is to study correlations of  different genre classifiers when given identical inputs [Liem and Mostert, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  12  Low correlations between classifiers point to problems of construct validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A  related topic is that of adversarial examples, which casts doubt on the con-  clusion that the high accuracy of an MIR system in some dataset reflects its  “perception” of the music in the waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Adversarial examples have first  been described in image analysis [Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2014], where imperceptible  perturbations of input data significantly degraded classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  For music genre classification systems, irrelevant audio filtering transforma-  tions of music signals are used in Sturm [2014b] to both deflate and inflate  classification accuracy to be no better than chance level or perfect 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The  irrelevance is ascertained with listening tests and real human subjects, with the  transformation being audible but not changing the clear impression of a certain  musical genre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The same problematic behavior has been documented concerning  music emotion classification [Sturm, 2014b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For deep music embedding a test  based on imperceptible audio transformations has been proposed [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=',  2019], essentially verifying distance consistency both in the input audio space  and corresponding latent deep space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Following these so-called untargeted attacks which try to change a predic-  tion to an arbitrary target, targeted attacks aiming at changing predictions to  specific classes have been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A targeted attack on genre recognition has  been reported [Kereliuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2015], where magnitude spectral frames com-  puted from audio are treated as images and attacked using approaches from  image object recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For music instrument classification a targeted attack  allowing to add perturbations directly to audio waveforms instead of spectro-  grams has also been presented [Prinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Quite similar to the results  by [Kereliuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2015], the attacks were able to reduce the accuracy close  to a random baseline and produce misclassifications to any desired instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Signal perturbations were almost imperceptible apart from some high-frequency  deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The authors also artificially boosted playcounts via an attack on a  real-world music recommender, thereby demonstrating that such attacks can be  a security issue in MIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Follow-up work presented lines of defence against such  malicious attacks [Hoedt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='2  Construct validity in the typical MIR experiment  When it comes to our typical MIR experiment in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  2, we are interested  in making construct inferences around the latent ability of rhythm recognition  we are supposedly measuring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  For instance, one construct inference is that  our features measure relevant aspects of rhythm in recorded music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In some  sense, by their definition from basic signal processing components, our features  come from temporal aspects that are certainly relevant to rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Our features  are also reliant on acoustic information, and in particular there being high-  contrast differences in onsets captured by spectral flux – hence limiting their  relationship to rhythms played by particular kinds of instruments with sharp  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' However, we have seen above that the features are also indicative of  tempo, which is not rhythm [Sethares, 2007], and that tempo is another path an  ML algorithm can use to get to the rhythm label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Hence we are left to question  13  the relationship of our features to the concept we are trying to operationalise,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The construction of the BALLROOM dataset, intended to reflect different  ballroom dance rhythms, is closely related to the validity of construct inferences  derived from its use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  How do the “Waltz” excerpts exemplify the “Waltz”  rhythm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Is there one “Waltz” rhythm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In BALLROOM, there are actually  two different labels for waltz: “Waltz” and “Viennese Waltz.” The distinction  between them is based in part on tempo, according to the World Sport Dance  Federation, a Viennese waltz is to be performed at a tempo between 174-180  BPM Federation [2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Having a system label any partition of the BALLROOM dataset provides no  reliable measure of a system’s ability to recognise rhythm without changing the  setting to control for other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' It is not as simple as choosing a different  feature, measure, cross-validation method, or using a particular statistical test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  One must change the experiment itself such that rhythm recognition is what  is actually being measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This means that BALLROOM can still be useful  to measuring the rhythm recognition of a ML system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Indeed, in the previ-  ous section we used it to disprove the causal claim that the good performance  of the ML systems of Table 1 is caused by their ability to recognize rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Might performance in BALLROOM also be an indication of performance in  other datasets focused on rhythm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This is where the notion of external validity  becomes relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  7  External Validity  External validity is “the validity of inferences about the extent to which a causal  relationship holds over variations in experimental units, settings, treatment  variables and measurement variables” [Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' More generally,  external validity is the truth of a generalised causal inference drawn from an ex-  periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' An example is inferring that medicine found to lower blood pressure  in patients living in Germany will also lower blood pressure in people living in  Mexico – a conclusion that can lack validity due to differences in diet, living  and working conditions, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Another example is that increasing the dose  of the medicine will cause blood pressure to lower further in the studied popu-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' If a causal inference we draw from an experiment lacks internal validity,  then generalising that inference to include variations not tested will not have  external validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 87) includes a table of five different  threats to external validity, which are in addition to the threats to internal va-  lidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The main threat is that variation of the components of the experiment  might destroy the causal inference that holds in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For instance, a  medication may work for the type of illness tested, but that type of illness may  not be generalisable to other closely related illnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  14  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='1  MIR concerns about external validity  Does my model generalize to out-of-sample data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The standard approach in  evaluating MIR classification systems is to use separate train and test sets in  cross-validation experiments to obtain seemingly unbiased estimates of perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' However, if such MIR systems are exposed to independent out-of-sample  data often severe loss of performance is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' One example are experiments  on genre recognition where accuracy results do not hold when evaluated across  different collections that are not part of the training sets [Bogdanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2016,  2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The results do not generalize to supposedly identical genre labels in dif-  ferent collections, which reflects a lack of external validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Genre labels like  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' ’Rock’ will be used differently by different annotators working on these  collections – which is a threat to construct validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Another example are how  different audio encodings affect subsequent computation of descriptors and clas-  sification results [Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2014], or how in general differences in software  implementations diminish replicability [McFee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Do different raters agree on a ground truth?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Human perception of music  is highly subjective resulting in possible low inter-rater agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Therefore  only a certain amount of agreement can be expected if several human subjects  are asked to rate the same song pairs according to their perceived similarity,  depending on a number of subjective factors [Schedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2013, Flexer and Grill,  2016] like personal taste, listening history, familiarity with the music, current  mood, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Concerning annotation of music, Seyerlehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2010] shows that  the performance of humans classifying songs into 19 genres ranges from modest  26% to 71%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Audio-based grounding of everyday musical terms shows the same  problematic results [Aucouturier, 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' It has even been argued [Wiggins, 2009]  that no such thing as an immovable ‘ground’ exists in the context of music,  because music itself is subjective, highly context-dependent and dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The lack of inter-rater agreement presents a problem of external validity be-  cause inferences from the experiment do not generalize from users or annotators  in the experiment to the intended target population of arbitrary users/annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  It is also a problem of reliability, since different groups of users or annotators  with their differing subjective opinions will impede repeatability of experimen-  tal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This lack of inter-rater agreement presents an upper bound for MIR  approaches, since it is not meaningful to have computational models going be-  yond the level of human agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Such upper bounds have been reported  [Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2007, Schedl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2013, Flexer and Grill, 2016] for the MIREX  tasks of ‘Audio Music Similarity and Retrieval’ (AMS) and ‘Music Structural  Segmentation’ (MSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For AMS the upper bound has already been reached in  2009, while for MSS the upper bound is within reach for at least some genres  of music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Comparable results exist concerning music structure analysis [Nieto  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2014] and chord estimation [Ni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2013, Koops et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Do raters agree with themselves at different points in time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Going beyond  the question of whether different annotators agree on a ground truth one can also  access what the level of agreement within one person is when faced with identical  annotation tasks at different points in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A high intra-rater agreement would  15  help to overcome the problem of upper bounds in MIR systems since it would  make personalization of models meaningful, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' to have separate models for  individual persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' However, at least for the task of general music similarity  it has been shown that intra-rater agreement is only slightly higher than inter-  rater agreement [Flexer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2021], with the absolute level also depending on  music material and mood of raters at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' An approach to personalize  chord labels for individual annotators via deep learning was more successful  [Koops et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Returning to the impact of irrelevant transformations [Sturm, 2014b, Kere-  liuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2015, Rodr´ıguez-Algarra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2016] and the existence of adversarial  examples [Prinz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2021], one can ask, does my model generalize to these  kinds of attacks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' More constructively, one can seek ways to lessen the impact of  these attacks, thus possibly increasing the generalization of the models [Hoedt  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='2  External validity in the typical MIR experiment  Considering the typical MIR experiment in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 2, we cannot validly conclude  that any of our models is recognizing rhythm in general because we do not know  if they are recognizing rhythm in BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Our dilation intervention ex-  periment in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='2 reveals that all of the models lose their supposed ability to  recognize rhythm in BALLROOM, so there is no reason to infer they will rec-  ognize rhythm elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' One causal conclusion we might generalise is that all  our models perform well in BALLROOM because they have learned something  about BALLROOM — a curated set of recordings downloaded from a specific  website in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Might our models have also learned something about other  recordings from that same website but collected many years later?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The extended BALLROOM dataset (X-BALLROOM) [Marchand and Peeters,  2016] consists of 3,484 audio recordings in the same eight dance styles or mu-  sic rhythms as BALLROOM, but downloaded from the same website over a  decade later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  This gives us a chance to test our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The figures of  merit measured from our models trained in BALLROOM but applied to all  of X-BALLROOM are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We still see significant covariation  between response and the use of ML with our features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' By and large, what-  ever concepts our ML models have learned about BALLROOM carry over to  X-BALLROOM – but we still do not know whether or not those concepts have  to do with rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  We can take this opportunity to test the generalizability of a conclusion  about Gaussian modelling outperforming nearest neighbor modelling from Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Training and testing the same ML models with a 70/30 random partition of  X-BALLROOM produces the results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' We still see QDA performing  the best, but the nearest neighbor models show large performance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  The cause of this change in performance should be investigated, and related  to the confounding known to exist in BALLROOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Does the confounding also  exist in X-BALLROOM, suggested by the results in Table 2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content='01  maj  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content='03  Table 3: As in Table 1, but models trained and tested in X-BALLROOM [Marc-  hand and Peeters, 2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  of avenues for exploration, sources of hypotheses, and not the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  8  Conclusion  This article provides a review of the notion of validity based on the typology  given in Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' It brings together the few sources in MIR that  mention anything to do with validity, and several sources that do not but are  related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  This article does not aim to prescribe how to design and perform  experiments such that valid conclusions can be drawn from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Instead it  aims to bring within the realm of MIR what validity means, why it is important,  and how it can be threatened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  In MIR the predominant experimental methodology is given by the Cran-  field paradigm: train a model on a partition of a dataset and count the number  of correct answers on another partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This kind of experiment is inexpen-  sive to do with the data conveniently at hand, and provides numbers that can  be compared in ways that convince peer reviewers that progress has been ac-  complished [Hand, 2006].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Despite various appeals [Peeters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2012, Schedl  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2013] and beseechings [Urbano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2012, 2013, Sturm, 2013, 2014b,  Flexer and Grill, 2016, Flexer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=', 2021], such an experimental approach is  17  still standard in the field and its serious flaws are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Any inference from  this experiment that is more general than “the system has learned something  about this dataset” lacks internal, construct and external validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' This does  not mean that all such inferences are false, just that they cannot follow from  the experiment as designed and implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Reproducing the ground truth  of a dataset represents a beginning and must be followed by a search for the  causes of the observed behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' One must resist the urge to conclude that a  system must be doing whatever is hoped for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] provides an established starting point for MIR, but there  exist other types of validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For instance, Lund [2021] revises the typology of  Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] to address ambiguities between causes and treatments, to  better define aspects of settings, and to establish a hierarchical ordering of five  types of validity: statistical conclusion, causal, construct, generalization and  theoretical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  An important distinction in this typology from that of Shadish  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] is its emphasis on a major aim of basic research: to contribute  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Other kinds of validity include ecological, convergent, and criterion  [Urbano, 2011];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' but these still deal with the kind of conclusion one is drawing  from evidence collected in some way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  As a final note, a frustration when encountering Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] as an  engineer is that of its 623 pages there are only five pages with at least one equa-  tion on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Instead, Shadish et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' [2002] describe experiments and how each  type of validity manifests in the conclusions drawn, with specific threats to the  reasoning of those conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Experiments, not to mention experimentalists,  are such complex assemblages that expressing them in formal ways that appear  to permit the computation of numbers that relate to each type of validity would  probably have very limited applicability, and then only be understood by a lim-  ited audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The language of validity is reason, and we hope this manuscript  will inspire MIR researchers to think creatively about the phenomena they ob-  serve to discover their causes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  Acknowledgments  We thank Juli´an Urbano and Hugo Maruri-Aguilar for helpful discussions during  the drafting of this manuscript, as well as the constructive criticisms of reviewers  from the Transactions of the Society for Music Information Retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The con-  tribution of Sturm is supported by a project that has received funding from the  European Research Council (ERC) under the European Union’s Horizon 2020  research and innovation programme (Grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 864189 MUSAiC:  Music at the Frontiers of Artificial Creativity and Criticism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The contribu-  tion of Flexer is supported by funding from the Austrian Science Fund (FWF,  project number P 31988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' For the purpose of open access, the authors have  applied a CC BY public copyright license to any author accepted manuscript  version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content=' G´omez Guti´errez, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content=' What  is the effect of audio quality on the robustness of mfccs and chroma features?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  In Proceedings of the 15th Conference of the International Society for Music  Information Retrieval, pages 573–578, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Voorhees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' The philosophy of information retrieval evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In Proceed-  ings of the Cross-Language Evaluation Forum, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' An industrial strength audio search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In Proceedings of the  4th Conference of the International Society for Music Information Retrieval,  pages 7–13, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  23  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Wiggins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' Semantic gap?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' schemantic schmap!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' methodological consid-  erations in the scientific study of music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' In Proceedings of the 11th IEEE  International Symposium on Multimedia, pages 477–482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content=' IEEE, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
+page_content='  24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AzT4oBgHgl3EQfnP2n/content/2301.01578v1.pdf'}
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+arXiv:2301.05181v1  [math-ph]  12 Jan 2023
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER
+MATRICES
+GIORGIO CIPOLLONI∗
+LÁSZLÓ ERD ˝OS†,#
+JOSCHA HENHEIK†,#
+OLEKSII KOLUPAIEV†
+ABSTRACT. The total energy of an eigenstate in a composite quantum system tends to be distributed equally
+among its constituents. We identify the quantum fluctuation around this equipartition principle in the simplest
+disordered quantum system consisting of linear combinations of Wigner matrices. As our main ingredient, we
+prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the
+bulk eigenvectors of Wigner matrices with an arbitrary deformation.
+1. INTRODUCTION
+The general principle of the equipartition of energy for a classical ergodic system asserts that in equilib-
+rium the total energy is equally distributed among all elementary degrees of freedom. A similar principle
+could be verified in some special quantum cases, see [6] and extensive literature therein. Motivated by
+E. Wigner’s original vision to model any sufficiently complex quantum system by random matrices, a
+particularly strong microcanonical version of the equipartition principle for Wigner matrices was first for-
+mulated and proven in [5]. In its simplest form, consider a fully mean field random Hamilton operator H in
+the high dimensional quantum state space CN that consists of the sum of two independent N × N Wigner
+matrices,
+H = W1 + W2 ,
+as two constituents of the system. Recall that Wigner matrices W = (wij) are real or complex Hermitian
+random matrices with independent (up to the symmetry constraint wij = ¯wji), identically distributed en-
+tries. Let u be a normalised eigenvector of H with eigenvalue (energy) λ = ⟨u, Hu⟩, then equipartition
+asserts that ⟨u, Wlu⟩ ≈ 1
+2λ for l = 1, 2. In [5, Theorem 3.4] even a precise error bound was proven, i.e.
+that
+���⟨u, Wlu⟩ − 1
+2λ
+��� ≤ N ǫ
+√
+N
+,
+l = 1, 2 ,
+(1.1)
+holds with very high probability for any fixed ǫ > 0; this estimate is optimal up the N ǫ factor. The main
+result of the current paper is to identify the fluctuation in (1.1), more precisely we will show that
+√
+N
+�
+⟨u, Wlu⟩ − 1
+2λ
+�
+converges to a centred normal distribution as N → ∞. We also compute its variance that turns out to be
+independent of the energy λ but depends on the symmetry class (real or complex). The result can easily be
+extended to the case when H is a more general linear combination of several independent Wigner matrices.
+The estimate (1.1) is reminiscent to the recently proven Eigenstate Thermalisation Hypothesis (ETH),
+also known as the Quantum Unique Ergodicity (QUE),1 for Wigner matrices in [15, Theorem 2.2] (see
+∗PRINCETON CENTER FOR THEORETICAL SCIENCE PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA
+†IST AUSTRIA, AM CAMPUS 1, 3400 KLOSTERNEUBURG, AUSTRIA
+E-mail address: gc4233@princeton.edu, lerdos@ist.ac.at, joscha.henheik@ist.ac.at,
+oleksii.kolupaiev@ist.ac.at.
+Date: January 13, 2023.
+2020 Mathematics Subject Classification. 60B20, 82B10, 58J51.
+Key words and phrases. Quantum unique ergodicity, Matrix Dyson equation, Local law.
+#Supported by ERC Advanced Grant “RMTBeyond” No. 101020331.
+1ETH for Wigner matrices was first conjectured by Deutsch [22]. Quantum ergodicity has a long history in the context of the
+quantisations of chaotic classical dynamical systems starting from the fundamental theorem by ˘Snirel’man [34]. For more background
+and related literature, see the Introduction of [15].
+1
+
+2
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+also [19, Theorem 2.6] for an improvement) which asserts that
+���⟨u, Au⟩ − ⟨A⟩
+��� ≤ N ǫ
+√
+N
+,
+⟨A⟩ := 1
+N TrA ,
+(1.2)
+holds for any bounded deterministic matrix A. In fact, even the Gaussian fluctuation of
+√
+N
+�
+⟨u, Au⟩ − ⟨A⟩
+�
+(1.3)
+was proven in [17, Theorem 2.2] and [19, Theorem 2.8], see also [9] and the recent generalisation [8] to
+off-diagonal elements as well as joint Gaussianity of several quadratic forms. Earlier results on ETH [23,
+29, 12, 10] and its fluctuations [13, 28, 35, 33] for Wigner matrices typically concerned rank one or finite
+rank observables A.
+Despite their apparent similarities, the quadratic form in (1.1) is essentially different from that in (1.2)
+since Wl is strongly correlated with u while A in (1.2) is deterministic. This explains the difference in the
+two leading terms; note that 1
+2λ in (1.1) is energy dependent and it is far from the value ⟨Wl⟩ ≈ 0, which
+might be erroneously guessed from (1.2). Still, the basic approach leading to ETH (1.2) is very useful to
+study ⟨u, Wlu⟩ as well. The basic idea is to condition on one of the Wigner matrices, say W2, and consider
+H = W1+W2 in the probability space of W1 as a Wigner matrix with an additive deterministic deformation
+W2. Assume that we can prove the generalisation of ETH (1.2) for such deformed Wigner matrices. In the
+space of W1 this would result in a concentration of ⟨u, W2u⟩ around some quantity f(W2) depending only
+on W2; however, the answer will nontrivially depend on the deformation, i.e. it will not be simply ⟨W2⟩.
+Once the correct form of f(W2) is established, we can find its concentration in the probability space of W2,
+yielding the final answer.
+To achieve these results we prove more general ETH and fluctuation results for eigenvector overlaps of
+deformed Wigner matrices of the general form H = W + D, where W is an N × N Wigner matrix and
+D is arbitrary, bounded deterministic matrix. The goal is to establish the concentration and the fluctuation
+of the quadratic form ⟨u, Au⟩ for a normalised eigenvector u of H with a bounded deterministic matrix A.
+We remark that for the special case of a rank one matrix A = |q⟩⟨q|, ETH is equivalent to the complete
+isotropic delocalisation of the eigenvector u, i.e. that |⟨q, u⟩| ≤ N ǫ/
+√
+N for any deterministic vector
+q with ∥q∥ = 1. For a diagonal deformation D this has been achieved in [31, 30] and for the general
+deformation D in [26]. The normal fluctuation of ⟨u, Au⟩ for a finite rank A and diagonal D was obtained
+in [7].
+It is well known that for very general mean-field type random matrices H their resolvent G(z) = (H −
+z)−1 concentrates around a deterministic matrix M = M(z); such results are called local laws, and we will
+recall them precisely in (4.1). Here M is a solution of the matrix Dyson equation (MDE), which, in case of
+H = W + D, reads as
+−
+1
+M(z) = z − D + ⟨M(z)⟩ .
+(1.4)
+Given M, it turns out that
+⟨ui, Auj⟩ ≈ δi,j
+⟨ℑM(λi)A⟩
+⟨ℑM(λi)⟩ ,
+(1.5)
+where λi is the eigenvalue corresponding to the normalised eigenvector ui, i.e. Hui = λiui. Since the
+eigenvalues are rigid, i.e. they fluctuate only very little, the right hand side of (1.5) is essentially determinis-
+tic and in general it depends on the energy. Similarly to (1.3), we will also establish the Gaussian fluctuation
+around the approximation (1.5). For zero deformation, D = 0, the matrix M is constant and (1.5) recov-
+ers (1.2)–(1.3) as a special case. For simplicity, in the current paper we establish all these results only in the
+bulk of the spectrum, but similar results may be obtained at the edge and at the possible cusp regime of the
+spectrum as well; the details are deferred to later works.
+We now comment on the new aspects of our methods. The proof of (1.5) relies on a basic observation
+about the local law for H = W + D. Its average form asserts that
+���
+�
+(G(z) − M(z))A
+���� ≤ N ǫ
+Nη ,
+η := |ℑz| ≫ 1
+N ,
+(1.6)
+holds with very high probability and the error is essentially optimal for any bounded deterministic matrix
+A. However, there is a one dimensional subspace of the matrices A for which the estimate improves to
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+3
+N ǫ/(N√η), gaining a √η factor in the relevant small η ≪ 1 regime. For Wigner matrices H = W
+without deformation, the traceless matrices A played this special role. The key idea behind the proof of
+ETH for Wigner matrices in [15] was to decompose any deterministic matrix as A =: ⟨A⟩ + ˚
+A into its
+tracial and traceless parts and prove multi-resolvent generalisations of the local law (1.6) with an error term
+distinguishing whether the deterministic matrices are traceless or not. For example, for a typical A we have
+⟨G(z)AG(z)∗A⟩ ∼ 1
+η
+but for the traceless part of A we have
+⟨G(z)˚
+AG(z)∗ ˚
+A⟩ ∼ 1 ,
+η ≫ 1
+N ,
+with appropriately matching error terms. In general, each traceless A improves the typical estimate by a
+factor √η. ETH then follows from the spectral theorem,
+1
+N
+N
+�
+i,j=1
+η
+(λi − e)2 + η2
+η
+(λj − e)2 + η2 |⟨ui, ˚
+Auj⟩|2 = ⟨ℑG(z)˚
+AℑG(z)˚
+A⟩ ≤ N ǫ ;
+(1.7)
+choosing z = e + iη appropriately with η ∼ N −1+ǫ we obtain that |⟨ui, ˚
+Auj⟩|2 ≤ N −1+3ǫ which even
+includes an off-diagonal version of (1.2).
+To extend this argument for deformed Wigner matrices requires to identify the appropriate singular
+(“tracial”) and regular (“traceless”) parts of an arbitrary matrix. It turns out that the improved local laws
+around an energy e = ℜz hold if A is orthogonal2 to ℑM(e), see (2.10) for the new definition of ˚
+A, which
+denotes the regular part of A. In this theory the matrix ℑM emerges as the critical eigenvector of a linear
+stability operator B = I − M⟨·⟩M ∗ related to the MDE (1.4). The major complication compared with the
+pure Wigner case in [15] is that now the regular part of a matrix becomes energy dependent. In particular,
+in a multi-resolvent chain ⟨G(z1)A1G(z2)A2 . . .⟩ it is a priori unclear at which spectral parameters the
+matrices Ai should be regularised; it turns out that the correct regularisation depends on both zi and zi+1,
+see (4.10) later. A similar procedure was performed for the Hermitisation of non-Hermitian i.i.d. matrices
+with a general deformation in [20], see [20, Appendix D] for a more conceptual presentation. Having
+identified the correct regularisation, we derive a system of master inequalities for the error terms in multi-
+resolvent local laws for regular observables; a similar strategy (with minor modifications) have been used
+in [18, 19] for Wigner matrices and in [20] for i.i.d. matrices. To keep the presentation short, here we will
+not aim at the most general local laws with optimal errors (unlike in [18, 19]). Although these would be
+achievable with our methods, here we prove only what is needed for our main results on the equipartition.
+The proof of the fluctuation around the ETH uses Dyson Brownian motion (DBM) techniques, namely
+the Stochastic Eigenstate Equation for quadratic forms of eigenvectors. This theory has been gradually
+developed for Wigner matrices in [13, 14, 33], we closely follow the presentation in [17, 19]. The extension
+of this technique to deformed Wigner matrices is fairly straightforward, so our presentation will be brief.
+The necessary inputs for this DBM analysis follow from the multi-resolvent local laws that we prove for
+deformed Wigner matrices.
+In a closing remark we mention that the original proof of (1.1) in [5] was considerably simpler than that
+of (1.2). This may appear quite surprising due to the complicated correlation between u and W, but a special
+algebraic cancellation helped in [5]. Namely, with the notation W := W1 − W2 and G(z) := (H − z)−1,
+z = e + iη ∈ C+, a relatively straightforward cumulant expansion showed that ⟨ℑG(z)WℑG(z)W⟩ is es-
+sentially bounded3 even for spectral parameters z very close to the real axis, η ≥ N −1+ǫ/2. Within this cu-
+mulant expansion an algebraic cancellation emerged due to the special form of W. Then, exactly as in (1.7)
+we obtain |⟨u, Wu⟩|2 ≲ N 1+ǫη2 = N −1+2ǫ. In particular, it shows that ⟨u, Wu⟩ = ⟨u, W1u⟩−⟨u, W2u⟩
+is essentially of order N ǫ/
+√
+N for every eigenvector of H. Since ⟨u, W1u⟩ + ⟨u, W2u⟩ = ⟨u, Hu⟩ = λ,
+we immediately obtain the equipartition (1.1). Similar idea proved the more general case (2.4). Note, how-
+ever, that this trick does not help in establishing the fluctuations of ⟨u, Wiu⟩. In fact, the full ETH analysis
+2The space of matrices is equipped with the usual Hilbert-Schmidt scalar product.
+3This means up to an Nǫ factor with arbitrary small ǫ.
+
+4
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+for deformed Wigner matrices must be performed to establish the necessary a priori bounds for the Dyson
+Brownian motion arguments.
+Notations and conventions. For positive quantities f, g we write f ≲ g and f ∼ g if f ≤ Cg or cg ≤
+f ≤ Cg, respectively, for some constants c, C > 0 which depend only on the constants appearing in the
+moment condition, see (2.1) later. For any natural number n we set [n] := {1, 2, . . ., n}.
+We denote vectors by bold-faced lower case Roman letters x, y ∈ CN, for some N ∈ N. Vector
+and matrix norms, ∥x∥ and ∥A∥, indicate the usual Euclidean norm and the corresponding induced matrix
+norm. For any N × N matrix A we use the notation ⟨A⟩ := N −1TrA to denote the normalised trace of A.
+Moreover, for vectors x, y ∈ CN and matrices A ∈ CN×N we define the scalar product
+⟨x, y⟩ :=
+N
+�
+i=1
+xiyi .
+Finally, we will use the concept of “with very high probability” (w.v.h.p.) meaning that for any fixed
+D > 0 the probability of an N-dependent event is bigger than 1 − N −D for N ≥ N0(D). We introduce
+the notion of stochastic domination (see e.g. [24]): given two families of non-negative random variables
+X =
+�
+X(N)(u) : N ∈ N, u ∈ U (N)�
+and
+Y =
+�
+Y (N)(u) : N ∈ N, u ∈ U (N)�
+indexed by N (and possibly some parameter u in some parameter space U (N)), we say that X is stochasti-
+cally dominated by Y , if for all ξ, D > 0 we have
+sup
+u∈U(N) P
+�
+X(N)(u) > N ξY (N)(u)
+�
+≤ N −D
+(1.8)
+for large enough N ≥ N0(ξ, D). In this case we use the notation X ≺ Y or X = O≺(|Y |). We also use
+the convention that ξ > 0 denotes an arbitrary small exponent which is independent of N.
+Acknowledgement. G.C. and L.E. gratefully acknowledge many discussions with Dominik Schröder at the
+preliminary stage of this project, especially his essential contribution to identify the correct generalisation
+of traceless observables to the deformed Wigner ensembles.
+2. MAIN RESULTS
+We consider N×N real symmetric or complex Hermitian Wigner matrices W = W ∗ having single-entry
+distributions wab = N −1/2χod, for a > b, and waa = N −1/2χd, where χod and χod are two independent
+random variables satisfying the following assumptions:
+Assumption 2.1. We assume that χd is a real centred random variable, that χod is a real or complex ran-
+dom variable such that Eχod = 0 and E|χod|2 = 1; additionally in the complex case we also assume that
+Eχ2
+od = 0. Customarily, we use the parameter β = 1, 2 to indicate the real or complex case, respectively.
+Furthermore, we assume that all the moments of χod and χd exist, i.e. for any p ∈ N there exists a constant
+Cp > 0 such that
+E|χod|p + E|χd|p ≤ Cp.
+(2.1)
+The equipartition principle concerns linear combinations of Wigner matrices. Fix k ∈ N and consider
+H := p1W1 + · · · + pkWk,
+(2.2)
+for some fixed N-independent vector p = (p1, ..., pk) ∈ Rk of weights and for k independent N × N
+Wigner matrices Wl, belonging to the same symmetry class (i.e. the off-diagonal random variables χod
+are either real or complex for each of the Wl, l ∈ [k]). Then, denoting by {λi}i∈[N] the eigenvalues of
+H, arranged in increasing order, with associated normalised eigenvectors {ui}i∈[N], it is expected that
+the total energy ⟨ui, Hui⟩ = λi of the composite system (2.2) is proportionally distributed among the k
+constituents, i.e.
+⟨ui, plWl ui⟩ ≈
+p2
+l
+∥p∥2 λi
+(2.3)
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+5
+for every l ∈ [k], where ∥p∥ :=
+� �k
+l=1 |pl|2�1/2 denotes the usual ℓ2-norm. This phenomenon, known as
+equipartition, was first proven in [5, Theorem 3.4], with an optimal error estimate:
+����⟨ui, plWluj⟩ − δi,j
+p2
+l
+∥p∥2 λi
+���� ≺
+1
+√
+N
+.
+(2.4)
+Our main result is the corresponding Central Limit Theorem to (2.4) for i = j, i.e. the proof of Gaussian
+fluctuations in Equipartition for Wigner matrices – for energies in the bulk of the spectrum of H.
+Theorem 2.2 (Gaussian Fluctuations in Equipartition).
+Fix k ∈ N. Let W1, . . . , Wk be independent Wigner matrices satisfying Assumption 2.1, all of which
+being in the same real (β = 1) or complex (β = 2) symmetry class, and p = (p1, . . . , pk) ∈ Rk be N-
+independent. Define H as in (2.2) and denote by {λi}i∈[N] the eigenvalues of H, arranged in increasing
+order, with associated normalised eigenvectors {ui}i∈[N]. Then, for fixed κ > 0, every index i ∈ [κN, (1 −
+κ)N] and l ∈ [k] it holds that
+�
+βN
+2
+∥p∥2
+p2
+l
+�
+∥p∥2 − p2
+l
+�
+�
+⟨ui, plWlui⟩ −
+p2
+l
+∥p∥2 λi
+�
+=⇒ N(0, 1)
+(2.5)
+in the sense of moments,4 where N(0, 1) denotes a real standard Gaussian.
+By polarisation we will also obtain the following:
+Corollary 2.3. Under the assumptions from Theorem 2.2, the random vector X = (X1, ..., Xk) ∈ Rk with
+Xl :=
+�
+βN
+2
+�
+⟨ui, plWlui⟩ −
+p2
+l
+∥p∥2 λi
+�
+,
+l ∈ [k] ,
+(2.6)
+is approximately (in the sense of moments) jointly Gaussian with covariance structure
+Cov(Xl, Xm) = p2
+l
+�
+δl,m∥p∥2 − p2
+m
+�
+∥p∥2
+.
+Theorem 2.2 and Corollary 2.3 will follow as a corollary to the Eigenstate Thermalisation Hypothesis
+(ETH) and its Gaussian fluctuations for deformed Wigner matrices, which we present as Theorem 2.6 and
+Theorem 2.8 in the following subsection.
+Remark 2.4. By a quick inspection of our proof of Theorem 2.2, given in Section 3, it is possible to
+generalise the Equipartition principle (2.4) as well as its Gaussian fluctuations (2.5) to linear combinations
+of deformed Wigner matrices, i.e. each Wl in (2.2) being replaced by Wl + Dl, where Dl = D∗
+l is an
+essentially arbitrary bounded deterministic matrix (see Assumption 2.7 later). However, for brevity of the
+current paper, we refrain from doing so explicitly.
+2.1. ETH and its fluctuations for deformed Wigner matrices. In this section, we consider deformed
+Wigner matrices, H = W + D, with increasingly ordered eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λN and corre-
+sponding orthonormal eigenvectors u1, . . . , uN. Here, D = D∗ ∈ CN×N is a self-adjoint matrix with
+uniformly bounded norm, i.e. ∥D∥ ≤ CD for some N-independent constant CD > 0. While the Eigen-
+state Thermalisation Hypothesis (ETH) will be shown to hold for general deformations D, we shall require
+slightly stronger assumptions for proving the Gaussian fluctuations (see Assumption 2.7 below).
+In order to state our results on the ETH and its fluctuations (Theorems 2.6 and 2.8, respectively), we
+need to introduce the concept of regular observables, first in a simple form in Definition 2.10 (later along
+the proofs we will need a more general version in Definition 4.2). For this purpose we introduce M(z)
+being the unique solution of the Matrix Dyson Equation (MDE)5:
+−
+1
+M(z) = z − D + ⟨M(z)⟩,
+ℑM(z)ℑz > 0.
+(2.7)
+4Given an N–dependent random variable, we say that XN converges to X∞ in the sense of moments if for any k ∈ N it holds
+E|XN|k = E|X∞|k + O(N−c(k)), for some small possibly k–dependent constant c(k) > 0.
+5The MDE for very general mean field random matrices has been introduced in [2] and further analysed in [3]. The properties we
+use here have been summarised in [20, Appendix A].
+
+6
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+The self consistent density of states (scDos) is then defined as
+ρ(e) := 1
+π lim
+η↓0⟨ℑM(e + iη)⟩ .
+(2.8)
+We point out that not only ⟨ℑM(e + iη)⟩ has an extension to the real axis, but the whole matrix M(e) :=
+limη↓0 M(e+iη) is well defined (see [20, Lemma A.1 (b)]). The scDos ρ is a compactly supported Hölder-
+1/3 continuous function on R which is real-analytic on the set {ρ > 0}6. Moreover, for any small κ > 0
+we define the κ-bulk of the scDos as
+Bκ =
+�
+x ∈ R : ρ(x) ≥ κ1/3�
+,
+(2.9)
+which is a finite union of disjoint compact intervals [20, Lemma A.2]. For ℜz ∈ Bκ it holds that ∥M(z)∥ ≲
+1, as easily follows by taking the imaginary part of (2.7).
+Definition 2.5 (Regular observables – One-point regularisation). Fix κ > 0 and an energy e ∈ Bκ in the
+bulk. Given a matrix A ∈ CN×N, we define its one-point regularisation w.r.t. the energy e, denoted by ˚
+Ae,
+as
+˚
+A = ˚
+Ae := A − ⟨ℑM(e)A⟩
+⟨ℑM(e)⟩ .
+(2.10)
+Moreover, we call A regular w.r.t. the energy e, if and only if A = ˚
+Ae.
+Notice that in the analysis of Wigner matrices without deformation, D = 0, in [15, 16, 18, 19], M was
+a constant matrix and the regular observables were simply given by traceless matrices, i.e. ˚
+A = A − ⟨A⟩.
+For deformed Wigner matrices the concept of regular observables depends on the energy.
+Next, we define the quantiles γi of the density ρ implicitly by
+� γi
+−∞
+ρ(x) dx = i
+N ,
+i ∈ [N].
+(2.11)
+We can now formulate the ETH in the bulk for deformed Wigner matrices which generalises the same result
+for Wigner matrices, D = 0, from [15].
+Theorem 2.6 (Eigenstate Thermalisation Hypothesis). Fix a bounded deterministic D = D∗ ∈ CN×N
+and κ > 0. Let H = W + D be a deformed Wigner matrix, where W satisfies Assumption 2.1, and denote
+the orthonormal eigenvectors of H by {ui}i∈[N]. Then, for any deterministic A ∈ CN×N with ∥A∥ ≲ 1, it
+holds that
+max
+i,j
+���⟨ui, ˚
+Aγiuj⟩
+��� = max
+i,j
+����⟨ui, Auj⟩ − δij
+⟨AℑM(γi)⟩
+⟨ℑM(γi)⟩
+���� ≺
+1
+√
+N
+,
+(2.12)
+where the maximum is taken over all i, j ∈ [N] such that the quantiles γi, γj ∈ Bκ defined in (2.11) are in
+the κ-bulk of the scDos ρ.
+This "Law of Large Numbers"-type result (2.12) is complemented by the corresponding Central Limit
+Theorem (2.13), which requires slightly strengthened assumptions on the deformation D.
+Assumption 2.7. We assume that D ∈ CN×N is a bounded self-adjoint deterministic matrix such that
+(i) the unique solution M(z) to (2.7) is uniformly bounded in norm, i.e. supz∈C ∥M(z)∥ ≤ CM for
+some N-independent constant CM > 0;
+(ii) the scDos ρ is Hölder-1/2 regular, i.e. it does not have any cusps (see Footnote 6).
+The requirements on D in Assumption 2.7 are natural and they hold for typical applications, see Re-
+mark 2.10 later for more details. We can now formulate our result on the Gaussian fluctuations in the ETH
+which generalises the analogous result for Wigner matrices, D = 0 from [17].
+6The scDos has been thoroughly analysed in increasing generality in [1, 2, 3]. It is supported on finitely many finite intervals.
+Roughly speaking there are three regimes: the bulk, where ρ is well separated away from 0, the edge where ρ vanishes as a square
+root at the edges of each supporting interval that are well separated, and the cusp where two supporting intervals (almost) meet and
+ρ behaves (almost) as a cubic root. Correspondingly, ρ is locally real analytic, Hölder-1/2, or Hölder-1/3 continuous, respectively.
+Near the singularities, it has an approximately universal shape. No other singularity type can occur and for typical deformation D
+there is no cusp regime.
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+7
+Theorem 2.8 (Fluctuations in ETH). Fix κ, σ > 0 N-independent constants and let H = W + D be a
+deformed Wigner matrix, where W satisfies Assumption 2.1 and D satisfies Assumption 2.7. Denote the
+orthonormal eigenvectors of H by {ui}i∈[N] and fix an index i ∈ [N], such that the quantile γi ∈ Bκ
+defined in (2.11) is in the bulk. Then, for any deterministic Hermitian matrix A ∈ CN×N with ∥A∥ ≲ 1
+(which we assume to be real in the case of a real Wigner matrix) satisfying ⟨(A − ⟨A⟩)2⟩ ≥ σ, it holds that
+�
+βN
+2 Varγi(A)
+�
+⟨ui, Aui⟩ − ⟨AℑM(γi)⟩
+⟨ℑM(γi)⟩
+�
+=⇒ N(0, 1)
+(2.13)
+in the sense of moments (see Footnote 4), where
+Varγi(A) :=
+1
+⟨ℑM(γi)⟩2
+�
+��˚
+AγiℑM(γi)
+�2�
+− 1
+2ℜ
+���
+M(γi)
+�2 ˚
+Aγi�2
+1 −
+��
+M(γi)
+�2�
+��
+.
+(2.14)
+This variance is strictly positive with an effective lower bound
+Varγi(A) ≥ c ⟨(A − ⟨A⟩)2⟩
+(2.15)
+for some constant c = c(κ, ∥D∥) > 0.
+In the following two remarks we comment on the condition ⟨(A − ⟨A⟩)2⟩ ≥ σ and on Assumption 2.7.
+Remark 2.9. The restriction to matrices satisfying ⟨(A − ⟨A⟩)2⟩ ≥ σ, i.e. A − ⟨A⟩ being of high rank, is
+technical. It is due to the fact that our multi-resolvent local laws for resolvent chains ⟨G(z1)A1G(z2)A2 . . .⟩
+in Proposition 4.4 are non-optimal in terms of the norm for the matrices Ai; they involve the Euclidean norm
+∥Ai∥ and not the smaller Hilbert-Schmidt norm
+�
+⟨|Ai|2⟩ which would be optimal. For the Wigner ensem-
+ble, this subtlety is the main difference between the main result in [17] for high rank observable matrices A
+and its extension to any low rank A in [19]. Following the technique in [19] it would be possible to achieve
+the estimate with the optimal norm of A also for deformed Wigner matrices. However, we refrain from
+doing so, since in our main application, Theorem 2.2, A itself will be a Wigner matrix which has high rank.
+Remark 2.10. We have several comments on Assumption 2.7.
+(i) The boundedness of ∥M(z)∥ is automatically fulfilled in the bulk Bκ (see remark below (2.9)) or
+when ℜz away from the support of the scDos ρ (see [3, Proposition 3.5]) without any further condi-
+tion. However, the uniform (in z) estimate formulated in Assumption 2.7 does not hold for arbitrary
+D. A sufficient condition for the boundedness of ∥M∥ in terms of the spectrum of D is given in
+[3, Lemma 9.1 (i)]. This especially applies if the eigenvalues {di}i∈[N] of D (in increasing order)
+form a piecewise Hölder-1/2 regular sequence7, see [3, Lemma 9.3]. In particular, by eigenvalue
+rigidity [2, 26], it is easy to see that any “Wigner-like" matrix D has Hölder-1/2 regular sequence
+of eigenvalues with very high probability. This is important for the applicability of Theorem 2.8
+below in the proof of our main result, Theorem 2.2, given in Section 3.
+(ii) The assumption that ρ does not have any cusps is an atypical condition and of technical nature (see
+Lemma A.2). In case that the sequence of matrices D = DN has a limiting density of states with
+single interval support, then also ρ, the scDos of W + D, also has single interval support [11],
+in particular, it has no cusps [3]. Again, this is important for the applicability of Theorem 2.8 in
+the proof of our main result, in which case D is a Wigner matrix with a semicircle as the limiting
+density of states.
+In the following Section 3, we will prove our main result, Theorem 2.2, assuming Theorems 2.6 and 2.8
+on deformed Wigner matrices as inputs. These, in turn, will be proven in Sections 4 and 5, respectively.
+Both proofs crucially rely on an averaged local law for two resolvents and two regular observables, Propo-
+sition 4.4, which we prove in Section 6. Several additional technical and auxiliary results are deferred to
+the Appendix.
+7In this context, Hölder-1/2 regularity means that |di − dj| ≤ C0(|i − j|/N)1/2 for some universal constant C0 > 0.
+
+8
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+3. FLUCTUATIONS IN EQUIPARTITION: PROOF OF THEOREM 2.2
+Note that it is sufficient to prove Theorem 2.2 only for k = 2 with p1, p2 ̸= 0, since we can view the
+sum (2.2) as the sum of p1W1 and
+k
+�
+l=2
+plWl
+d=
+� k
+�
+l=2
+p2
+l
+�1/2
+�
+W ,
+where �
+W is a Wigner matrix independent of W1 and the equality is understood in distribution.
+As a main step, we shall prove the following lemma, where we condition on W2.
+Lemma 3.1. Under the assumptions of Theorem 2.2 with k = 2 it holds that
+EW1⟨ui, p2W2ui⟩ =
+p2
+2
+∥p∥2 γi + O≺
+�
+N −1/2−ǫ�
+,
+(3.1)
+βN
+2 VarW1[⟨ui, p2W2ui⟩] = p2
+1p2
+2
+∥p∥2 + O≺
+�
+N −ǫ�
+,
+(3.2)
+for some ǫ > 0, where γi is the ith quantile of the semicircular density with radius 2∥p∥, i.e.
+1
+2π∥p∥2
+� γi
+−∞
+�
+[4∥p∥2 − x]+dx = i
+N .
+Expectation and variance are taken in the probability space of W1, conditioned on W2, while the stochastic
+domination in the error term is understood in the probability space of W2.
+Proof of Theorem 2.2. First of all, we note that all requirements for applying Theorem 2.8 to H = p1W1 +
+D, with D = p2W2 for some fixed realisation of W2 in a very high probability event, are satisfied. This
+follows from Remark 2.10 and ⟨(W2−⟨W2⟩)2⟩ ≳ 1 with very high probability. Next, observe that replacing
+γi in (3.1) by the eigenvalue λi appearing in (2.5) is trivial by the usual eigenvalue rigidity |γi − λi| ≺ 1/N
+for Wigner matrices in the bulk [25]. Hence, Theorem 2.8 shows that, conditioned on a fixed realisation of
+W2,
+�
+βN
+2
+�
+⟨ui, p2W2ui⟩ −
+p2
+2
+∥p∥2 λi
+�
+(3.3)
+is approximately Gaussian with an approximately constant variance (independent of W2) given in (3.2).
+Since this holds with very high probability w.r.t. W2, this proves (2.5) for l = 2; the proof for l = 1 is the
+same.
+□
+Proof of Corollary 2.3. We formulated Theorem 2.8 as a CLT for overlaps ⟨ui, Aui⟩ for a single determin-
+istic matrix A, but by standard polarisation it also shows the joint approximate Gaussianity of any p-vector
+�
+⟨ui, A1ui⟩, ⟨ui, A2ui⟩, . . . , ⟨ui, Apui⟩
+�
+(3.4)
+for any fixed k and deterministic observables A1, A2, . . . Ap satisfying ⟨(Aj − ⟨Aj⟩)2⟩ ≥ c, j ∈ [p].
+Namely, using Theorem 2.8 to compute the moments of ⟨ui, A(t)ui⟩ for the linear combination A(t) =
+�
+j tjAj with any real vector t = (t1, t2, . . . , tp), we can identify any joint moments of the coordinates of
+the vector in (3.4) and we find that they satisfy the (approximate) Wick theorem.
+Now we can follow the above proof of Theorem 2.2, but without the simplification k = 2. Conditioning
+on W2, . . . , Wk and working in the probability space of W1, by the polarisation argument above we find that
+not only (3.3) each Xl from (2.6) is asymptotically Gaussian with a variance independent of W2, . . . , Wk,
+but they are jointly Gaussian for l = 2, 3, . . ., k. This is sufficient for the joint Gaussianity of the entire
+vector X since �
+l Xl = 0. This completes the proof of Corollary 2.3.
+□
+The proof of Lemma 3.1 is divided into the computation of the expectation (3.1) and the variance (3.2).
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+9
+3.1. Computation of the expectation (3.1). As in the proof of Theorem 2.2 above, we condition on W2
+and work in the probability space of W1 i.e. we consider p2W2 as a deterministic deformation of p1W1.
+This allows us to use Theorem 2.8 as8
+EW1⟨ui, p2W2ui⟩ = ⟨p2W2ℑM2(γi,2))⟩
+⟨ℑM2(γi,2)⟩
++ O≺
+�
+N −1/2−ǫ�
+(3.5)
+for some constant ǫ > 0. Here M2(z), depending on W2, is the unique solution of the MDE
+−
+1
+M2(z) = z − p2W2 + p2
+1⟨M2(z)⟩ ,
+(3.6)
+corresponding to the matrix p1W1 + p2W2, where p2W2 is considered a deformation, and γi,2 is the ith
+quantile of the scDos ρ2 corresponding to (3.6). The subscript ‘2’ for M2, ρ2 and γi,2 in (3.5) and (3.6)
+indicates that these objects are dependent on W2 and hence random.
+The Stieltjes transform m2(z) of ρ2 is given by the implicit equation
+m2(z) := ⟨M2(z)⟩ = 1
+p2
+·
+�
+1
+W2 − 1
+p2 (z + p2
+1m2(z))
+�
+with the usual side condition ℑz · ℑm2(z) > 0. Applying the standard local law for the resolvent of W2 on
+the right hand side shows that
+���m2(z) − 1
+p2
+msc(w2)
+��� ≺
+1
+N|ℑw2|,
+w2 := 1
+p2
+(z + p2
+1m2(z)).
+(3.7)
+where msc is the Stieltjes transform of the standard semicircle law, i.e. it satisfies the quadratic equation
+msc(w)2 + wmsc(w) + 1 = 0
+(3.8)
+with the side condition ℑw · ℑmsc(w) > 0. Note that in (3.7) w2 is random, it depends on W2, but the
+local law for ⟨(W2 − w)−1⟩ holds uniformly in the spectral parameter |ℑw| ≥ N −1, hence a standard
+grid argument and the Lipschitz continuity of the resolvents shows that it holds for any (random) w with
+|ℑw| ≥ N −1+ξ with any fixed ξ > 0.
+Applying (3.8) at w = w2 together with (3.7) implies that
+−
+1
+∥p∥ m2(z) =
+z
+∥p∥ + ∥p∥ m2(z) + O≺
+�
+1
+N|ℑw2|
+�
+.
+(3.9)
+We view this relation as a small additive perturbation of the exact equation
+−
+1
+msc
+�
+z
+∥p∥
+� =
+z
+∥p∥ + msc
+�
+z
+∥p∥
+�
+(3.10)
+to conclude
+���∥p∥ m2(z) − msc
+� z
+∥p∥
+���� ≺
+1
+N|ℑz| ,
+|ℑz| ≥ N −1+ξ ,
+(3.11)
+using that |ℑw2| ≳ |ℑz| from the definition of w2 in (3.7) and that ℑz ·ℑm2(z) > 0. The conclusion (3.11)
+requires a standard continuity argument, starting from a z with a large imaginary part and continuously
+reducing the imaginary part by keeping the real part fixed (the same argument is routinely used in the proof
+of the local law for Wigner matrices, see, e.g., [25]).
+The estimate (3.11) implies that the quantiles of ρ2 satisfy the usual rigidity estimate, i.e.
+|γ2,i − γi| ≺ 1
+N
+(3.12)
+for bulk indices i ∈ [κN, (1 − κ)N] with any N-independent κ > 0. Moreover, (3.11) also implies that
+for any z in the bulk of the semicircle, i.e. |ℑmsc(z)| ≥ c > 0 for some c > 0, independent of N, we
+have |ℑm2(z)| ≥ c/2 as long as |ℑz| ≥ N −1+ξ. Using the definition of w2 in (3.7) again, this shows
+|ℑw2| ∼ |ℑw| for the deterministic w :=
+1
+p2 (z + p2
+1msc(z)) for any z with |ℑz| ≥ N −1+ξ. Feeding this
+8Note that Theorem 2.6 alone would prove (3.5) only with ǫ = 0, but the convergence in the sense of moments from Theorem 2.8
+gains a factor N−ǫ with a positive ǫ.
+
+10
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+information into (3.9) and viewing it again as a perturbation of (3.10) but with the improved deterministic
+bound O≺(1/(N|ℑw|)), we obtain
+���∥p∥ m2(z) − msc
+�
+z
+∥p∥
+���� ≺
+1
+N|ℑw| ,
+with
+w = 1
+p2
+(z + p2
+1msc(z)) ,
+(3.13)
+uniformly in |ℑz| ≥ N −1+ξ. In particular, when z is in the bulk of the semicircle, then we have that
+���∥p∥ m2(z) − msc
+�
+z
+∥p∥
+���� ≺ 1
+N
+and this relation holds even down to the real axis by the Lipschitz continuity (in fact, real analyticity) of the
+Stieltjes transform m2(z) in the bulk.
+In the following, we will use the shorthand notation A ≈ B for two (families of) random variables A
+and B if and only if |A − B| ≺ N −1. Evaluating (3.6) at z = γi,2, we have
+M2(γi,2) = 1
+p2
+·
+1
+W2 − wi,2
+,
+wi,2 := 1
+p2
+(γi,2 + p2
+1 m2(γi,2)) ,
+(3.14)
+and note that wi,2 ≈ wi :=
+1
+p2 (γi + p2
+1 msc(γi)) by (3.13) and since γi,2 ≈ γi in the bulk by rigidity (3.12).
+Now we are ready to evaluate the rhs. of (3.5). By elementary manipulations using (3.14), we can now
+write the rhs. of (3.5) as
+⟨p2W2ℑM2(γi,2))⟩
+⟨ℑM2(γi,2)⟩
+= γi,2 + p2
+1
+p2
+ℑ
+�
+⟨(W2 − wi,2)−1⟩2�
+ℑ ⟨(W2 − wi,2)−1⟩
+.
+(3.15)
+Using (3.13), we obtain
+⟨(W2 − wi,2)−1⟩ ≈ ⟨(W2 − wi)−1⟩ ≈ msc(wi)
+(3.16)
+with very high W2-probability. Continuing with (3.15) and using γi,2 ≈ γi, we thus find
+⟨p2W2ℑM2(γi,2))⟩
+⟨ℑM2(γi,2)⟩
+≈ γi + p2
+1
+p2
+ℑ
+�
+msc(wi)2�
+ℑmsc(wi)
+.
+(3.17)
+Next, we combine (3.8) with p2m2(γi) ≈ msc(wi) from (3.7), (3.14) and (3.16) and find that
+msc(wi)2 ≈ −
+p2
+2
+p2
+1 + p2
+2
+�
+1 + 1
+p2
+γi msc(wi)
+�
+.
+(3.18)
+Hence, plugging (3.18) into (3.17) we deduce
+⟨p2W2ℑM2(γi,2))⟩
+⟨ℑM2(γi,2)⟩
+≈
+�
+1 −
+p2
+1
+p2
+1 + p2
+2
+�
+γi =
+p2
+2
+∥p∥2 γi .
+This completes the proof of (3.1).
+3.2. Computation of the variance (3.2). As in the calculation of the expectation in Section 3.1, we first
+condition on W2 and work in the probability space of W1. So, we apply Theorem 2.8 to the matrix p1W1 +
+p2W2, where the second term is considered a fixed deterministic deformation. Indeed, using the same
+notations as in Section 3.1, this gives that the lhs. of (3.2) equals
+p2
+2 Varγi,2 (W2) = p2
+2
+1
+⟨ℑM2(γi,2)⟩2
+�
+�� ˚
+W γi,2
+2
+ℑM2(γi,2)
+�2�
+− p2
+1
+2 ℜ
+���
+M2(γi,2)
+�2 ˚
+W γi,2
+2
+�2
+1 − p2
+1
+��
+M2(γi,2)
+�2�
+��
+(3.19)
+up to an additive error of order O≺
+�
+N −ǫ�
+, which will appear on the rhs. of (3.2). The factor p2
+1 in the
+second term of (3.19) is a natural rescaling caused by applying Theorem 2.8 to a deformation of p1W1
+instead of a Wigner matrix W1. Further we express M2 in terms of a Wigner resolvent G := (W2 − w)−1
+and use local laws not only for a single resolvent ⟨G⟩ but also their multi-resolvent versions for ⟨G2⟩ and
+⟨GG∗⟩ (see [18]). With a slight abuse of notation we shall henceforth drop the subscript ‘2’ in γi,2 and wi,2
+and replace them by their deterministic values γi and wi, respectively, at a negligible error of order N −1
+exactly as in Section 3.1. Note that ℑwi ≳ 1 for bulk indices i, so all resolvents below are stable and all
+denominators are well separated away from zero; this is needed to justify the ≈ relations below.
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+11
+The first term in (3.19) can be rewritten as (here G = G(wi) and msc := msc(wi) for brevity)
+�� ˚
+W γi
+2 ℑM2(γi)
+�2�
+≈ 1
+2ℜ
+�����
+wi
+p2
+−
+γi
+p2
+1 + p2
+2
+����
+2
+⟨GG∗⟩ −
+�wi
+p2
+−
+γi
+p2
+1 + p2
+2
+�2
+⟨G2⟩
+�
+≈ 1
+2ℜ
+�����
+wi
+p2
+−
+γi
+p2
+1 + p2
+2
+����
+2
+|msc|2
+1 − |msc|2 −
+�wi
+p2
+−
+γi
+p2
+1 + p2
+2
+�2
+m2
+sc
+1 − m2sc
+�
+≈
+p4
+1
+2p2
+2(p2
+1 + p2
+2)2 ℜ
+�
+1
+1 − |msc|2 −
+1
+1 − m2sc
+�
+,
+(3.20)
+where in the last step we used (3.18). Similarly for the second term in (3.19), we have
+��
+M2(γi)
+�2 ˚
+W γi
+2
+�2
+1 − p2
+1
+��
+M(γi)
+�2� ≈
+�
+1
+p2
+�
+1
+p2 G +
+�
+wi
+p2 −
+γi
+p2
+1+p2
+2
+�
+G2� �2
+1 − p2
+1
+p2
+2 ⟨G2⟩
+≈ m2
+sc(p2
+2 − (p2
+1 + p2
+2)m2
+sc)
+p2
+2(p2
+1 + p2
+2)2(1 − m2sc) .
+(3.21)
+Plugging (3.20) and (3.21) into (3.19) we obtain
+p2
+2 Varγi(W2) ≈ 2p2
+2 + p2γiℜmsc
+(ℑmsc)2
+·
+p2
+1p2
+2
+2(p2
+1 + p2
+2)2 .
+(3.22)
+Taking the imaginary part of (3.18), we find that |msc|2 ≈
+p2
+2
+p2
+1+p2
+2 and hence, using (3.8) again, we infer
+1
+p2
+1 + p2
+2
+2p2
+2 + p2γiℜmsc
+(ℑmsc)2
+≈ ℜ[|msc|2 − m2
+sc]
+(ℑmsc)2
+= 2 .
+(3.23)
+Combining (3.22) and (3.23) with (3.19), this completes the proof of (3.2). This proves Lemma 3.1.
+4. MULTI–RESOLVENT LOCAL LAWS: PROOF OF THEOREM 2.6
+To study the eigenvectors of H we analyse its resolvent G(z) := (H − z)−1, with z ∈ C \ R. It is
+well known [26, 4] that G(z) becomes approximately deterministic in the large N limit. Its deterministic
+approximation (as a matrix) is given by M(z), the unique solution of (2.7), in the following averaged and
+isotropic sense:
+|⟨(G(z) − M(z))B⟩| ≺
+1
+N|ℑz|,
+|⟨x , (G(z) − M(z))y⟩| ≺
+1
+�
+N|ℑz|
+,
+(4.1)
+uniformly in deterministic vectors ∥x∥ + ∥y∥ ≲ 1 and deterministic matrices ∥B∥ ≲ 1. To be precise,
+while the local laws (4.1) hold for ℜz ∈ Bκ and dist(ℜz, supp(ρ)) ≳ 1 for arbitrary bounded self-
+adjoint deformations D = D∗ (see [26, Theorem 2.2]), the complementary regime requires the strengthened
+Assumption 2.7 on D (see [4]). Note that cusps for ρ have been excluded in Assumption 2.7, hence the
+complementary regime only consists of edges, which are covered in [4, Theorem 2.6], under the requirement
+that ∥M∥ is bounded – which was also supposed in Assumption 2.7.
+The isotropic bound ⟨x, ℑG(z)x⟩ ≺ 1 from (4.1) immediately gives an (almost) optimal bound on the
+delocalisation of eigenvalues: |⟨ui, x⟩| ≺ N −1/2 [31, 30, 26, 4, 9]. However, these estimates are not precise
+enough to conclude optimal bounds for eigenvector overlaps and generic matrices A as in Theorem 2.6; in
+fact by (4.1) we can only obtain the trivial bound |⟨ui, Auj⟩| ≺ 1. Instead of the single resolvent local law
+(4.1), we rely on the fact that (see (4.17) below)
+N
+��⟨ui, Auj⟩
+��2 ≲ ⟨ℑG(γi + iη)AℑG(γj + iη)A∗⟩ ,
+(4.2)
+for η ∼ N −1+ǫ, where ǫ > 0 is small but fixed, and γi, γj ∈ Bκ are in the bulk and we estimate the rhs.
+of (4.2). In particular, to prove Theorem 2.6 we will use the multi-resolvent local laws from Proposition 4.4
+below.
+Multi-resolvent local laws are natural generalisations of (4.1) and they assert that longer products
+G1B1G2 · · · Gk−1Bk−1Gk
+(4.3)
+
+12
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+of resolvents Gi := G(zi) and deterministic matrices9 B1, ..., Bk−1 also become approximately determin-
+istic both in average and isotropic sense in the large N limit as long as N|ℑzi| ≫ 1. The deterministic
+approximation to the chain (4.3) is denoted by
+M(z1, B1, z2, ..., zk−1, Bk−1, zk).
+(4.4)
+It is not simply M(z1)B1M(z2)B2 . . ., i.e. it cannot be obtained by mechanically replacing each G with M
+as (4.1) might incorrectly suggest. Instead, it is defined recursively in the length k of the chain as follows
+(see [20, Definition C.1]):
+Definition 4.1. Fix k ∈ N and let z1, ..., zk ∈ C \ R be spectral parameters. As usual, the correspond-
+ing solutions to (2.7) are denoted by M(zj), j ∈ [k]. Then, for deterministic matrices B1, ..., Bk−1 we
+recursively define
+M(z1, B1, ...Bk−1, zk) =
+�
+B1k
+�−1
+�
+M(z1)B1M(z2, ..., zk)
+(4.5)
++
+k−1
+�
+l=2
+M(z1)⟨M(z1, ..., zl)⟩M(zl, ..., zk)
+�
+,
+where we introduced the shorthand notation
+Bmn ≡ B(zm, zn) = 1 − M(zm)⟨·⟩M(zn)
+(4.6)
+for the stability operator acting on the space of N × N matrices.
+It turns out that the size of M(z1, B1, z2, . . . , zk) in the relevant regime of small η := minj |ℑzj| is
+roughly η−k+1 in the worst case, with a matching error term in the corresponding local law. This blow-up
+in the small η regime comes recursively from the large norm of the inverse of the stability operator B1k
+in (4.5). However, for a special subspace of observable matrices Bi, called regular matrices, the size of
+M(z1, B1, z2, . . . , zk) is much smaller. For Wigner matrices, i.e. for D = 0, the regular observables are
+simply the traceless matrices, i.e. observables B such that ⟨B⟩ = 0. In [15, 16, 18, 19] it was shown that
+when the matrices Bi are all traceless, then M(z1, B1, z2, . . . , zk) hence (4.3) are smaller by an ηk/2-factor
+than for general Bi;’s.
+The situation for deformed Wigner matrices is more complicated, since the concept of regular observ-
+ables will be dependent on the precise location in the spectrum of H, i.e. dependent on the energy. More
+precisely, we will require that the trace of A tested against a deterministic energy dependent matrix has to
+vanish; this reflects the inhomogeneity introduced by D. Analogously to the Wigner case, in Proposition 4.4
+below we will show that resolvent chains (4.3) are much smaller when the deterministic matrices Bi are
+regular.
+Next, we give the definition of regular matrices in the chain (4.3). Using the notation A for regular
+matrices, we will consider chains of resolvents and deterministic matrices of the form
+⟨G1A1 · · · GkAk⟩
+(4.7)
+in the averaged case, or
+�
+G1A1 · · · AkGk+1
+�
+xy
+(4.8)
+in the isotropic case, with Gi := G(zi) and Ai being regular matrices according to the following Defini-
+tion 4.2 (cf. [20, Definition 4.1]), which generalises the earlier Definition 2.5.
+Definition 4.2 (Regular observables – Two-point regularisation in chains). Fix a parameter κ > 0 and let
+δ = δ(κ, ∥D∥) > 0 be small enough (see the discussion below). Consider one of the two expressions (4.7)
+or (4.8) for some fixed length k ∈ N and bounded matrices ∥Ai∥ ≲ 1 and let z1, ..., zk+1 ∈ C \ R be
+spectral parameters with ℜzj ∈ Bκ. For any j ∈ [k], we denote
+1δ(zj, zj+1) := φδ(ℜzj − ℜzj+1) φδ(ℑzj) φδ(ℑzj+1)
+(4.9)
+9We will use the the notational convention, that the letter B denotes arbitrary (generic) matrices, while A is reserved for regular
+matrices, in the sense of Definition 4.2 below.
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+13
+where 0 ≤ φδ ≤ 1 is a smooth symmetric bump function on R satisfying φδ(x) = 1 for |x| ≤ δ/2 and
+φδ(x) = 0 for |x| ≥ δ. Here and in the following, in case of (4.7), the indices in (4.9) are understood
+cyclically modulo k.
+(a) For j ∈ [k], denoting sj := −sgn(ℑzjzj+1), we define the (two-point) regularisation of Aj from
+(4.7) or (4.8) w.r.t. the spectral parameters (zj, zj+1) as
+˚
+Azj,zj+1
+j
+:= Aj − 1δ(zj, zj+1)⟨M(ℜzj + iℑwj)AM(ℜzj+1 + sjiℑzj+1)⟩
+⟨M(ℜzj + iℑzj)M(ℜzj+1 + sjiℑzj+1)
+.
+(4.10)
+(b) Moreover, we call Aj regular w.r.t. (zj, zj+1) if and only if ˚
+Azj,zj+1
+j
+= Aj.
+As already indicated above, the two-point regularisation generalises Definition 2.5 in the sense that
+˚
+Ae±iη,e±iη −→ ˚
+Ae ,
+and
+˚
+Ae±iη,e∓iη −→ ˚
+Ae ,
+as
+η ↓ 0 ,
+(4.11)
+with a linear speed of convergence, for e ∈ Bκ and any bounded deterministic A ∈ CN×N, where we used
+that, by taking the imaginary part of (2.7), M(z)M(z)∗ = ℑM(z)/(⟨ℑM(z)⟩ + ℑz).
+Moreover, we point out, that the above Definition 4.2 of the regularisation is identical to [20, Defs. 3.1
+and 4.1] when dropping the summand with sjτ = −1 in Equation (3.6) of [20]. In particular, for spectral
+parameters zj, zj+1 satisfying 1δ(zj, zj+1) > 0 (for some δ > 0 small enough), it holds that the denomina-
+tor in (4.10) is bounded away from zero, which shows that the linear map A �→ ˚
+A is bounded. Additionally,
+we have the following Lipschitz property (see [20, Lemma 3.3]):
+˚
+Az1,z2 = ˚
+Aw1,w2 + O
+�
+|z1 − w1| + |z2 − w2|
+�
+I,
+(4.12)
+for any z1, z2, w1, w2 ∈ C \ R such that ℑziℑwi > 0. It is important that the error in (4.12) is a constant
+times the identity matrix, indicated by O(·)I.
+Next, we give bounds on the size of M(z1, A1, ...Ak−1, zk), the deterministic approximation to the chain
+G1A1 · · · Ak−1Gk introduced in Definition 4.1; the proof of this lemma is presented in Appendix A.
+Lemma 4.3. Fix κ > 0. Let k ∈ [4] and z1, ..., zk+1 ∈ C \ R be spectral parameters with ℜzj ∈ Bκ. Set
+η := minj |ℑzj|. Then, for bounded regular deterministic matrices A1, ..., Ak (according to Definition 4.2),
+we have the bounds
+∥M(z1, A1, ..., Ak, zk+1)∥ ≲
+�
+1
+η⌊k/2⌋
+if η ≤ 1
+1
+ηk+1
+if η > 1 ,
+(4.13)
+|⟨M(z1, A1, ..., Ak−1, zk)Ak⟩| ≲
+�
+1
+η⌊k/2⌋−1 ∨ 1
+if η ≤ 1
+1
+ηk
+if η > 1 .
+(4.14)
+For the presentation of Proposition 4.4, the main technical result underlying the proof of Theorem 2.6,
+we would only need (4.13) and (4.14) for k ∈ [2] from the previous lemma. However, the remaining bounds
+covered by Lemma 4.3 will be instrumental in several parts of our proofs (see Section 6 and Appendix A).
+Proposition 4.4. Fix ǫ > 0, κ > 0, k ∈ [2], and consider z1, . . . , zk+1 ∈ C \ R with ℜzj ∈ Bκ. Consider
+regular matrices A1, . . . , Ak with ∥Ai∥ ≤ 1, deterministic vectors x, y with ∥x∥ + ∥y∥ ≲ 1, and set
+Gi := G(zi). Then, uniformly in η := minj |ℑzj| ≥ N −1+ǫ, we have the averaged local law
+��⟨
+�
+G1A1 . . . Gk − M(z1, A1, ..., zk)
+�
+Ak⟩
+�� ≺
+�
+N k/2−1
+√Nη
+if η ≤ 1
+1
+Nηk+1
+if η > 1
+(4.15a)
+and the isotropic local law
+��⟨x,
+�
+G1A1 . . . Gk+1 − M(z1, A1, ..., zk+1)
+�
+y⟩
+�� ≺
+
+
+
+N (k−1)/2
+√
+Nη2
+if η ≤ 1
+1
+√
+Nηk+2
+if η > 1
+.
+(4.15b)
+In Section 6, we will carry out the proof of Proposition 4.4 in the much more involved η ≤ 1 regime.
+For η > 1, the bound simply follows by induction on the number of resolvents in a chain by invoking the
+trivial estimate ∥M(z)∥ ≲ 1/|ℑz|. The detailed argument has been carried out in [18, Appendix B] for the
+case of Wigner matrices. Having Proposition 4.4 at hand, we can now prove Theorem 2.6.
+
+14
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+Proof of Theorem 2.6. By (4.15a) and (4.14) for k = 2 it follows that
+|⟨G1A1G2A2⟩| ≺ 1 ,
+(4.16)
+for arbitrary regular matrices A1 = ˚
+Az1,z2
+1
+and A2 = ˚
+Az2,z1
+2
+. Now, using that (see [20, Lemma 3.6] for an
+analogous statement; see also (4.11) and (4.12))
+˚
+Aγi = ˚
+Aγi±iη,γj±2iη + O
+�
+|γi − γj| + η
+�
+I = ˚
+Aγi±iη,γj∓2iη + O
+�
+|γi − γj| + η
+�
+I ,
+and analogously for (˚
+A∗)γi, we obtain (cf. [20, Proof of Theorem 2.7])
+⟨ℑG(γi + iη)˚
+AγiℑG(γj + 2iη)(˚
+A∗)γi⟩ ≺ 1 .
+Moreover, by spectral decomposition, together with the rigidity of eigenvalues (see e.g. [2, 26]) it follows
+that (cf. [20, Lemma 3.5])
+N|⟨ui, ˚
+Aγiuj⟩|2 ≺ (Nη)2⟨ℑG(γi + iη)˚
+AγiℑG(γj + 2iη)(˚
+A∗)γi⟩ ≺ (Nη)2 .
+(4.17)
+Choosing η = N −1+ξ/2 for some arbitrary small ξ > 0, we conclude the desired.
+□
+5. DYSON BROWNIAN MOTION: PROOF OF THEOREM 2.8
+Given the optimal a priori bound (2.12), the proof of Theorem 2.8 is very similar to the analysis of the
+Stochastic Eigenstate Equation (SEE) in [17, Sections 3-4] and [19, Section 4]. Even if very similar to
+those papers, to make the presentation clearer, here we write out the main steps of the proof and explain the
+differences, but we do not write the details; we defer the interested reader to [17]. We also remark that the
+proof in [17, 19] heavily relies on the analysis of SEE developed in [13] and extend in [14, 33].
+Similarly to [17, 19] we only consider the real case, the complex case is completely analogous and so
+omitted. We prove Theorem 2.8 dynamically, i.e. we consider the flow
+dWt = d �Bt
+√
+N
+,
+W0 = W ,
+(5.1)
+with �Bt a real symmetric matrix valued Brownian motion (see e.g. [13, Definition 2.1]). Note that Wt has
+a Gaussian component of size
+√
+t, i.e.
+Wt
+d= W0 +
+√
+tU ,
+with U being a GOE matrix independent of W0. Denoting by λi(t) the eigenvalues of Wt (labeled in
+increasing order) and by ui(t) the corresponding orthonormal eigenvectors, we will prove Theorem 2.8 for
+the eigenvectors ui(T ), with T = N −1+ω, for some small fixed ω > 0. Since T is very small, the Gaussian
+component added in the flow (5.1) can easily be removed by a standard Green function comparison (GFT)
+argument as in [19, Appendix B].
+By [13], it is known that the eigenvalues λi(t) and the eigenvectors ui(t) are the unique strong solution
+of the following system of stochastic differential equations (SDEs):
+dλi(t) = dBii(t)
+√
+N
++ 1
+N
+�
+j̸=i
+1
+λi(t) − λj(t)dt
+(5.2)
+dui(t) =
+1
+√
+N
+�
+j̸=i
+dBij(t)
+λi(t) − λj(t)uj(t) −
+1
+2N
+�
+j̸=i
+ui
+(λi(t) − λj(t))2 dt ,
+(5.3)
+where the matrix B(t) = (Bij(t))N
+i,j=1 is a standard real symmetric Brownian motion (see e.g. [13, Defi-
+nition 2.1]).
+Even if in Theorem 2.8 we want to prove a CLT only for diagonal overlaps ⟨ui, Aui⟩, by (5.3), it follows
+that there is no closed equation for such quantities. For this reason, following [14, Section 2.3], we study
+the evolution of the perfect matching observable (see (5.5) below) along the flow (5.3).
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+15
+5.1. Perfect matching observable and proof of Theorem 2.8. We introduce the notation
+pij = pij(t) = ⟨ui, Auj⟩ − δijC0 ,
+(5.4)
+with A being a fixed real symmetric deterministic matrix A and C0 being a fixed constant independent of i.
+Note that compared to [17, 19] in (5.4) we define the diagonal pii without subtracting their expectation (see
+(2.12) above), but rather a generic constant C0 which we will choose later (see (5.46) below). The reason
+behind this choice is that in the current setting, unlike in the Wigner case [17, 19], the expectation of pii is
+now i–dependent, hence the flow (5.8) below would not be satisfied if we had defined (5.5) with the centred
+pii’s.
+To study moments of the pij’s we use the particle representation introduced in [13] and further developed
+in [14, 33].. A particle configuration, corresponding to a certain monomials of pij’s, can be encoded by a
+function η : [N] → N0. The image ηj = η(j) denotes the particle at the site j, and �
+j ηj = n denotes
+the total number of particles. Additionally, given a particle configuration η, by ηij, with i ̸= j, we denote
+a new particle configuration in which a particle at the site i moved to a new site j, if there is no particle in i
+then ηij = η. We denote the set of such configuration by Ωn.
+Fix a configuration η, then we define the perfect matching observable (see [14, Section 2.3]):
+fλ,t,C0,C1(η) :=
+N n/2
+[2C1]n/2
+1
+(n − 1)!!
+1
+M(η)E
+
+ �
+G∈Gη
+P(G)
+�����λ
+
+ ,
+M(η) :=
+N
+�
+i=1
+(2ηi − 1)!! ,
+(5.5)
+with n being the total number of particles in the configuration η. The sum in (5.5) is taken over Gη, which
+denotes the set of perfect matchings on the complete graph with vertex set
+Vη := {(i, a) : 1 ≤ i ≤ n, 1 ≤ a ≤ 2ηi} .
+We also introduced the short–hand notation
+P(G) :=
+�
+e∈E(G)
+p(e),
+p(e) := pi1i2 ,
+(5.6)
+where e = {(i1, a1), (i2, a2)} ∈ V2
+η, and E(G) denotes the edges of G. Note that in (5.5) we took the
+conditional expectation with respect to the entire trajectories of the eigenvalues, λ = {λ(t)}t∈[0,T ] for
+some fixed 0 < T ≪ 1. We also remark that the definition (5.5) differs slightly from [17, Eq. (3.9)] and
+[19, Eq. (4.6)], since we now do not normalise by ⟨(A − ⟨A⟩)2⟩ but using a different constant C1 which we
+will choose later in the proof (see (5.46) below); this is a consequence of the fact that the diagonal overlaps
+pii are not correctly centred and normalised. Note that we did not incorporate the factor 2 in (5.5) into the
+constant C1, since C1 will be chosen has a normalisation constant to compensate the size of the matrix A,
+whilst the factor 2 represents the fact that diagonal overlaps, after the proper centering and normalisation
+depending on A and i, would be centred Gaussian random variable of variance two. Furthermore, we
+consider eigenvalues paths {λ(t)}t∈[0,T ] which lie in the event
+�Ω = �Ωξ, :=
+�
+sup
+0≤t≤T
+max
+i∈[N] ηf(γi(t))−1|λi(t) − γi(t)| ≤ N ξ�
+(5.7)
+for any ξ > 0, where ηf(γi(t)) is the local fluctuation scale defined as in [27, Definition 2.4]. In most
+instances we will use this rigidity estimate in the bulk regime when ηf(γi(t)) ∼ N −1; at the edges
+ηf(γi(t)) ∼ N −2/3. We recall that here γi(t) denote the quantiles of ρt defined as in (2.11). The fact
+that the event �Ω holds with very high probability follows by [4, Corollary 2.9].
+By [14, Theorem 2.6] it follows that fλ,t is a solution of the parabolic discrete partial differential equation
+(PDE):
+∂tfλ,t = B(t)fλ,t ,
+(5.8)
+B(t)fλ,t =
+�
+i̸=j
+cij(t)2ηi(1 + 2ηj)
+�
+fλ,t(ηkl) − fλ,t(η)
+�
+.
+(5.9)
+where
+cij(t) :=
+1
+N(λi(t) − λj(t))2 .
+(5.10)
+
+16
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+In the remainder of this section we may often omit λ from the notation since the paths of the eigenvalues
+are fixed within this proof.
+The main result of this section is the following Proposition 5.1, which will readily prove Theorem 2.8.
+For this purpose we define a version of ft(η) with centred and rescaled pii:
+qλ,t(η) :=
+� N
+�
+i=1
+1
+Varγi(A)ηi/2
+�
+N n/2
+2n/2(n − 1)!!
+1
+M(η)E
+
+ �
+G∈Gη
+Q(G)
+�����λ
+
+
+(5.11)
+with ˚
+Aγi denoting the regular component of A defined as in (2.10):
+˚
+Aγi := A − ⟨AℑM(γi)⟩
+⟨ℑM(γi)⟩ ,
+and
+Q(G) :=
+�
+e∈E(G)
+q(e),
+q(e) := ⟨ui1, ˚
+Aγi1 ui2⟩.
+(5.12)
+Note that the definition in (5.12) is not asymmetric for i1 ̸= i2, since in this case ⟨ui1, ˚
+Aγi1 ui2⟩ =
+⟨ui1, ˚
+Aγi2ui2⟩.
+We now comment on the main difference between qt and ft from (5.11) and (5.5), respectively. First of
+all we notice the q(e)’s in (5.12) are slightly different compared with the p(e)’s from (5.4). In particular, we
+choose the q(e)’s in such a way that the diagonal overlaps have very small expectation (i.e. much smaller
+than their fluctuations size). The price to pay for this choice is that the centering is i–dependent, hence qt
+is not a solution of an equation of the form (5.8)–(5.9). We also remark that later within the proof, C0 from
+(5.4) will be chosen as
+C0 = ⟨AℑM(γi0)⟩
+⟨ℑM(γi0)⟩
+for some fixed i0 such that γi0 ∈ Bκ is in the bulk (recall (2.9)). The idea behind this choice is that the
+analysis of the flow (5.8)–(5.9) will be completely local, we can thus fix a base point i0 and ensure that
+the corresponding overlap is exactly centred, then the nearby overlaps for indices |i − i0| ≤ K, for some
+N–dependent K > 0, will not be exactly centred, but their expectation will be very small compared to the
+size of their fluctuations:
+⟨AℑM(γi0)⟩
+⟨ℑM(γi0)⟩ − ⟨AℑM(γi)⟩
+⟨ℑM(γi)⟩ = O
+�K
+N
+�
+.
+A consequence of this choice is also that the normalisation for qt and ft is different: for qt we chose a nor-
+malisation that is i’s dependent, whilst for ft the normalisation C1 is i–independent and later, consistently
+with the choice of C0, it will be chosen as
+C1 = Varγi0(A)n/2,
+which is exactly the normalisation that makes ft(η) = 1 when η is such that ηi0 = n and zero otherwise.
+Proposition 5.1. For any n ∈ N there exists c(n) > 0 such that for any ǫ > 0, and for any T ≥ N −1+ǫ it
+holds
+sup
+η |qT (η) − 1(n even)| ≲ N −c(n) ,
+(5.13)
+with very high probability. The supremum is taken over configurations η supported on bulk indices and the
+implicit constant in (5.13) depends on n and ǫ.
+Proof of Theorem 2.8. Fix n ∈ N, an index i such that γi ∈ Bκ is in the bulk, and choose a configuration
+η such that ηi = n and ηj = 0 for any j ̸= i. Then Proposition 5.1, together with a standard GFT argument
+(see e.g. [19, Appendix B]), readily implies (2.13).
+The lower bound on the variance (2.15) is an explicit calculation relying on the the definition of M
+from (2.7). In particular, we use that
+(i) A and hence ˚
+Aγi are self-adjoint;
+(ii) ℑM(γi) ≥ g for some g = g(κ, ∥D∥) > 0 since we are in the bulk;
+(iii) ⟨˚
+AγiℑM(γi)⟩ = 0 by definition of the regularisation;
+(iv) [ℜM(γi), ℑM(γi)] = 0 from (2.7).
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+17
+Then, after writing Varγi(A) as a sum of squares and abbreviating ℑM = ℑM(γi), we find
+Varγi(A) ≥
+��√
+ℑM
+�
+A − ⟨A(ℑM)2⟩
+⟨(ℑM)2⟩
+�√
+ℑM
+�2�
+⟨(ℑM)2⟩
+≥ g2
+��
+A − ⟨A(ℑM)2⟩
+⟨(ℑM)2⟩
+�2�
+⟨(ℑM)2⟩
+≥
+g2
+⟨(ℑM)2⟩⟨(A − ⟨A⟩)2⟩ ,
+where in the last step we used the trivial variational principle ⟨(A − ⟨A⟩)2⟩ = inft∈R⟨(A − t)2⟩. This
+completes the proof of Theorem 2.8.
+□
+5.2. DBM analysis. Similarly to [17, Section 4.1] and [19, Section 4.2] we introduce an equivalent particle
+representation to encode moments of the pij’s. In particular, here, and previously in [17, 19], we relied on
+the particle representation (5.14)–(5.16) below since our arguments heavily builds on [33], which use this
+latter representation.
+Consider a particle configuration η ∈ Ωn, for some fixed n ∈ N, i.e. η is such that �
+j ηj = n. We now
+define the new configuration space
+Λn := {x ∈ [N]2n : ni(x) is even for every i ∈ [N]
+�
+,
+(5.14)
+where
+ni(x) := |{a ∈ [2n] : xa = i}|
+(5.15)
+for all i ∈ N.
+By the correspondence
+η ↔ x
+ηi = ni(x)
+2
+.
+(5.16)
+it is easy to see that these two representations are basically equivalent. The only difference is that x uniquely
+determines η, but η determines only the coordinates of x as a multi-set and not its ordering.
+From now on, given a function f defined on Ωn, we will always consider functions g on Λn ⊂ [N]2n
+defined by
+f(η) = f(φ(x)) = g(x),
+with φ: Λn → Ωn, φ(x) = η being the projection from the x-configuration space to the η-configuration
+space using (5.16). We thus defined the observable
+gt(x) = gλ,t(x) := fλ,t(φ(x)) ,
+(5.17)
+with fλ,t from (5.5). Note that gt(x) is equivariant under permutation of the arguments, i.e. it depends on
+x only as a multi–set. Similarly we define
+rt(x) = rλ,t(x) := qλ,t(φ(x)) .
+(5.18)
+We remark that gt and rt are the counterpart of ft and qt, respectively, in the x-configuration space.
+We can thus now write the flow (5.8)–(5.9) in the x–configuration space:
+∂tgt(x) = L(t)gt(x)
+(5.19)
+L(t) :=
+�
+j̸=i
+Lij(t),
+Lij(t)g(x) : = cij(t)nj(x) + 1
+ni(x) − 1
+�
+a̸=b∈[2n]
+�
+g(xij
+ab) − g(x)
+�
+,
+(5.20)
+where
+xij
+ab := x + δxaiδxbi(j − i)(ea + eb),
+(5.21)
+with ea ∈ R2n denoting the unit vector. We remark that this flow is map on functions defined on Λn ⊂
+[N]2n which preserves equivariance.
+For the following analysis it is convenient to define the scalar product and the natural measure on Λn:
+⟨f, g⟩Λn = ⟨f, g⟩Λn,π :=
+�
+x∈Λn
+π(x) ¯f(x)g(x),
+π(x) :=
+N
+�
+i=1
+((ni(x) − 1)!!)2,
+(5.22)
+as well as the norm on Lp(Λn):
+∥f∥p = ∥f∥Lp(Λn,π) :=
+� �
+x∈Λn
+π(x)|f(x)|p
+�1/p
+.
+(5.23)
+
+18
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+The operator L = L(t) is symmetric with respect to the measure π and it is a negative in L2(Λn), with
+associated Dirichlet form (see [32, Appendix A.2]):
+D(g) = ⟨g, (−L)g⟩Λn = 1
+2
+�
+x∈Λn
+π(x)
+�
+i̸=j
+cij(t)nj(x) + 1
+ni(x) − 1
+�
+a̸=b∈[2n]
+��g(xij
+ab) − g(x)
+��2.
+Finally, by U(s, t) we denote the semigroup associated to L, i.e. for any 0 ≤ s ≤ t it holds
+∂tU(s, t) = L(t)U(s, t),
+U(s, s) = I.
+(5.24)
+5.3. Short range approximation. As a consequence of the singularity of the coefficients cij(t) in (5.20),
+the main contribution to the flow (5.19) comes from nearby eigenvalues, hence its analysis will be com-
+pletely local. For this purpose we define the sets
+J = Jκ := {i ∈ [N] : γi(0) ∈ Bκ},
+(5.25)
+which correspond to indices with quantiles γi(0) (recall (2.11)) in the bulk.
+Fix a point y ∈ J 2n, and an N-dependent parameter K such that 1 ≪ K ≪
+√
+N. We remark that
+y ∈ J 2n will be fixed for the rest of the analysis. Next, we define the averaging operator as a simple
+multiplication operator by a “smooth” cut-off function:
+Av(K, y)h(x) := Av(x; K, y)h(x),
+Av(x; K, y) := 1
+K
+2K−1
+�
+j=K
+1(∥x − y∥1 < j),
+(5.26)
+with ∥x − y∥1 := �2n
+a=1 |xa − ya|. For notational simplicity we may often omit K, y from the notation
+since they are fixed throughout the proof:
+Av(x) = Av(x; K, y)h(x),
+Avh(x) = Av((x))h((x)).
+(5.27)
+Additionally, fix an integer ℓ with 1 ≪ ℓ ≪ K, and define the short range coefficients
+cS
+ij(t) :=
+�
+cij(t)
+if i, j ∈ J and |i − j| ≤ ℓ
+0
+otherwise,
+(5.28)
+where cij(t) is defined in (5.10). The parameter ℓ is the length of the short range interaction.
+We now define a short–range approximation of rt, with rt defined in (5.18). Note that in the definition
+of the short–range flow (5.29) below there is a slight notational difference compared to [17, Section 4.2]
+and [19, Section 4.2.1]: we now choose an initial condition h0 which depends on r0 rather than g0. This
+minor difference is caused by the fact that in [17, 19] the observable gt was already centred and rescaled,
+while in the current case the centred and rescaled version of gt is given by rt, hence the definition in (5.29)
+is still conceptually the same as the one in [17, 19] (see also the paragraph above (5.45) for a more detailed
+explanation). We point out that we make this choice to ensure that the infinite norm of the short range
+approximation is always bounded by N ξ (see below (5.30)). The short range approximation ht = ht(x) is
+defined as the unique solution of the parabolic equation
+∂tht(x; ℓ, K, y) = S(t)ht(x; ℓ, K, y)
+h0(x; ℓ, K, y) = h0(x; K, y) : = Av(x; K, y)(r0(x) − 1(n even)),
+(5.29)
+where
+S(t) :=
+�
+j̸=i
+Sij(t),
+Sij(t)h(x) := cS
+ij(t)nj(x) + 1
+ni(x) − 1
+�
+a̸=b∈[2n]
+�
+h(xij
+ab) − h(x)
+�
+.
+(5.30)
+In the remainder of this section we may often omit K, y and ℓ from the notation, since they are fixed for
+the rest of the proof. We conclude this section defining the transition semigroup US(s, t) = US(s, t; ℓ)
+associated to the short range generator S(t). Note that ∥ht∥∞ ≤ N ξ, for any t ≥ 0 and any small ξ > 0,
+since US(s, t) is a contraction and ∥h0∥∞ ≤ N ξ by (2.12), as a consequence of ht(x) being supported on
+x ∈ J 2n.
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+19
+5.4. L2–estimates. To prove the L∞–bound in Proposition 5.1, we first prove an L2–bound in Proposi-
+tion 5.3 below and then use an ultracontractivity argument for the parabolic PDE (5.19) (see [17, Section
+4.4]) to get an L∞–bound. To get an L2–bound we will analyse ht, the short–range version of the observ-
+able gt from (5.17), and then we will show that ht and gt are actually close to each other using the following
+finite speed of propagation (see [17, Proposition 4.2, Lemmas 4.3–4.4]):
+Lemma 5.2. Let 0 ≤ s1 ≤ s2 ≤ s1 + ℓN −1, and f be a function on Λn, then for any x ∈ Λn supported
+on J it holds
+���(U(s1, s2) − US(s1, s2; ℓ))f(x)
+��� ≲ N 1+nξ s2 − s1
+ℓ
+∥f∥∞ ,
+(5.31)
+for any small ξ > 0. The implicit constant in (5.13) depends on n, ǫ, δ.
+To estimate several terms in the analysis of (5.29) we will rely on the multi–resolvent local laws from
+Proposition 4.4 (in combination with the extensions in Lemma A.1 in Lemma A.2). For this purpose, for a
+small ω > 2ξ > 0, we define the very high probability event (see Lemmas A.1–A.2)
+�Ω = �Ωω,ξ :=
+�
+ei∈Bκ,
+|ℑzi|≥N −1+ω
+�
+n
+�
+k=2
+�
+sup
+0≤t≤T
+���⟨Gt(z1)˚
+A1 . . . Gt(zk)˚
+Ak⟩ − 1(k = 2)⟨M(z1, ˚
+A1, z2)˚
+A2⟩
+��� ≤ N ξ+k/2−1
+√Nη
+�
+∩
+�
+sup
+0≤t≤T
+��⟨Gt(z1)˚
+A1⟩
+�� ≤
+N ξ
+N
+�
+|ℑz1|
+� �
+�
+z1,z2:e1∈Bκ,
+|e1−e2|≥c1,|ℑzi|≥N−1+ω
+�
+sup
+0≤t≤T
+��⟨(Gt(z1)B1Gt(z2)B2⟩
+�� ≤ N ξ
+�
+,
+(5.32)
+where ˚
+A1, . . . , ˚
+Ak are regular matrices defined as in Definition 4.2 (here we used the short–hand notation
+˚
+Ai = ˚
+Azi,zi+1
+i
+),
+⟨M(z1, ˚
+A1, z2)˚
+A2⟩ = ⟨M(z1)˚
+A1M(z2)˚
+A2⟩ + ⟨M(z1)˚
+A1M(z2)⟩⟨M(z2)˚
+A2M(z1)⟩
+1 − ⟨M(z1)M(z2)⟩
+,
+(5.33)
+η := min{|ℑzi| : i ∈ [k]}, ρi,t := |ℑ⟨Mt(zi)⟩|, c1 > 0 is a fixed small constant, and B1, B2 are norm
+bounded deterministic matrices. We remark that for |e1 − e2| ≥ c1 we have the norm bound ∥M B1
+12 ∥ ≲ 1.
+Then, by standard arguments (see e.g. [19, Eq. (4.30)]), we conclude the bound (recall that �Ωξ from (5.7)
+denotes the rigidity event)
+max
+i,j∈J |⟨ui(t), Auj(t)⟩| ≤ N ω
+√
+N
+on �Ωω,ξ ∩ �Ωξ ,
+(5.34)
+simultaneously for all i, j ∈ J and 0 ≤ t ≤ T . Additionally, on �Ωω,ξ ∩ �Ωξ it also holds that
+|⟨ui(t), Auj(t)⟩| ≤ N ω
+�
+⟨ℑG(γi(t) + iη)AℑG(γj(t) + iη)A⟩
+Nρi,tρj,t
+≲ N ω
+N 1/4 ,
+(5.35)
+when one among i and j is in the bulk and |i − j| ≥ cN, for some small constant c depending on c1
+from (5.32). Here we used that for the index in the bulk, say i, we have ρi,t ∼ 1 and for the other index
+ρj,t ≳ N −1/2 as a consequence of η ≫ N −1. We point out that this non optimal bound N −1/4, instead of
+the optimal N −1/2, follows from the fact that the bound from Lemma A.2 is not optimal when one of the
+two spectral parameters in close to an edge; this is exactly the same situation as in [19, Eq. (4.31)] where
+we get an analogous non optimal bound for overlaps of eigenvectors that are not in the bulk.
+We are now ready to prove the main technical proposition of this section. Note the additional term
+KN −1/2 in the error E in (5.37) compared to [17, Proposition 4.2] and [19, Proposition 4.4]; this is a
+consequence of the fact the pii’s in (5.4) are not correctly centred. We stress that the base point y in
+Proposition 5.3 is fixed throughout the remainder of this section.
+
+20
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+Proposition 5.3. For any parameters satisfying N −1 ≪ η ≪ T1 ≪ ℓN −1 ≪ KN −1 ≪ N −1/2, and any
+small ǫ, ξ > 0 it holds
+∥hT1(·; ℓ, K, y)∥2 ≲ Kn/2E,
+(5.36)
+with
+E := N nξ
+�N ǫℓ
+K
++ NT1
+ℓ
++ Nη
+ℓ
++
+N ǫ
+√Nη +
+1
+√
+K
++ K
+√
+N
+�
+,
+(5.37)
+uniformly in particle configurations y such that ya = i0, for any a ∈ [2n] and i0 ∈ J , and eigenvalue
+trajectory λ in the high probability event �Ωξ ∩ �Ωω,ξ.
+Proof. The proof of this proposition is very similar to the one of [17, Proposition 4.2] and [19, Proposi-
+tion 4.4], we thus only explain the main differences here. In the following by the star over � we denote that
+the summation runs over two n-tuples of fully distinct indices. The key idea in this proof is that in order to
+rely on the multi–resolvent local laws (5.32) we replace the operator S(t) in (5.29) with the new operator
+A(t) :=
+∗
+�
+i,j∈[N]n
+Aij(t),
+Aij(t)h(x) := 1
+η
+� n
+�
+r=1
+aS
+ir,jr(t)
+�
+∗
+�
+a,b∈[2n]n
+(h(xij
+ab) − h(x)),
+(5.38)
+where
+aij = aij(t) :=
+η
+N((λi(t) − λj(t))2 + η2),
+(5.39)
+and aS
+ij are their short range version defined as in (5.28) with cij(t) replaced with aij(t), and
+xij
+ab := x +
+� n
+�
+r=1
+δxar irδxbr ir
+�
+n
+�
+r=1
+(jr − ir)(ear + ebr).
+(5.40)
+The main idea behind this replacement is that infinitesimally S(t) averages only in one direction at a time,
+whilst A(t) averages in all direction simultaneously. This is expressed by the fact that xij
+ab from (5.21)
+changes two entries of x per time, instead xij
+ab changes all the coordinates of x at the same time, i.e. let
+i := (i1, . . . , in), j := (j1, . . . , jn) ∈ [N]n, with {i1, . . . , in} ∩ {j1, . . . , jn} = ∅, then xij
+ab ̸= x if and
+only if for all r ∈ [n] it holds that xar = xbr = ir. Technically, the replacement of S(t) by A(t) can be
+performed at the level of Dirichlet forms.
+Lemma 5.4 (Lemma 4.6 of [17]). Let S(t), A(t) be defined in (5.30) and (5.38), respectively, and let µ
+denote the uniform measure on Λn for which A(t) is reversible. Then there exists a constant C(n) > 0
+such that
+⟨h, S(t)h⟩Λn,π ≤ C(n)⟨h, A(t)h⟩Λn,µ ≤ 0,
+(5.41)
+for any h ∈ L2(Λn), on the very high probability event �Ωξ ∩ �Ωω,ξ.
+We start noticing the fact that by (5.29) it follows
+∂t∥ht∥2
+2 = 2⟨ht, S(t)ht⟩Λn.
+(5.42)
+Then, combining this with (5.41), and using that xij
+ab = x unless xar = xbr = ir for all r ∈ [n], we
+conclude that
+∂t∥ht∥2
+2 ≤ C(n)⟨ht, A(t)ht⟩Λn,µ
+= C(n)
+2η
+�
+x∈Λn
+∗
+�
+i,j∈[N]n
+� n
+�
+r=1
+aS
+irjr(t)
+�
+∗
+�
+a,b∈[2n]n
+ht(x)
+�
+ht(xij
+ab) − ht(x)
+�
+� n
+�
+r=1
+δxar irδxbr ir
+�
+.
+(5.43)
+Then, proceeding as in the proof of [17, Proposition 4.5] (see also [19, Eq. (4.40)]), we conclude that
+∂t∥ht∥2
+2 ≤ −C1(n)
+2η
+⟨ht⟩2
+2 + C3(n)
+η
+E2Kn,
+(5.44)
+which implies ∥hT1∥2
+2 ≤ C(n)E2Kn, by a simple Gronwall inequality, using that T1 ≫ η.
+We point out that to go from (5.43) to (5.44) the proof is completely analogous to [17, Proof of Propo-
+sition 4.5], with the only exception being the proof of [17, Eqs. (4.41), (4.43)]. We thus now explain how
+to obtain the analog of [17, Eqs. (4.41), (4.43)] in the current case as well. The fact that we now have the
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+21
+bound (5.35) rather than the stronger bound N −1/3 as in [19, Eq. (4.31)] does not cause any difference in
+the final estimate. We thus focus on the main new difficulty in the current analysis, i.e. that in (5.29) we
+choose the initial condition depending on r0 rather than g0. We recall that the difference between r0 and g0
+is that r0 is defined in such a way all the eigenvector overlaps are precisely centred and normalised in an
+i–dependent way, whilst for g0 we can choose the i–independent constant C0, C1 so that only the overlap
+corresponding to a certain base point i0 is exactly centred and normalised, whilst the nearby overlaps are
+centred and normalised only modulo a negligible error K/N (see also the paragraph below (5.12) for a
+detailed explanation). This additional difficulty requires that to prove the analog of [17, Eqs. (4.41), (4.43)]
+we need to estimate the error produced by this mismatch.
+Using that the function f(x) ≡ 1(n even) is in the kernel of L(t), for any fixed x ∈ Γ, and for any fixed
+i, a, b, we conclude (recall the notation from (5.27))
+ht(xij
+ab)
+= US(0, t)
+�
+(Avr0)(xij
+ab) − (Av1(n even))(xij
+ab)
+�
+= Av(xij
+ab)
+�
+US(0, t)r0(xij
+ab) − 1(n even)
+�
++ O
+�N ǫ+nξℓ
+K
+�
+=
+�
+Av(x) + O
+� ℓ
+K
+�� �
+U(0, t)r0(xij
+ab) − 1(n even) + O
+�N 1+nξt
+ℓ
+��
++ O
+�N ǫ+nξℓ
+K
+�
+= Av(x)
+�
+gt(xij
+ab) − 1(n even)
+�
++ O
+�N ǫ+nξℓ
+K
++ N 1+nξt
+ℓ
++ N ξK
+√
+N
+�
+,
+(5.45)
+where in the definition of gt from (5.5), (5.17) we chose
+C0 := ⟨AℑM(γi0)⟩
+⟨ℑM(γi0)⟩ ,
+C1 := Varγi0 (A),
+(5.46)
+and the error terms are uniform in x ∈ Γ. Here i0 is the index defined below (5.37). The first three
+inequalities are completely analogous to [17, Eq. (4.41)]. We now explain how to obtain the last inequality
+at the price of the additional negligible error KN −1/2. Recall the definition of rt from (5.11), (5.18), then
+we note that for any x supported in the bulk it holds
+∥r0(x) − g0(x)∥∞ ≲ N ξK
+√
+N
+,
+(5.47)
+for C0, C1 chosen as in (5.46). Using (5.47) together with U(0, t)g0 = gt we prove the last equality in
+(5.45). We point out that to prove (5.47) we used that
+˚
+Aγi0 − ˚
+Aγir = O(|γi0 − γir|)I = O(KN −1)I.
+In particular, we used this approximation to compensate the mismatch that only the diagonal overlaps
+corresponding to the index i0 are properly centred and normalised in the definition of g0, whilst for nearby
+indices we use this approximation to replace the approximate centering C0 and normalisation C1 from
+(5.46) with the correct one, which is the one in the definition of r0.
+Then proceeding as in the proof of [19, Eq. (4.41)] we conclude the analog of [17, Eq. (4.43)]:
+∗
+�
+j
+� n
+�
+r=1
+aS
+irjr(t)
+�
+�
+qt(xij
+ab) − 1(n even)
+�
+=
+�
+j
+� n
+�
+r=1
+airjr(t)
+� 
+
+N n/2
+Varγi0 (A)n/22n/2(n − 1)!!
+�
+G∈Gηj
+P(G) − 1(n even)
+
+
++ O
+�N nξ
+Nη + N 1+nξη
+ℓ
++ N ξK
+√
+N
+�
+.
+(5.48)
+
+22
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+Given (5.48), the remaining part of the proof is completely analogous to [17, Eqs. (4.44)–(4.51)] except for
+the slightly different computation
+⟨ℑG(λir1 + iη)˚
+Aγi0 ℑG(λir2 + iη)˚
+Aγi0 ⟩
+Varγi0 (A)
+= −
+1
+4Varγi0 (A)
+�
+σ,τ∈{+,−}
+⟨G(λir1 + σiη)˚
+Aγi0 G(λir2 + τiη)˚
+Aγi0 ⟩
+= −
+1
+4Varγi0 (A)
+�
+σ,τ∈{+,−}
+�
+G(λir1 + iση)˚
+Aγir1 +iση,γir2 +iτηG(λir2 + iτη)˚
+Aγir2 +iτη,γir1 +iση�
++ O
+�
+K
+N 2η3/2 + K2
+N 2η +
+1
+N√η
+�
+= ⟨ℑM(γi0)⟩2 + O
+�
+K
+N 2η3/2 + K2
+N 2η +
+1
+√Nη
+�
+,
+(5.49)
+which replaces [17, Eqs. (4.47)]. Here ˚
+Aγir1 ±iη,γir2 ±iη is defined as in Definition 4.2. We also point out
+that in the second equality we used the approximation (see (4.12))
+˚
+Aγi0 = ˚
+Aγir1 ±iη,γir2 ±iη + O
+�
+|γir1 − γi0| + |γir2 − γi0| + η
+�
+I = ˚
+Aγir1 ,γir2 + O
+�K
+N + η
+�
+I,
+(5.50)
+together with (here we present the estimate only for one representative term, the other being analogous)
+⟨G(λir1 + iη)G(λir2 + iη)˚
+Aγi0 ⟩ = ⟨G(λir1 + iη)G(λir2 + iη)˚
+Aγir2 +iη,γir2 +iη⟩ + O
+�
+1 + K
+Nη
+�
+= ⟨M(γir2 + iη, ˚
+Aγir2 +iη,γir1 +iη, γir1 + iη)⟩ + O
+�
+1 +
+1
+Nη3/2 + K
+Nη
+�
+= O
+�
+1 +
+1
+Nη3/2 + K
+Nη
+�
+,
+(5.51)
+which follows by (5.32) and Lemma 4.3 to estimate the deterministic term, together with the integral repre-
+sentation from [20, Lemma 5.1] (see also (6.15) later), to bound the error terms arising from the replacement
+(5.50). Additionally, in the third equality we used the local for two resolvents from (5.32):
+− 1
+4
+�
+σ,τ∈{+,−}
+�
+G(λir1 + iση)˚
+Aγir1 +iση,γir2 +iτηG(λir2 + iτη)˚
+Aγir2 +iτη,γir1 +iση�
+= −1
+4
+�
+σ,τ∈{+,−}
+�
+M(γir1 + iση, ˚
+Aγir1 +iση,γir2 +iτη, γir2 + iτη)˚
+Aγir2 +iτη,γir1 +iση�
++ O
+�
+1
+√Nη
+�
+= ⟨ℑM(γi0)⟩2Varγi0(A) + O
+�
+1
+√Nη + K
+N
+�
+,
+with the deterministic term defined in (5.33), and in the second equality we used the approximation (5.50)
+again. We remark that here we presented this slightly different (compared to [17, 19]) computation only
+for chain of length k = 2, the computation for longer chains is completely analogous and so omitted. This
+concludes the proof of (5.36).
+□
+We conclude this section with the proof of Proposition 5.1.
+Proof of Proposition 5.1. Combining the L2–bound on ht from Proposition 5.3 and the finite speed of prop-
+agation estimates in Lemma 5.2 we can enhance this L2–bound to an L∞–bound completely analogously
+to the proof of [17, Proposition 3.2 ] presented in [17, Section 4.4].
+□
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+23
+6. PROOF OF PROPOSITION 4.4
+Our strategy for proving Proposition 4.4 (in the much more involved η ≤ 1 regime) is to derive a system
+of master inequalities (Proposition 6.3) for the errors in the local laws by cumulant expansion, then use
+an iterative scheme to gradually improve their estimates. The cumulant expansion naturally introduces
+longer resolvent chains, potentially leading to an uncontrollable hierarchy, so our master inequalities are
+complemented by a set of reduction inequalities (Lemma 6.4) to estimate longer chain in terms of shorter
+ones. We have used a similar strategy in [18, 19] for Wigner matrices, but now, analogously to [20], dealing
+with non-Hermitian i.i.d. matrices, many new error terms due to several adjustments of the z-dependent two-
+point regularisations need to be handled. By the strong analogy to [20], our proof of the master inequalities
+formulated in Proposition 6.3 and given in Section 6.2 will be rather short and focus on the main differences
+between [20] and the current setup.
+As the basic control quantities, analogously to [18, 20], in the sequel of the proof, we introduce the
+normalised differences
+Ψav
+k (zk, Ak) := Nηk/2|⟨G1A1 · · · GkAk − M(z1, A1, ..., zk)Ak⟩| ,
+(6.1)
+Ψiso
+k (zk+1, Ak, x, y) :=
+�
+Nηk+1
+���
+�
+G1A1 · · · AkGk+1 − M(z1, A1, ..., Ak, zk+1)
+�
+xy
+���
+(6.2)
+for k ∈ N, where we used the short hand notations
+Gi := G(zi) ,
+η := min
+i
+|ℑzi| ,
+zk := (z1, ..., zk) ,
+Ak := (A1, ..., Ak) .
+The deterministic matrices ∥Ai∥ ≤ 1, i ∈ [k], are assumed to be regular (i.e., Ai = ˚
+Azi,zi+1, see Defini-
+tion 4.2) and the deterministic counterparts used in (6.1) and (6.2) are given recursively in Definition 4.1.
+For convenience, we extend the above definitions to k = 0 by
+Ψav
+0 (z) := Nη|⟨G(z) − M(z)⟩| ,
+Ψiso
+0 (z, x, y) :=
+�
+Nη
+���
+G(z) − M(z)
+�
+xy
+��
+and observe that
+Ψav
+0 + Ψiso
+0
+≺ 1
+(6.3)
+is the usual single-resolvent local law from (4.1), where here and in the following the arguments of Ψav/iso
+k
+shall occasionally be omitted. We remark that the index k counts the number of regular matrices in the
+sense of Definition 4.2.
+Throughout the entire argument, let ǫ > 0 and κ > 0 be arbitrary but fixed, and let
+D(ǫ,κ) :=
+�
+z ∈ C : ℜz ∈ Bκ , N 100 ≥ |ℑz| ≥ N −1+ǫ�
+(6.4)
+be the spectral domain, where the κ-bulk Bκ has been introduced in (2.9). Strictly speaking, we would need
+to define an entire (finite) family of slightly enlarged spectral domains along which the above mentioned
+iterative scheme for proving Proposition 4.4 is conducted. Since this has been carried out in detail in [20]
+(see, in particular, [20, Figure 2]), we will neglect this technicality and henceforth assume all bounds on
+Ψav/iso
+k
+to be uniform on D(ǫ,κ) in the following sense.
+Definition 6.1 (Uniform bounds in the spectral domain). Let ǫ > 0 and κ > 0 as above and let k ∈ N. We
+say that the bounds
+��⟨G(z1)B1 · · · G(zk)Bk − M(z1, B1, ..., zk)Bk⟩
+�� ≺ Eav ,
+���
+�
+G(z1)B1 · · · BkG(zk+1) − M(z1, B1, ..., Bk, zk+1)
+�
+xy
+��� ≺ Eiso
+(6.5)
+hold (ǫ, κ)-uniformly (or simply uniformly)for some deterministic control parameters Eav/iso = Eav/iso(N, η),
+depending only on N and η := mini |ℑzi|, if the implicit constant in (6.5) are uniform in bounded deter-
+ministic matrices ∥Bj∥ ≤ 1, deterministic vectors ∥x∥, ∥y∥ ≤ 1, and admissible spectral parameters
+zj ∈ D(ǫ,κ) satisfying 1 ≥ η := minj |ℑzj|.
+Moreover, we may allow for additional restrictions on the deterministic matrices. For example, we may
+talk about uniformity under the additional assumption that some (or all) of the matrices are regular (in the
+sense of Definition 4.2).
+
+24
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+Note that (6.5) is stated for a fixed choice of spectral parameters zj in the left hand side, but it is in
+fact equivalent to an apparently stronger statement, when the same bound holds with a supremum over the
+spectral parameters (with the same constraints). While one implication is trivial, the other direction follows
+from (6.5) by a standard grid argument (see, e.g., the discussion after [18, Definition 3.1]).
+We can now formulate Proposition 4.4, in the language of our basic control quantities Ψav/iso
+k
+.
+Lemma 6.2 (Estimates on Ψav/iso
+1
+and Ψav/iso
+2
+). For any ǫ > 0 and κ > 0 we have
+Ψav
+1 + Ψiso
+1
+≺ 1
+and
+Ψav
+2 + Ψiso
+2
+≺
+�
+Nη
+(6.6)
+(ǫ, κ)-uniformly in regular matrices.
+Proof of Proposition 4.4. The η ≥ 1 case was already explained right after Proposition 4.4. The more
+critical η ≤ 1 case immediately follows from Lemma 6.2.
+□
+6.1. Master inequalities and reduction lemma: Proof of Lemma 6.2. We now state the relevant part of
+a non-linear infinite hierarchy of coupled master inequalities for Ψav
+k and Ψiso
+k . In fact, for our purposes, it
+is sufficient to have only the inequalities for k ∈ [2]. Slightly simplified versions of this master inequalities
+will be used in Appendix A for general k ∈ N. The proof of Proposition 6.3 is given in Section 6.2
+Proposition 6.3 (Master inequalities, see Proposition 4.8 in [20]). Assume that for some deterministic
+control parameters ψav/iso
+j
+we have that
+Ψav/iso
+j
+≺ ψav/iso
+j
+,
+j ∈ [4] ,
+(6.7)
+holds uniformly in regular matrices. Then we have
+Ψav
+1 ≺ 1 + ψav
+1
+Nη + ψiso
+1
++ (ψav
+2 )1/2
+(Nη)1/2
++ (ψiso
+2 )1/2
+(Nη)1/4 ,
+(6.8a)
+Ψiso
+1
+≺ 1 + ψiso
+1
++ ψav
+1
+(Nη)1/2
++ (ψiso
+2 )1/2
+(Nη)1/4 ,
+(6.8b)
+Ψav
+2 ≺ 1 + (ψav
+1 )2 + (ψiso
+1 )2 + ψav
+2
+Nη
++ ψiso
+2
++ (ψav
+4 )1/2
+(Nη)1/2
++ (ψiso
+3 )1/2 + (ψiso
+4 )1/2
+(Nη)1/4
+,
+(6.8c)
+Ψiso
+2
+≺ 1 + ψiso
+1
++ ψav
+1 ψiso
+1
++ (ψiso
+1 )2
+Nη
++ ψiso
+2
++ (ψiso
+1 ψiso
+3 )1/2
+(Nη)1/2
++ (ψiso
+3 )1/2 +(ψiso
+4 )1/2
+(Nη)1/4
+,
+(6.8d)
+again uniformly in regular matrices.
+As shown in the above proposition, resolvent chains of length k = 1, 2 are estimated by resolvent chains
+up to length 2k. In order to avoid the indicated infinite hierarchy of master inequalities with higher and
+higher k indices, we will need the following reduction lemma.
+Lemma 6.4 (Reduction inequalities, see Lemma 4.8 in [20]). As in (6.7), assume that Ψav/iso
+j
+≺ ψav/iso
+j
+holds for 1 ≤ j ≤ 4 uniformly in regular matrices. Then we have
+Ψav
+4 ≺ (Nη)2 + (ψav
+2 )2 ,
+(6.9)
+uniformly in regular matrices, and
+Ψiso
+3
+≺ Nη
+�
+1 + ψiso
+2
+√Nη
+� �
+1 + ψav
+2
+Nη
+�1/2
+,
+Ψiso
+4
+≺ (Nη)3/2
+�
+1 + ψiso
+2
+√Nη
+� �
+1 + ψav
+2
+Nη
+�
+(6.10)
+again uniformly in regular matrices.
+Proof. This is completely analogous to [20, Lemma 4.9] and hence omitted. The principle idea is to write
+out the lhs. of (6.9) and (6.10) by spectral decomposition and tacitly employ a Schwarz inequality. This
+leaves us with shortened chains, where certain resolvents G are replaced with absolute values |G|, which
+can be handled by means of a suitable integral representation [20, Lemma 6.1]
+□
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+25
+Now the estimates (6.6) follow by combining Proposition 6.3 and Lemma 6.4 in an iterative scheme,
+which has been carried out in detail in [20, Section 4.3]. This completes the proof of Lemma 6.2.
+6.2. Proof of the master inequalities in Proposition 6.3. The proof of Proposition 6.3 is very similar to
+the proof of the master inequalities in [20, Proposition 4.8]. Therefore, we shall only elaborate on (6.8a) as
+a showcase in some detail and briefly discuss (6.8b)–(6.8d) afterwards.
+First, we notice that [20, Lemma 5.2] also holds for deformed Wigner matrices (see Lemma 6.5 below).
+In order to formulate it, recall the definition of the second order renormalisation, denoted by underline,
+from [20, Equation (5.3)]. For a function f(W) of the Wigner matrix W, we define
+Wf(W) := Wf(W) − �E
+��
+W(∂�
+W f)(W)
+�
+,
+(6.11)
+where ∂�
+W denotes the directional derivative in the direction of �
+W, which is a GUE/GOE matrix (depending
+on the symmetry class of W) that is independent of W. The expectation is taken w.r.t. the matrix �
+W.
+Note that, if W itself a GUE/GOE matrix, then EWf(W) = 0, while for W with general single entry
+distributions, this expectation is independent of the first two moments of W. In other words, the underline
+renormalises the product Wf(W) to second order.
+We note that �E�
+WR�
+W = ⟨R⟩ and furthermore, that the directional derivative of the resolvent is given by
+∂�
+W G = −G�
+WG. For example, in the special case and f(W) = (W + D − z)−1 = G, we thus have
+WG = WG + ⟨G⟩G
+by definition of the underline in (6.11).
+Lemma 6.5. Under the assumption (6.7), for any regular matrix A = ˚
+A we have that
+⟨(G − M)˚
+A⟩ = −⟨WG˚
+A′⟩ + O≺ (Eav
+1 ) ,
+(6.12)
+for some other regular matrix A′ = ˚
+A′, which linearly depends on A (see (6.18) for an explicit formula).
+Here G = G(z) and η := |ℑz|. For the error term we used the shorthand notation
+Eav
+1
+:=
+1
+Nη1/2
+�
+1 + ψav
+1
+Nη
+�
+.
+(6.13)
+By simple complex conjugation of (6.12), we may henceforth assume that z = e + iη with η > 0. The
+representation (6.12) will be verified later. Now, using (6.12) we compute the even moments of ⟨(G−M)˚
+A⟩
+as
+E |⟨(G − M)A⟩|2p =
+��−E⟨WGA′⟩⟨(G − M)A⟩p−1⟨(G − M)∗A∗⟩p�� + O≺
+�
+(Eav
+1 )2p�
+and then apply a cumulant expansion to the first summand. The term corresponding to the second cumulant
+(also called the Gaussian term) is bounded from above by (a p-dependent constant times)
+E
+�|⟨GGA′GA⟩| + |⟨G∗GA′G∗A∗⟩|
+N 2
+|⟨(G − M)A⟩|2p−2
+�
+.
+(6.14)
+The main technical tool to estimate (6.14) is the following contour integral representation for the square of
+resolvent (see [20, Lemma 5.1]). This is given by
+G(z)2 =
+1
+2πi
+�
+Γ
+G(ζ)
+(ζ − z)2 dζ ,
+(6.15)
+where the contour Γ = Γ(z) is the boundary of a half band J × [i˜η, i∞) which is parametrised counter-
+clockwise. Here the closed interval J is chosen as Bκ′ for a suitable κ′ ∈ (0, κ) and hence contains e = ℜz;
+the parameter ˜η is chosen to be smaller than η/2.
+Applying the integral identity (6.15) to the product GG in the term ⟨GGA′GA⟩ gives us that
+|⟨GGA′GA⟩| ≲
+������
+�
+Γ
+⟨G(ζ)A′GA⟩
+(ζ − z)2
+dζ
+������
+.
+(6.16)
+Now, the parameter κ′ in the definition of J = Bκ′ is chosen in such a way that the distance between z
+and the vertical parts of the contour Γ is greater than δ > 0 from Definition 4.2 (see also [20, Figure 4]).
+Hence, for ζ on the vertical parts of Γ, every matrix is considered regular w.r.t. (z, ζ), which allows us to
+
+26
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+use the upper bounds for products of two resolvents and two regular matrices. Therefore, estimating this
+contribution with the aid of Lemma 4.3, we find that
+|⟨GGA′GA⟩| ≺
+�
+1 + ψav
+2
+Nη
+�
++
+�
+J
+|⟨G(x + i˜η)A′G(e + iη)A⟩|
+(x − e)2 + η2
+dx .
+In this integral over J, the horizontal part of Γ, we decompose A and A′ in accordance to the spectral
+parameters of the adjacent resolvents and use the following regularity property:
+A = ˚
+Ae+iη,e+iη = ˚
+Ae+iη,x+i˜η + O(|x − e| + η)I ,
+A′ =
+˚
+(A′)
+e+iη,e+iη =
+˚
+(A′)
+x+i˜η,e+iη + O(|x − e| + η)I .
+Now the integral over J is represented as a sum of four integrals: one of them contains two regular matrices,
+the rest have at least one identity matrix with a small error factor. For the first one we use the same estimates
+as for the integral over the vertical part of Γ. For the other terms, thanks to the identity matrix, we use a
+resolvent identity, e.g. for G(x + i˜η)IG(e + iη) and note that the
+�
+|x − e| + η
+�
+-error improves the original
+1/η-blow up of
+�
+J
+1
+(x−e)2+η2 dx to an only | log η|-divergent singularity, which is incorporated into ‘≺’.
+For the term ⟨G∗GA′G∗A∗⟩ from (6.14), we use a similar strategy: After an application of the Ward
+identity G∗G = ℑG/η, we decompose the deterministic matrices A, A′ according to the spectral parameters
+of their neighbouring resolvents in the product. This argument gives us that each of terms ⟨GGA′GA⟩ and
+⟨G∗GA′G∗A∗⟩ is stochastically dominated by
+1
+η
+�
+1 + ψav
+2
+Nη
+�
+.
+Contributions stemming from higher order cumulants in (6.14) are estimated exactly in the same way as
+in [20, Section 5.5]. The proof of (6.8a) is concluded by applying Young inequalities to (6.14) (see [20,
+Section 5.1]).
+Finally we show that (6.12) holds:
+Proof of Lemma 6.5. The proof of this representation is simpler than the proof of its analogue [20, Lemma 5.2]
+because in the current setting all terms in [20, Lemma 5.2] containing the particular chiral symmetry matrix
+E− are absent. In the same way as in [20] we arrive at the identity
+⟨(G − M)A⟩ = −⟨WGX [A] M⟩ + ⟨G − M⟩⟨(G − M)X[A]M⟩ ,
+(6.17)
+where we introduced the bounded linear operator X[B] :=
+�
+1 − ⟨M · M⟩
+�−1[B]. Indeed, boundedness
+follows from the explicit formula
+X[B] = B + ⟨MBM⟩
+1 − ⟨M 2⟩
+by means of the lower bound
+|1 − ⟨M 2⟩| = |(1 − ⟨MM ∗⟩) − 2i⟨MℑM⟩| ≥ 2⟨ℑM⟩2 ≳ 1 ,
+obtained by taking the imaginary part of the MDE (2.7), in combination with ∥M∥ ≲ 1.
+Next, completely analogously to [20, Lemma 5.4], we find the decomposition
+X[A]M =
+�
+X[A]M
+�◦ + O(η)I = ˚
+A′ + O(η)I ,
+A′ := X[A]M .
+(6.18)
+Plugging this into (6.17), we thus infer
+⟨(G − M)A⟩ = −⟨WGA′⟩ + ⟨G − M⟩⟨(G − M)˚
+A′⟩ +
+�
+−⟨WG⟩ + ⟨G − M⟩2�
+O(η),
+(6.19)
+The second term in the rhs. of (6.19) is obviously bounded by ψav
+1 /(N 2η3/2) and in the third term we use
+the usual local law (4.1) to estimate it by O≺
+�
+N −1�
+. Combining these information gives (6.12).
+□
+Notice that the above arguments leading to (6.8a) are completely identical to the ones required in the
+proof of the analogous master inequality in [20, Proposition 4.8] with one minor but key modification:
+Every term involving the chiral symmetry matrix E− in [20] is simply absent and hence, with the notation
+of [20], all sums over signs �
+σ=± · · · collapse to a single summand with E+ ≡ I. With this recipe, the
+proofs of (6.8b)–(6.8d) are completely analogous to the ones given in [20, Sections 5.2–5.5 and Appendix E]
+and hence omitted.
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+27
+APPENDIX A. ADDITIONAL TECHNICAL LEMMAS
+In this appendix, we prove several technical lemmas underlying the proofs our main results.
+A.1. Bounds on averaged multi-resolvent chains. For the proof of Theorem 2.8, especially controlling
+the event �Ω in (5.32), we need to extend the key estimate
+���
+G(z1)˚
+A1G(z2)˚
+A2
+��� ≺ 1 for ℜz1, ℜz2 ∈ Bκ
+from (4.16), which underlies the proof of the ETH in Theorem 2.6, in two directions: First, we consider
+chains with an arbitrary number k of resolvents in Lemma A.1, but we will only need a quite weak sub-
+optimal bound which makes its proof quite direct and short. Second, in Lemma A.2, we no longer restrict
+ℜz1, ℜz2 ∈ Bκ to the bulk, but assume that |ℜz1 − ℜz2| + |ℑz1| + |ℑz2| ≥ ν for some N-independent
+constant ν > 0 and additionally allow for arbitrary (non-regular) matrices A1, A2. This second extension
+requires Assumption 2.7 on the deformation D, i.e. the boundedness of M(z) also for ℜz /∈ Bκ; this is
+a slightly stronger requirement than just the boundedness of D assumed in Theorem 2.6. Both extensions
+are relevant for constructing the high probability event �Ω in (5.32) for the DBM analysis. The proofs of
+Lemma A.1 and Lemma A.2 are simple extensions and slight adjustments of the arguments leading to
+Proposition 4.4.
+Lemma A.1. Fix ǫ > 0, κ > 0, k ∈ N, and consider z1, . . . , zk ∈ C\R with ℜzj ∈ Bκ. Consider regular
+matrices A1, . . . , Ak with ∥Ai∥ ≤ 1, deterministic vectors x, y with ∥x∥ + ∥y∥ ≲ 1, and set Gi := G(zi).
+Define
+Gk := �G1A1 . . . Ak−1 �GkAk,
+�Gj ∈ {Gj, |Gj|} .
+(A.1)
+Then, uniformly in η := minj |ℑzj| ≥ N −1+ǫ, we have
+��⟨Gk⟩
+�� ≺ N k/2−1
+√Nη .
+(A.2)
+Proof. To keep the notation simple we only consider the case when Gk = G1A1 . . . Ak−1GkAk, the general
+case is completely analogous and so omitted. We split the proof into three parts. In Step (i) we first prove
+the slightly weaker bound
+��⟨Gk⟩
+�� ≺ N k/2−1 for any k ≥ 3 and a similar bound in isotropic sense; then,
+using Step (i) as an input, we will prove the better estimate (A.2) for k = 3, 4; finally, similarly to Step (i),
+we prove (A.2) for any k ≥ 3.
+Step (i): Similarly to the proof of the reduction inequalities in Lemma 6.4 (see [20, Lemma 4.9]) we readily
+see that for k = 2j:
+⟨(GA)2j⟩ ≲ N
+�
+⟨|G|A(GA)j/2−1|G|A(G∗A)j/2−1⟩2
+j even,
+⟨|G|A(GA)(j−1)/2|G|A(G∗A)(j−1)/2⟩⟨|G|A(GA)(j−3)/2|G|A(G∗A)(j−3)/2⟩
+j odd,
+(A.3)
+and for k = 2j − 1:
+⟨(GA)2j−1⟩ ≲ ⟨|G|A(GA)j−2|G|A(G∗A)j−2⟩1/2⟨|G|A(GA)j−1|G|A(G∗A)j−1⟩1/2.
+(A.4)
+We proceed by induction. First we use (A.3) for j = 2, together with ⟨GAGA⟩ ≺ 1 from Proposition 4.4,
+to get the bound ⟨G4⟩ ≺ N, and then use this bound as an input to obtain ⟨G3⟩ ≺ N 1/2 using (A.4). Then
+proceeding exactly in the same way we prove that if ⟨Gk⟩ ≺ N k/2−1 holds then the same bound holds for
+chains of length k + 1 and k + 2 as well. Similarly, in the isotropic chains we prove ⟨x, Gky⟩ ≺ N (k−1)/2;
+this concludes Step (i).
+Step (ii): Given the bounds ⟨Gk⟩ ≺ N k/2−1, ⟨x, Gky⟩ ≺ N (k−1)/2, the estimate in (A.2) for k = 3, 4
+immediately follows by writing the equation for Gk, performing cumulant expansion and using the corre-
+sponding bounds on M(z1, ..., zk) from Lemma 4.3. This was done in [18, Proof of Proposition 3.5], hence
+we omit the details.
+Step (iii): The proof of (A.2) for k ≥ 5 proceeds by induction. We first show that it holds for k = 5, 6 and
+then we prove that if it holds for k − 2 and k − 1 then it holds for k and k + 1 as well. To keep the notation
+short with a slight abuse of notation we denote (GA)k = G1A1 . . . Ak−1Gk.
+By Step (ii), it follows that (A.2) holds for k = 3, 4, and for k = 2 we have ⟨GAGA⟩ ≺ 1. Then by
+(A.3) we immediately conclude that the same bound is true for k = 6, which together with (A.4) also imply
+the desired bound for k = 5. The key point is that (A.3) splits a longer k-chain (k even) into a product of
+shorter chains of length k1, k2 with k1 + k2 = k. As long as k ≥ 5, at least one of the shorter chain has
+
+28
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+already length at least three, so we gain the factor (Nη)−1/2. In fact chains of length k = 2, from which
+we do not gain any extra factor, |⟨GAGA⟩| ≺ 1, appear only once when we apply (A.3) for k = 6. But in
+this case the other factor is a chain of length four with a gain of a (Nη)−1/2 factor. Similarly, (A.4) splits
+the long k chain (k odd) into the square root of two chains of length k − 1, and k + 1, and for k ≥ 5 we
+have the (Nη)−1/2 factor from both. The induction step then readily follows by using again (A.3)–(A.4)
+as explained above; this concludes the proof. In fact, in most steps of the induction we gain more than one
+factor (Nη)−1/2; this would allow us to improve the bound (A.2), but for the purpose of the present paper
+the suboptimal estimate (A.2) is sufficient.
+□
+We now turn to the second extension of (4.16) allowing for arbitrary spectral parameters, i.e. not neces-
+sarily in the bulk, but separated by a safe distance ν > 0.
+Lemma A.2. Fix ǫ, ν > 0 and let the deformation D ∈ CN×N satisfy Assumption 2.7. Let z1, z2 ∈ C \ R
+be spectral parameters with ∆ := |ℜz1 − ℜz2| + |ℑz1| + |ℑz2| ≥ ν > 0 and B1, B2 ∈ CN×N bounded
+deterministic matrices. Then, uniformly in η := min
+�
+|ℑz1|, |ℑz2|
+�
+≥ N −1+ǫ, it holds that
+���
+G(z1)B1G(z2)B2
+��� ≺ 1 .
+(A.5)
+Proof. The proof is very similar to that of Proposition 4.4, relying on a system of master inequalities
+(Proposition 6.3) complemented by the reduction inequalities (Lemma 6.4), we just comment on the minor
+differences.
+Recall that the naive size in averaged sense for a chain (4.3) with k resolvents and arbitrary deterministic
+matrices in between is of order η−k+1; generically this is the size of the corresponding deterministic term
+in the usual multi-resolvent local law (see [18, Theorem 2.5] with a = 0 for the case of Wigner matrices)
+|⟨G1B1 · · · GkBk − M(z1, B1, ..., zk)Bk⟩| ≺
+1
+Nηk
+���
+�
+G1B1 · · · BkGk+1 − M(z1, B1, ..., Bk, zk+1)
+�
+xy
+��� ≺
+1
+√Nη ηk
+(A.6)
+with the customary short hand notations
+Gi := G(zi) ,
+η := min
+i
+|ℑzi| ,
+zk := (z1, ..., zk) ,
+Bk := (B1, ..., Bk) .
+In the following, we will consider every deterministic matrix Bj together with its neighbouring resolvents,
+GjBjGj+1, which we will call the unit of Bj. For every distinct10 unit GjBjGj+1 in the initial resolvent
+chain with a regular Bj, the naive size of M gets reduced by an η-factor, yielding the bound η−⌊k/2⌋+1
+in (4.14) when all matrices are regular, with an (almost) matching improvement in the error (for odd k,
+the error is bigger by a factor η−1/2). However, if the spectral parameters zj and zj+1 are far away (as
+quantified by the parameter ν > 0 in (A.9) below), then any matrix Bj in the chain . . . GjBjGj+1 . . .
+behaves as if it were regular because the corresponding stability operator has no singular direction (see
+(A.10) below). In fact, when Bj is the identity, then this effect is seen even easier by the resolvent identity
+GjGj+1 = (Gj − Gj+1)/(zj − zj+1). Therefore, when mimicking the proof of Proposition 4.4, instead
+of counting well separated regular matrices (i.e. their corresponding units are distinct), we need to count
+distinct units within the chain (4.3) for which the corresponding spectral parameters are far away; their
+overall effects are the same – modulo the difference, that now the errors for odd k do not get increased by
+η−1/2 when compared to the M-bound.
+To be more precise, in our new setup, where all consecutive spectral parameters are separated by a safe
+distance ν > 0, we introduce the modified11 normalised differences
+�Ψav
+k (zk, Bk) := Nη⌊k/2⌋|⟨G1B1 · · · GkBk − M(z1, B1, ..., zk)Bk⟩| ,
+(A.7)
+�Ψiso
+k (zk+1, Bk, x, y) :=
+�
+Nη η⌊k/2⌋ ���
+�
+G1B1 · · · BkGk+1 − M(z1, B1, ..., Bk, zk+1)
+�
+xy
+���
+(A.8)
+for k ∈ N as a new set of basic control quantities (cf. (6.1) and (6.2)). The deterministic counterparts
+used in (A.7) and (A.8) are again given recursively in Definition 4.1, and contrarily to (6.1) and (6.2), the
+10This means that sharing a resolvent is not allowed. But in the averaged case, one of the k resolvents can be “reused” in this
+counting of units.
+11Notice that for odd k, the η-power in the prefactor is slightly different from those in (6.1) and (6.2).
+
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+29
+deterministic matrices ∥Bj∥ ≤ 1, j ∈ [k], are not assumed to be regular. This “lack of regularity” is
+compensated by the requirement that consecutive spectral parameters zj, zj+1 of the unit of Bj satisfy
+∆j := |ℜzj − ℜzj+1| + |ℑzj| + |ℑzj+1| ≥ ν > 0 .
+(A.9)
+Just as in Definition 4.2, in case of (A.7), the indices in (A.9) are understood cyclically modulo k. Chains
+satisfying (A.9) for all j ∈ [k] are called “good”. Hence, in a “good” chain one can potentially gain a factor
+η from every unit GjBjGj+1. Therefore, analogous to the regularity requirement for all deterministic
+matrices in Definition 6.1, the normalised differences �Ψav/iso
+k
+in (A.7) and (A.8) will only be used for
+“good” chains.
+As already indicated above, the analogy between our new setup and the setup of Proposition 4.4 is not
+perfect due to the following reason: For k = 1 the error bounds in (A.6) improve by √η for B1 being a
+regular matrix, but for ∆1 ≥ ν > 0, the improvement is by a full power of η.12 This discrepancy causes
+slightly different η-powers for odd k in all estimates (cf. (A.7) and (A.8)).
+We now claim that for “good” chains, the requirement (A.9) for all j ∈ [k] reduces the naive sizes of the
+errors in the usual multi-resolvent local laws (A.6) at least by a factor η⌈k/2⌉ for k = 1, 2. Previously, in the
+proof of Proposition 4.4, these sizes got reduced by a factor √η for every matrix Bj which was regular in
+the sense of Definition 4.2. Now, compared to this regularity gain, the main effect for our new, say, ν-gain
+for “good” chains is that for every j ∈ [k], the inverse of the stability operator (4.6) (explicitly given in
+(A.12) and (A.13) below) is bounded, i.e.
+∥B−1
+j,j+1[R]∥ ≲ ∥R∥
+for all
+j ∈ [k] ,
+R ∈ CN×N .
+(A.10)
+Armed with (A.10), completely analogously to Proposition 4.4, one then starts a proof of the mas-
+ter inequalities (similar to those in Proposition 6.3): First, one establishes suitable underlined lemmas
+(cf. Lemma 6.5 and [20, Lemmas 5.2, 5.6, 5.8, and 5.9]), where now no splittings of observables into
+singular and regular parts (see, e.g., (6.18) and [20, Equation (5.35)]) are necessary, since the bounded ma-
+trices Bj are arbitrary. Afterwards, the proof proceeds by cumulant expansion (see (6.14)), where resolvent
+chains of length k are estimated by resolvent chains of length up to 2k. This indicated infinite hierarchy
+is handled by suitable reduction inequalities, completely analogously to Lemma 6.4. However, along this
+procedure, we also create non-“good" chains, but a direct inspection shows that there are always sufficiently
+many "good" chains providing the necessary improvements, exactly as in the proof of Proposition 4.4. For
+example, in the Gaussian term appearing in the cumulant expansion of (A.7) for k = 2, analogous to (6.14),
+we encounter the term
+|⟨G2B2G1B1G2G1B1G2B2⟩|
+N 2
+≺ |⟨G2B2G1B1G2B1G2B2⟩| + |⟨G2B2G1B1G1B1G2B2⟩|
+N 2
+,
+(A.11)
+where we used a resolvent identity for G2G1 and that ∆ = |ℜz1 −ℜz2|+|ℑz1|+|ℑz2| ≥ ν > 0. Now, the
+two chains on the rhs. of (A.11) cannot be cast in the form (A.7), since the indices of spectral parameters
+are not alternating (these chains are not “good"). However, after application of a reduction inequality (see
+(A.3)), we find that
+|⟨G2B2G1B1G2B1G2B2⟩| ≺ N
+�
+⟨|G2|B2|G1|B∗
+2⟩⟨|G1|B1|G2|B∗
+1⟩⟨|G2|B1|G2|B∗
+1⟩⟨|G2|B2|G2|B∗
+2⟩
+�1/2
+and analogously for the second summand in (A.11). Estimating the two shorter chains involving only G2
+by 1/η via a trivial Schwarz inequality, this yields that
+|⟨G2B2G1B1G2G1B1G2B2⟩|
+N 2
+≺
+�
+⟨|G2|B2|G1|B∗
+2⟩⟨|G1|B1|G2|B∗
+1⟩
+�1/2
+Nη
+,
+where the remaining shorter chains are again “good” and can hence eventually be cast in terms of �Ψav
+2 . A
+similar mechanism works for any other term. This completes the discussion of the discrepancies between
+the current setup and Proposition 4.4.
+Notice that this argument always assumes that we have a single resolvent local law and that M’s are
+bounded. At potential cusps in the scDos ρ we do not have a single resolvent local law (see the discussion
+below (4.1)) and the estimates on ∥M(z)∥ for ℜz close to edges (and cusps) of ρ may deteriorate for
+general deformation D. However, these two phenomena are simply excluded by Assumption 2.7 on the
+12For the averaged case, this improvement is really artificial, since the ∆1 ≥ ν-requirement means that η ≳ 1.
+
+30
+GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES
+deformation D (see also Remark 2.10). In particular, this assumption allows us to show exactly same
+estimates on M(z1, ..., zk) as given in Lemma 4.3, which serve as an input in the proof sketched above.
+To conclude, similarly to Proposition 4.4, our method again shows that
+����
+G(z1)B1G(z2) − M(z1, B1, z2)
+�
+B2
+��� ≺
+1
+(Nη)1/2
+which, together with the corresponding bound
+��⟨M(z1, B1, z2)B2⟩
+�� ≲ 1, immediately yields the desired
+bound and completes the proof of Lemma A.2.
+□
+A.2. Proof of Lemma 4.3. The proof is completely analogous to the proof of Lemma 4.2 from [20], hence
+we only show how the three main technical aspects of the latter should be adjusted to our setup of deformed
+Wigner matrices.
+Recursive Relations: The principal idea is to derive several different recursive relations for M(z1, ..., zk)
+(which itself is given by one of those in Definition 4.1) by a so-called meta argument [21, 20], which can
+then be employed to prove Lemma 4.3 iteratively in the number of spectral parameters. These recursive
+relations are identical to those in [20, Lemma C.2] when dropping the σ = −1 terms in [20, Eqs. (C.2) and
+(C.3)] and writing the N × N identity instead of E+.
+Stability Operator: The inverse of the stability operator (4.6) can be expressed in the following explicit form
+B−1
+12 [R] = R +
+⟨R⟩
+1 − ⟨M1M2⟩M1M2 = R +
+1
+β12
+⟨R⟩M1M2 ,
+(A.12)
+where β12 := 1 − ⟨M1M2⟩ is the only non-trivial eigenvalue of B12. Completely analogously to [20,
+Lemma A.5 (b)], it holds that
+|β12| ≳
+�
+|ℜz1 − ℜz2| + |ℑz1| + |ℑz2|
+�
+∧ 1 ,
+(A.13)
+which, in combination with (A.12), in particular implies (4.13) for k = 1 and (4.14) for k = 2.
+Longer Chains: In order to prove (4.13) for k = 3, similarly to [20] we verify at first (4.13) for k = 2 in
+the case when exactly one observable is regular. For this purpose we again use the recursive relation of the
+form [20, Equation (C.2)]:
+M(z1, A1, z2, A2, z3) = M(z1, X12[A1]M2A2, z2) + M(z1, X12[A1]M2, z3)⟨M(z2, A2, z3)⟩ ,
+where we denoted the linear operator Xmn as
+Xmn[R] :=
+�
+1 − ⟨Mm · Mn⟩
+�−1[R]
+for
+R ∈ CN×N .
+(A.14)
+Now, similarly to the arguments around [20, Equation (C.14)], we observe a balancing cancellation in the
+last term, which comes from the continuity with respect to one of spectral parameters of the regular part of
+a deterministic matrix when another spectral parameter is fixed (see (4.12)).
+□
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+
diff --git a/atE4T4oBgHgl3EQfoA2t/content/tmp_files/load_file.txt b/atE4T4oBgHgl3EQfoA2t/content/tmp_files/load_file.txt
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@@ -0,0 +1,1981 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf,len=1980
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='05181v1  [math-ph]  12 Jan 2023  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER  MATRICES  GIORGIO CIPOLLONI∗  LÁSZLÓ ERD ˝OS†,#  JOSCHA HENHEIK†,#  OLEKSII KOLUPAIEV†  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The total energy of an eigenstate in a composite quantum system tends to be distributed equally  among its constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We identify the quantum fluctuation around this equipartition principle in the simplest  disordered quantum system consisting of linear combinations of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' As our main ingredient, we  prove the Eigenstate Thermalisation Hypothesis and Gaussian fluctuation for general quadratic forms of the  bulk eigenvectors of Wigner matrices with an arbitrary deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' INTRODUCTION  The general principle of the equipartition of energy for a classical ergodic system asserts that in equilib-  rium the total energy is equally distributed among all elementary degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' A similar principle  could be verified in some special quantum cases, see [6] and extensive literature therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Motivated by  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Wigner’s original vision to model any sufficiently complex quantum system by random matrices, a  particularly strong microcanonical version of the equipartition principle for Wigner matrices was first for-  mulated and proven in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In its simplest form, consider a fully mean field random Hamilton operator H in  the high dimensional quantum state space CN that consists of the sum of two independent N × N Wigner  matrices,  H = W1 + W2 ,  as two constituents of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Recall that Wigner matrices W = (wij) are real or complex Hermitian  random matrices with independent (up to the symmetry constraint wij = ¯wji), identically distributed en-  tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let u be a normalised eigenvector of H with eigenvalue (energy) λ = ⟨u, Hu⟩, then equipartition  asserts that ⟨u, Wlu⟩ ≈ 1  2λ for l = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4] even a precise error bound was proven, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  that  ���⟨u, Wlu⟩ − 1  2λ  ��� ≤ N ǫ  √  N  ,  l = 1, 2 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)  holds with very high probability for any fixed ǫ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this estimate is optimal up the N ǫ factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The main  result of the current paper is to identify the fluctuation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1), more precisely we will show that  √  N  �  ⟨u, Wlu⟩ − 1  2λ  �  converges to a centred normal distribution as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We also compute its variance that turns out to be  independent of the energy λ but depends on the symmetry class (real or complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The result can easily be  extended to the case when H is a more general linear combination of several independent Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The estimate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) is reminiscent to the recently proven Eigenstate Thermalisation Hypothesis (ETH),  also known as the Quantum Unique Ergodicity (QUE),1 for Wigner matrices in [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] (see  ∗PRINCETON CENTER FOR THEORETICAL SCIENCE PRINCETON UNIVERSITY, PRINCETON, NJ 08544, USA  †IST AUSTRIA, AM CAMPUS 1, 3400 KLOSTERNEUBURG, AUSTRIA  E-mail address: gc4233@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='edu, lerdos@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='at, joscha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='henheik@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='at,  oleksii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='kolupaiev@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' 60B20, 82B10, 58J51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Quantum unique ergodicity, Matrix Dyson equation, Local law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  #Supported by ERC Advanced Grant “RMTBeyond” No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' 101020331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  1ETH for Wigner matrices was first conjectured by Deutsch [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Quantum ergodicity has a long history in the context of the  quantisations of chaotic classical dynamical systems starting from the fundamental theorem by ˘Snirel’man [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For more background  and related literature, see the Introduction of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  1  2  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  also [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6] for an improvement) which asserts that  ���⟨u, Au⟩ − ⟨A⟩  ��� ≤ N ǫ  √  N  ,  ⟨A⟩ := 1  N TrA ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  holds for any bounded deterministic matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In fact, even the Gaussian fluctuation of  √  N  �  ⟨u, Au⟩ − ⟨A⟩  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)  was proven in [17, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] and [19, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8], see also [9] and the recent generalisation [8] to  off-diagonal elements as well as joint Gaussianity of several quadratic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Earlier results on ETH [23,  29, 12, 10] and its fluctuations [13, 28, 35, 33] for Wigner matrices typically concerned rank one or finite  rank observables A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Despite their apparent similarities, the quadratic form in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) is essentially different from that in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  since Wl is strongly correlated with u while A in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) is deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This explains the difference in the  two leading terms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' note that 1  2λ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) is energy dependent and it is far from the value ⟨Wl⟩ ≈ 0, which  might be erroneously guessed from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Still, the basic approach leading to ETH (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) is very useful to  study ⟨u, Wlu⟩ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The basic idea is to condition on one of the Wigner matrices, say W2, and consider  H = W1+W2 in the probability space of W1 as a Wigner matrix with an additive deterministic deformation  W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Assume that we can prove the generalisation of ETH (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) for such deformed Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In the  space of W1 this would result in a concentration of ⟨u, W2u⟩ around some quantity f(W2) depending only  on W2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' however, the answer will nontrivially depend on the deformation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' it will not be simply ⟨W2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Once the correct form of f(W2) is established, we can find its concentration in the probability space of W2,  yielding the final answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  To achieve these results we prove more general ETH and fluctuation results for eigenvector overlaps of  deformed Wigner matrices of the general form H = W + D, where W is an N × N Wigner matrix and  D is arbitrary, bounded deterministic matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The goal is to establish the concentration and the fluctuation  of the quadratic form ⟨u, Au⟩ for a normalised eigenvector u of H with a bounded deterministic matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We remark that for the special case of a rank one matrix A = |q⟩⟨q|, ETH is equivalent to the complete  isotropic delocalisation of the eigenvector u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' that |⟨q, u⟩| ≤ N ǫ/  √  N for any deterministic vector  q with ∥q∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For a diagonal deformation D this has been achieved in [31, 30] and for the general  deformation D in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The normal fluctuation of ⟨u, Au⟩ for a finite rank A and diagonal D was obtained  in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  It is well known that for very general mean-field type random matrices H their resolvent G(z) = (H −  z)−1 concentrates around a deterministic matrix M = M(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' such results are called local laws, and we will  recall them precisely in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here M is a solution of the matrix Dyson equation (MDE), which, in case of  H = W + D, reads as  −  1  M(z) = z − D + ⟨M(z)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  Given M, it turns out that  ⟨ui, Auj⟩ ≈ δi,j  ⟨ℑM(λi)A⟩  ⟨ℑM(λi)⟩ ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5)  where λi is the eigenvalue corresponding to the normalised eigenvector ui, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Hui = λiui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Since the  eigenvalues are rigid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' they fluctuate only very little, the right hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) is essentially determinis-  tic and in general it depends on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Similarly to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3), we will also establish the Gaussian fluctuation  around the approximation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For zero deformation, D = 0, the matrix M is constant and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) recov-  ers (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For simplicity, in the current paper we establish all these results only in the  bulk of the spectrum, but similar results may be obtained at the edge and at the possible cusp regime of the  spectrum as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the details are deferred to later works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We now comment on the new aspects of our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) relies on a basic observation  about the local law for H = W + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Its average form asserts that  ���  �  (G(z) − M(z))A  ���� ≤ N ǫ  Nη ,  η := |ℑz| ≫ 1  N ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  holds with very high probability and the error is essentially optimal for any bounded deterministic matrix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, there is a one dimensional subspace of the matrices A for which the estimate improves to  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  3  N ǫ/(N√η), gaining a √η factor in the relevant small η ≪ 1 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For Wigner matrices H = W  without deformation, the traceless matrices A played this special role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The key idea behind the proof of  ETH for Wigner matrices in [15] was to decompose any deterministic matrix as A =: ⟨A⟩ + ˚  A into its  tracial and traceless parts and prove multi-resolvent generalisations of the local law (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) with an error term  distinguishing whether the deterministic matrices are traceless or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For example, for a typical A we have  ⟨G(z)AG(z)∗A⟩ ∼ 1  η  but for the traceless part of A we have  ⟨G(z)˚  AG(z)∗ ˚  A⟩ ∼ 1 ,  η ≫ 1  N ,  with appropriately matching error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In general, each traceless A improves the typical estimate by a  factor √η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ETH then follows from the spectral theorem,  1  N  N  �  i,j=1  η  (λi − e)2 + η2  η  (λj − e)2 + η2 |⟨ui, ˚  Auj⟩|2 = ⟨ℑG(z)˚  AℑG(z)˚  A⟩ ≤ N ǫ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  choosing z = e + iη appropriately with η ∼ N −1+ǫ we obtain that |⟨ui, ˚  Auj⟩|2 ≤ N −1+3ǫ which even  includes an off-diagonal version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  To extend this argument for deformed Wigner matrices requires to identify the appropriate singular  (“tracial”) and regular (“traceless”) parts of an arbitrary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' It turns out that the improved local laws  around an energy e = ℜz hold if A is orthogonal2 to ℑM(e), see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10) for the new definition of ˚  A, which  denotes the regular part of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In this theory the matrix ℑM emerges as the critical eigenvector of a linear  stability operator B = I − M⟨·⟩M ∗ related to the MDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The major complication compared with the  pure Wigner case in [15] is that now the regular part of a matrix becomes energy dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular,  in a multi-resolvent chain ⟨G(z1)A1G(z2)A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='⟩ it is a priori unclear at which spectral parameters the  matrices Ai should be regularised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' it turns out that the correct regularisation depends on both zi and zi+1,  see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10) later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' A similar procedure was performed for the Hermitisation of non-Hermitian i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' matrices  with a general deformation in [20], see [20, Appendix D] for a more conceptual presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Having  identified the correct regularisation, we derive a system of master inequalities for the error terms in multi-  resolvent local laws for regular observables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' a similar strategy (with minor modifications) have been used  in [18, 19] for Wigner matrices and in [20] for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' To keep the presentation short, here we will  not aim at the most general local laws with optimal errors (unlike in [18, 19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Although these would be  achievable with our methods, here we prove only what is needed for our main results on the equipartition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The proof of the fluctuation around the ETH uses Dyson Brownian motion (DBM) techniques, namely  the Stochastic Eigenstate Equation for quadratic forms of eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This theory has been gradually  developed for Wigner matrices in [13, 14, 33], we closely follow the presentation in [17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The extension  of this technique to deformed Wigner matrices is fairly straightforward, so our presentation will be brief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The necessary inputs for this DBM analysis follow from the multi-resolvent local laws that we prove for  deformed Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In a closing remark we mention that the original proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) in [5] was considerably simpler than that  of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This may appear quite surprising due to the complicated correlation between u and W, but a special  algebraic cancellation helped in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Namely, with the notation W := W1 − W2 and G(z) := (H − z)−1,  z = e + iη ∈ C+, a relatively straightforward cumulant expansion showed that ⟨ℑG(z)WℑG(z)W⟩ is es-  sentially bounded3 even for spectral parameters z very close to the real axis, η ≥ N −1+ǫ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Within this cu-  mulant expansion an algebraic cancellation emerged due to the special form of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, exactly as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  we obtain |⟨u, Wu⟩|2 ≲ N 1+ǫη2 = N −1+2ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, it shows that ⟨u, Wu⟩ = ⟨u, W1u⟩−⟨u, W2u⟩  is essentially of order N ǫ/  √  N for every eigenvector of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Since ⟨u, W1u⟩ + ⟨u, W2u⟩ = ⟨u, Hu⟩ = λ,  we immediately obtain the equipartition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Similar idea proved the more general case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note, how-  ever, that this trick does not help in establishing the fluctuations of ⟨u, Wiu⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In fact, the full ETH analysis  2The space of matrices is equipped with the usual Hilbert-Schmidt scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  3This means up to an Nǫ factor with arbitrary small ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  4  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  for deformed Wigner matrices must be performed to establish the necessary a priori bounds for the Dyson  Brownian motion arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Notations and conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For positive quantities f, g we write f ≲ g and f ∼ g if f ≤ Cg or cg ≤  f ≤ Cg, respectively, for some constants c, C > 0 which depend only on the constants appearing in the  moment condition, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For any natural number n we set [n] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We denote vectors by bold-faced lower case Roman letters x, y ∈ CN, for some N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Vector  and matrix norms, ∥x∥ and ∥A∥, indicate the usual Euclidean norm and the corresponding induced matrix  norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For any N × N matrix A we use the notation ⟨A⟩ := N −1TrA to denote the normalised trace of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Moreover, for vectors x, y ∈ CN and matrices A ∈ CN×N we define the scalar product  ⟨x, y⟩ :=  N  �  i=1  xiyi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Finally, we will use the concept of “with very high probability” (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=') meaning that for any fixed  D > 0 the probability of an N-dependent event is bigger than 1 − N −D for N ≥ N0(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We introduce  the notion of stochastic domination (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [24]): given two families of non-negative random variables  X =  �  X(N)(u) : N ∈ N, u ∈ U (N)�  and  Y =  �  Y (N)(u) : N ∈ N, u ∈ U (N)�  indexed by N (and possibly some parameter u in some parameter space U (N)), we say that X is stochasti-  cally dominated by Y , if for all ξ, D > 0 we have  sup  u∈U(N) P  �  X(N)(u) > N ξY (N)(u)  �  ≤ N −D  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)  for large enough N ≥ N0(ξ, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In this case we use the notation X ≺ Y or X = O≺(|Y |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We also use  the convention that ξ > 0 denotes an arbitrary small exponent which is independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' gratefully acknowledge many discussions with Dominik Schröder at the  preliminary stage of this project, especially his essential contribution to identify the correct generalisation  of traceless observables to the deformed Wigner ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' MAIN RESULTS  We consider N×N real symmetric or complex Hermitian Wigner matrices W = W ∗ having single-entry  distributions wab = N −1/2χod, for a > b, and waa = N −1/2χd, where χod and χod are two independent  random variables satisfying the following assumptions:  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We assume that χd is a real centred random variable, that χod is a real or complex ran-  dom variable such that Eχod = 0 and E|χod|2 = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' additionally in the complex case we also assume that  Eχ2  od = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Customarily, we use the parameter β = 1, 2 to indicate the real or complex case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Furthermore, we assume that all the moments of χod and χd exist, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' for any p ∈ N there exists a constant  Cp > 0 such that  E|χod|p + E|χd|p ≤ Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)  The equipartition principle concerns linear combinations of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix k ∈ N and consider  H := p1W1 + · · · + pkWk,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  for some fixed N-independent vector p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', pk) ∈ Rk of weights and for k independent N × N  Wigner matrices Wl, belonging to the same symmetry class (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the off-diagonal random variables χod  are either real or complex for each of the Wl, l ∈ [k]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, denoting by {λi}i∈[N] the eigenvalues of  H, arranged in increasing order, with associated normalised eigenvectors {ui}i∈[N], it is expected that  the total energy ⟨ui, Hui⟩ = λi of the composite system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) is proportionally distributed among the k  constituents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  ⟨ui, plWl ui⟩ ≈  p2  l  ∥p∥2 λi  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  5  for every l ∈ [k], where ∥p∥ :=  � �k  l=1 |pl|2�1/2 denotes the usual ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This phenomenon, known as  equipartition, was first proven in [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4], with an optimal error estimate:  ����⟨ui, plWluj⟩ − δi,j  p2  l  ∥p∥2 λi  ���� ≺  1  √  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  Our main result is the corresponding Central Limit Theorem to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) for i = j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the proof of Gaussian  fluctuations in Equipartition for Wigner matrices – for energies in the bulk of the spectrum of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 (Gaussian Fluctuations in Equipartition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Fix k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , Wk be independent Wigner matrices satisfying Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, all of which  being in the same real (β = 1) or complex (β = 2) symmetry class, and p = (p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , pk) ∈ Rk be N-  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Define H as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) and denote by {λi}i∈[N] the eigenvalues of H, arranged in increasing  order, with associated normalised eigenvectors {ui}i∈[N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, for fixed κ > 0, every index i ∈ [κN, (1 −  κ)N] and l ∈ [k] it holds that  �  βN  2  ∥p∥2  p2  l  �  ∥p∥2 − p2  l  �  �  ⟨ui, plWlui⟩ −  p2  l  ∥p∥2 λi  �  =⇒ N(0, 1)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5)  in the sense of moments,4 where N(0, 1) denotes a real standard Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  By polarisation we will also obtain the following:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Under the assumptions from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, the random vector X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Xk) ∈ Rk with  Xl :=  �  βN  2  �  ⟨ui, plWlui⟩ −  p2  l  ∥p∥2 λi  �  ,  l ∈ [k] ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  is approximately (in the sense of moments) jointly Gaussian with covariance structure  Cov(Xl, Xm) = p2  l  �  δl,m∥p∥2 − p2  m  �  ∥p∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 will follow as a corollary to the Eigenstate Thermalisation Hypothesis  (ETH) and its Gaussian fluctuations for deformed Wigner matrices, which we present as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6 and  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' By a quick inspection of our proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, given in Section 3, it is possible to  generalise the Equipartition principle (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) as well as its Gaussian fluctuations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) to linear combinations  of deformed Wigner matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' each Wl in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) being replaced by Wl + Dl, where Dl = D∗  l is an  essentially arbitrary bounded deterministic matrix (see Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7 later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, for brevity of the  current paper, we refrain from doing so explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ETH and its fluctuations for deformed Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In this section, we consider deformed  Wigner matrices, H = W + D, with increasingly ordered eigenvalues λ1 ≤ λ2 ≤ · · · ≤ λN and corre-  sponding orthonormal eigenvectors u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here, D = D∗ ∈ CN×N is a self-adjoint matrix with  uniformly bounded norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ∥D∥ ≤ CD for some N-independent constant CD > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' While the Eigen-  state Thermalisation Hypothesis (ETH) will be shown to hold for general deformations D, we shall require  slightly stronger assumptions for proving the Gaussian fluctuations (see Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In order to state our results on the ETH and its fluctuations (Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8, respectively), we  need to introduce the concept of regular observables, first in a simple form in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10 (later along  the proofs we will need a more general version in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For this purpose we introduce M(z)  being the unique solution of the Matrix Dyson Equation (MDE)5:  −  1  M(z) = z − D + ⟨M(z)⟩,  ℑM(z)ℑz > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  4Given an N–dependent random variable, we say that XN converges to X∞ in the sense of moments if for any k ∈ N it holds  E|XN|k = E|X∞|k + O(N−c(k)), for some small possibly k–dependent constant c(k) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  5The MDE for very general mean field random matrices has been introduced in [2] and further analysed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The properties we  use here have been summarised in [20, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  6  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  The self consistent density of states (scDos) is then defined as  ρ(e) := 1  π lim  η↓0⟨ℑM(e + iη)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)  We point out that not only ⟨ℑM(e + iη)⟩ has an extension to the real axis, but the whole matrix M(e) :=  limη↓0 M(e+iη) is well defined (see [20, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1 (b)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The scDos ρ is a compactly supported Hölder-  1/3 continuous function on R which is real-analytic on the set {ρ > 0}6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Moreover, for any small κ > 0  we define the κ-bulk of the scDos as  Bκ =  �  x ∈ R : ρ(x) ≥ κ1/3�  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)  which is a finite union of disjoint compact intervals [20, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For ℜz ∈ Bκ it holds that ∥M(z)∥ ≲  1, as easily follows by taking the imaginary part of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5 (Regular observables – One-point regularisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix κ > 0 and an energy e ∈ Bκ in the  bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Given a matrix A ∈ CN×N, we define its one-point regularisation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the energy e, denoted by ˚  Ae,  as  ˚  A = ˚  Ae := A − ⟨ℑM(e)A⟩  ⟨ℑM(e)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10)  Moreover, we call A regular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the energy e, if and only if A = ˚  Ae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Notice that in the analysis of Wigner matrices without deformation, D = 0, in [15, 16, 18, 19], M was  a constant matrix and the regular observables were simply given by traceless matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ˚  A = A − ⟨A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  For deformed Wigner matrices the concept of regular observables depends on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Next, we define the quantiles γi of the density ρ implicitly by  � γi  −∞  ρ(x) dx = i  N ,  i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)  We can now formulate the ETH in the bulk for deformed Wigner matrices which generalises the same result  for Wigner matrices, D = 0, from [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6 (Eigenstate Thermalisation Hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix a bounded deterministic D = D∗ ∈ CN×N  and κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let H = W + D be a deformed Wigner matrix, where W satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, and denote  the orthonormal eigenvectors of H by {ui}i∈[N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, for any deterministic A ∈ CN×N with ∥A∥ ≲ 1, it  holds that  max  i,j  ���⟨ui, ˚  Aγiuj⟩  ��� = max  i,j  ����⟨ui, Auj⟩ − δij  ⟨AℑM(γi)⟩  ⟨ℑM(γi)⟩  ���� ≺  1  √  N  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12)  where the maximum is taken over all i, j ∈ [N] such that the quantiles γi, γj ∈ Bκ defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11) are in  the κ-bulk of the scDos ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  This "Law of Large Numbers"-type result (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) is complemented by the corresponding Central Limit  Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13), which requires slightly strengthened assumptions on the deformation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We assume that D ∈ CN×N is a bounded self-adjoint deterministic matrix such that  (i) the unique solution M(z) to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) is uniformly bounded in norm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' supz∈C ∥M(z)∥ ≤ CM for  some N-independent constant CM > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (ii) the scDos ρ is Hölder-1/2 regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' it does not have any cusps (see Footnote 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The requirements on D in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7 are natural and they hold for typical applications, see Re-  mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10 later for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We can now formulate our result on the Gaussian fluctuations in the ETH  which generalises the analogous result for Wigner matrices, D = 0 from [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  6The scDos has been thoroughly analysed in increasing generality in [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' It is supported on finitely many finite intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Roughly speaking there are three regimes: the bulk, where ρ is well separated away from 0, the edge where ρ vanishes as a square  root at the edges of each supporting interval that are well separated, and the cusp where two supporting intervals (almost) meet and  ρ behaves (almost) as a cubic root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Correspondingly, ρ is locally real analytic, Hölder-1/2, or Hölder-1/3 continuous, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Near the singularities, it has an approximately universal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' No other singularity type can occur and for typical deformation D  there is no cusp regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  7  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 (Fluctuations in ETH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix κ, σ > 0 N-independent constants and let H = W + D be a  deformed Wigner matrix, where W satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1 and D satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Denote the  orthonormal eigenvectors of H by {ui}i∈[N] and fix an index i ∈ [N], such that the quantile γi ∈ Bκ  defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11) is in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, for any deterministic Hermitian matrix A ∈ CN×N with ∥A∥ ≲ 1  (which we assume to be real in the case of a real Wigner matrix) satisfying ⟨(A − ⟨A⟩)2⟩ ≥ σ, it holds that  �  βN  2 Varγi(A)  �  ⟨ui, Aui⟩ − ⟨AℑM(γi)⟩  ⟨ℑM(γi)⟩  �  =⇒ N(0, 1)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13)  in the sense of moments (see Footnote 4), where  Varγi(A) :=  1  ⟨ℑM(γi)⟩2  �  ��˚  AγiℑM(γi)  �2�  − 1  2ℜ  ���  M(γi)  �2 ˚  Aγi�2  1 −  ��  M(γi)  �2�  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)  This variance is strictly positive with an effective lower bound  Varγi(A) ≥ c ⟨(A − ⟨A⟩)2⟩  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15)  for some constant c = c(κ, ∥D∥) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In the following two remarks we comment on the condition ⟨(A − ⟨A⟩)2⟩ ≥ σ and on Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The restriction to matrices satisfying ⟨(A − ⟨A⟩)2⟩ ≥ σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' A − ⟨A⟩ being of high rank, is  technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' It is due to the fact that our multi-resolvent local laws for resolvent chains ⟨G(z1)A1G(z2)A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='⟩  in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 are non-optimal in terms of the norm for the matrices Ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' they involve the Euclidean norm  ∥Ai∥ and not the smaller Hilbert-Schmidt norm  �  ⟨|Ai|2⟩ which would be optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For the Wigner ensem-  ble, this subtlety is the main difference between the main result in [17] for high rank observable matrices A  and its extension to any low rank A in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Following the technique in [19] it would be possible to achieve  the estimate with the optimal norm of A also for deformed Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, we refrain from  doing so, since in our main application, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, A itself will be a Wigner matrix which has high rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We have several comments on Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (i) The boundedness of ∥M(z)∥ is automatically fulfilled in the bulk Bκ (see remark below (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)) or  when ℜz away from the support of the scDos ρ (see [3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5]) without any further condi-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, the uniform (in z) estimate formulated in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7 does not hold for arbitrary  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' A sufficient condition for the boundedness of ∥M∥ in terms of the spectrum of D is given in  [3, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1 (i)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This especially applies if the eigenvalues {di}i∈[N] of D (in increasing order)  form a piecewise Hölder-1/2 regular sequence7, see [3, Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, by eigenvalue  rigidity [2, 26], it is easy to see that any “Wigner-like" matrix D has Hölder-1/2 regular sequence  of eigenvalues with very high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This is important for the applicability of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8  below in the proof of our main result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (ii) The assumption that ρ does not have any cusps is an atypical condition and of technical nature (see  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In case that the sequence of matrices D = DN has a limiting density of states with  single interval support, then also ρ, the scDos of W + D, also has single interval support [11],  in particular, it has no cusps [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Again, this is important for the applicability of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 in  the proof of our main result, in which case D is a Wigner matrix with a semicircle as the limiting  density of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In the following Section 3, we will prove our main result, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, assuming Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8  on deformed Wigner matrices as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' These, in turn, will be proven in Sections 4 and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Both proofs crucially rely on an averaged local law for two resolvents and two regular observables, Propo-  sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, which we prove in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Several additional technical and auxiliary results are deferred to  the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  7In this context, Hölder-1/2 regularity means that |di − dj| ≤ C0(|i − j|/N)1/2 for some universal constant C0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  8  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' FLUCTUATIONS IN EQUIPARTITION: PROOF OF THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2  Note that it is sufficient to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 only for k = 2 with p1, p2 ̸= 0, since we can view the  sum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) as the sum of p1W1 and  k  �  l=2  plWl  d=  � k  �  l=2  p2  l  �1/2  �  W ,  where �  W is a Wigner matrix independent of W1 and the equality is understood in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  As a main step, we shall prove the following lemma, where we condition on W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Under the assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 with k = 2 it holds that  EW1⟨ui, p2W2ui⟩ =  p2  2  ∥p∥2 γi + O≺  �  N −1/2−ǫ�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)  βN  2 VarW1[⟨ui, p2W2ui⟩] = p2  1p2  2  ∥p∥2 + O≺  �  N −ǫ�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  for some ǫ > 0, where γi is the ith quantile of the semicircular density with radius 2∥p∥, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  1  2π∥p∥2  � γi  −∞  �  [4∥p∥2 − x]+dx = i  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Expectation and variance are taken in the probability space of W1, conditioned on W2, while the stochastic  domination in the error term is understood in the probability space of W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' First of all, we note that all requirements for applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 to H = p1W1 +  D, with D = p2W2 for some fixed realisation of W2 in a very high probability event, are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This  follows from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10 and ⟨(W2−⟨W2⟩)2⟩ ≳ 1 with very high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Next, observe that replacing  γi in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) by the eigenvalue λi appearing in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) is trivial by the usual eigenvalue rigidity |γi − λi| ≺ 1/N  for Wigner matrices in the bulk [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Hence, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 shows that, conditioned on a fixed realisation of  W2,  �  βN  2  �  ⟨ui, p2W2ui⟩ −  p2  2  ∥p∥2 λi  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)  is approximately Gaussian with an approximately constant variance (independent of W2) given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Since this holds with very high probability w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' W2, this proves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) for l = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the proof for l = 1 is the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We formulated Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 as a CLT for overlaps ⟨ui, Aui⟩ for a single determin-  istic matrix A, but by standard polarisation it also shows the joint approximate Gaussianity of any p-vector  �  ⟨ui, A1ui⟩, ⟨ui, A2ui⟩, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , ⟨ui, Apui⟩  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  for any fixed k and deterministic observables A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Ap satisfying ⟨(Aj − ⟨Aj⟩)2⟩ ≥ c, j ∈ [p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Namely, using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 to compute the moments of ⟨ui, A(t)ui⟩ for the linear combination A(t) =  �  j tjAj with any real vector t = (t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , tp), we can identify any joint moments of the coordinates of  the vector in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) and we find that they satisfy the (approximate) Wick theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Now we can follow the above proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, but without the simplification k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Conditioning  on W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , Wk and working in the probability space of W1, by the polarisation argument above we find that  not only (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) each Xl from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) is asymptotically Gaussian with a variance independent of W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , Wk,  but they are jointly Gaussian for l = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This is sufficient for the joint Gaussianity of the entire  vector X since �  l Xl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This completes the proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  The proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1 is divided into the computation of the expectation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) and the variance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Computation of the expectation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' As in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 above, we condition on W2  and work in the probability space of W1 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' we consider p2W2 as a deterministic deformation of p1W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  This allows us to use Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 as8  EW1⟨ui, p2W2ui⟩ = ⟨p2W2ℑM2(γi,2))⟩  ⟨ℑM2(γi,2)⟩  + O≺  �  N −1/2−ǫ�  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5)  for some constant ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here M2(z), depending on W2, is the unique solution of the MDE  −  1  M2(z) = z − p2W2 + p2  1⟨M2(z)⟩ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  corresponding to the matrix p1W1 + p2W2, where p2W2 is considered a deformation, and γi,2 is the ith  quantile of the scDos ρ2 corresponding to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The subscript ‘2’ for M2, ρ2 and γi,2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  indicates that these objects are dependent on W2 and hence random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The Stieltjes transform m2(z) of ρ2 is given by the implicit equation  m2(z) := ⟨M2(z)⟩ = 1  p2 �  1  W2 − 1  p2 (z + p2  1m2(z))  �  with the usual side condition ℑz · ℑm2(z) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Applying the standard local law for the resolvent of W2 on  the right hand side shows that  ���m2(z) − 1  p2  msc(w2)  ��� ≺  1  N|ℑw2|,  w2 := 1  p2  (z + p2  1m2(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  where msc is the Stieltjes transform of the standard semicircle law, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' it satisfies the quadratic equation  msc(w)2 + wmsc(w) + 1 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)  with the side condition ℑw · ℑmsc(w) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) w2 is random, it depends on W2, but the  local law for ⟨(W2 − w)−1⟩ holds uniformly in the spectral parameter |ℑw| ≥ N −1, hence a standard  grid argument and the Lipschitz continuity of the resolvents shows that it holds for any (random) w with  |ℑw| ≥ N −1+ξ with any fixed ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Applying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) at w = w2 together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) implies that  −  1  ∥p∥ m2(z) =  z  ∥p∥ + ∥p∥ m2(z) + O≺  �  1  N|ℑw2|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)  We view this relation as a small additive perturbation of the exact equation  −  1  msc  �  z  ∥p∥  � =  z  ∥p∥ + msc  �  z  ∥p∥  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10)  to conclude  ���∥p∥ m2(z) − msc  � z  ∥p∥  ���� ≺  1  N|ℑz| ,  |ℑz| ≥ N −1+ξ ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)  using that |ℑw2| ≳ |ℑz| from the definition of w2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) and that ℑz ·ℑm2(z) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The conclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)  requires a standard continuity argument, starting from a z with a large imaginary part and continuously  reducing the imaginary part by keeping the real part fixed (the same argument is routinely used in the proof  of the local law for Wigner matrices, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11) implies that the quantiles of ρ2 satisfy the usual rigidity estimate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  |γ2,i − γi| ≺ 1  N  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12)  for bulk indices i ∈ [κN, (1 − κ)N] with any N-independent κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Moreover, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11) also implies that  for any z in the bulk of the semicircle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' |ℑmsc(z)| ≥ c > 0 for some c > 0, independent of N, we  have |ℑm2(z)| ≥ c/2 as long as |ℑz| ≥ N −1+ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Using the definition of w2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) again, this shows  |ℑw2| ∼ |ℑw| for the deterministic w :=  1  p2 (z + p2  1msc(z)) for any z with |ℑz| ≥ N −1+ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Feeding this  8Note that Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6 alone would prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) only with ǫ = 0, but the convergence in the sense of moments from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8  gains a factor N−ǫ with a positive ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  10  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  information into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) and viewing it again as a perturbation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10) but with the improved deterministic  bound O≺(1/(N|ℑw|)), we obtain  ���∥p∥ m2(z) − msc  �  z  ∥p∥  ���� ≺  1  N|ℑw| ,  with  w = 1  p2  (z + p2  1msc(z)) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13)  uniformly in |ℑz| ≥ N −1+ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, when z is in the bulk of the semicircle, then we have that  ���∥p∥ m2(z) − msc  �  z  ∥p∥  ���� ≺ 1  N  and this relation holds even down to the real axis by the Lipschitz continuity (in fact, real analyticity) of the  Stieltjes transform m2(z) in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In the following, we will use the shorthand notation A ≈ B for two (families of) random variables A  and B if and only if |A − B| ≺ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Evaluating (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) at z = γi,2, we have  M2(γi,2) = 1  p2 1  W2 − wi,2  ,  wi,2 := 1  p2  (γi,2 + p2  1 m2(γi,2)) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)  and note that wi,2 ≈ wi :=  1  p2 (γi + p2  1 msc(γi)) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) and since γi,2 ≈ γi in the bulk by rigidity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Now we are ready to evaluate the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' By elementary manipulations using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14), we can now  write the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) as  ⟨p2W2ℑM2(γi,2))⟩  ⟨ℑM2(γi,2)⟩  = γi,2 + p2  1  p2  ℑ  �  ⟨(W2 − wi,2)−1⟩2�  ℑ ⟨(W2 − wi,2)−1⟩  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15)  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13), we obtain  ⟨(W2 − wi,2)−1⟩ ≈ ⟨(W2 − wi)−1⟩ ≈ msc(wi)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16)  with very high W2-probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Continuing with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15) and using γi,2 ≈ γi, we thus find  ⟨p2W2ℑM2(γi,2))⟩  ⟨ℑM2(γi,2)⟩  ≈ γi + p2  1  p2  ℑ  �  msc(wi)2�  ℑmsc(wi)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17)  Next, we combine (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) with p2m2(γi) ≈ msc(wi) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16) and find that  msc(wi)2 ≈ −  p2  2  p2  1 + p2  2  �  1 + 1  p2  γi msc(wi)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18)  Hence, plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17) we deduce  ⟨p2W2ℑM2(γi,2))⟩  ⟨ℑM2(γi,2)⟩  ≈  �  1 −  p2  1  p2  1 + p2  2  �  γi =  p2  2  ∥p∥2 γi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  This completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Computation of the variance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' As in the calculation of the expectation in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, we first  condition on W2 and work in the probability space of W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' So, we apply Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 to the matrix p1W1 +  p2W2, where the second term is considered a fixed deterministic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Indeed, using the same  notations as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, this gives that the lhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) equals  p2  2 Varγi,2 (W2) = p2  2  1  ⟨ℑM2(γi,2)⟩2  �  �� ˚  W γi,2  2  ℑM2(γi,2)  �2�  − p2  1  2 ℜ  ���  M2(γi,2)  �2 ˚  W γi,2  2  �2  1 − p2  1  ��  M2(γi,2)  �2�  ��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19)  up to an additive error of order O≺  �  N −ǫ�  , which will appear on the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The factor p2  1 in the  second term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19) is a natural rescaling caused by applying Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 to a deformation of p1W1  instead of a Wigner matrix W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Further we express M2 in terms of a Wigner resolvent G := (W2 − w)−1  and use local laws not only for a single resolvent ⟨G⟩ but also their multi-resolvent versions for ⟨G2⟩ and  ⟨GG∗⟩ (see [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' With a slight abuse of notation we shall henceforth drop the subscript ‘2’ in γi,2 and wi,2  and replace them by their deterministic values γi and wi, respectively, at a negligible error of order N −1  exactly as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that ℑwi ≳ 1 for bulk indices i, so all resolvents below are stable and all  denominators are well separated away from zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this is needed to justify the ≈ relations below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  11  The first term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19) can be rewritten as (here G = G(wi) and msc := msc(wi) for brevity)  �� ˚  W γi  2 ℑM2(γi)  �2�  ≈ 1  2ℜ  �����  wi  p2  −  γi  p2  1 + p2  2  ����  2  ⟨GG∗⟩ −  �wi  p2  −  γi  p2  1 + p2  2  �2  ⟨G2⟩  �  ≈ 1  2ℜ  �����  wi  p2  −  γi  p2  1 + p2  2  ����  2  |msc|2  1 − |msc|2 −  �wi  p2  −  γi  p2  1 + p2  2  �2  m2  sc  1 − m2sc  �  ≈  p4  1  2p2  2(p2  1 + p2  2)2 ℜ  �  1  1 − |msc|2 −  1  1 − m2sc  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='20)  where in the last step we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Similarly for the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19), we have  ��  M2(γi)  �2 ˚  W γi  2  �2  1 − p2  1  ��  M(γi)  �2� ≈  �  1  p2  �  1  p2 G +  �  wi  p2 −  γi  p2  1+p2  2  �  G2� �2  1 − p2  1  p2  2 ⟨G2⟩  ≈ m2  sc(p2  2 − (p2  1 + p2  2)m2  sc)  p2  2(p2  1 + p2  2)2(1 − m2sc) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='21)  Plugging (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='20) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='21) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19) we obtain  p2  2 Varγi(W2) ≈ 2p2  2 + p2γiℜmsc  (ℑmsc)2 p2  1p2  2  2(p2  1 + p2  2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='22)  Taking the imaginary part of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18), we find that |msc|2 ≈  p2  2  p2  1+p2  2 and hence, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) again, we infer  1  p2  1 + p2  2  2p2  2 + p2γiℜmsc  (ℑmsc)2  ≈ ℜ[|msc|2 − m2  sc]  (ℑmsc)2  = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='23)  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='23) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19), this completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This proves Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' MULTI–RESOLVENT LOCAL LAWS: PROOF OF THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6  To study the eigenvectors of H we analyse its resolvent G(z) := (H − z)−1, with z ∈ C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' It is  well known [26, 4] that G(z) becomes approximately deterministic in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Its deterministic  approximation (as a matrix) is given by M(z), the unique solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), in the following averaged and  isotropic sense:  |⟨(G(z) − M(z))B⟩| ≺  1  N|ℑz|,  |⟨x , (G(z) − M(z))y⟩| ≺  1  �  N|ℑz|  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)  uniformly in deterministic vectors ∥x∥ + ∥y∥ ≲ 1 and deterministic matrices ∥B∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' To be precise,  while the local laws (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) hold for ℜz ∈ Bκ and dist(ℜz, supp(ρ)) ≳ 1 for arbitrary bounded self-  adjoint deformations D = D∗ (see [26, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2]), the complementary regime requires the strengthened  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7 on D (see [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that cusps for ρ have been excluded in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7, hence the  complementary regime only consists of edges, which are covered in [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6], under the requirement  that ∥M∥ is bounded – which was also supposed in Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The isotropic bound ⟨x, ℑG(z)x⟩ ≺ 1 from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) immediately gives an (almost) optimal bound on the  delocalisation of eigenvalues: |⟨ui, x⟩| ≺ N −1/2 [31, 30, 26, 4, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, these estimates are not precise  enough to conclude optimal bounds for eigenvector overlaps and generic matrices A as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' in  fact by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) we can only obtain the trivial bound |⟨ui, Auj⟩| ≺ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Instead of the single resolvent local law  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1), we rely on the fact that (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17) below)  N  ��⟨ui, Auj⟩  ��2 ≲ ⟨ℑG(γi + iη)AℑG(γj + iη)A∗⟩ ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  for η ∼ N −1+ǫ, where ǫ > 0 is small but fixed, and γi, γj ∈ Bκ are in the bulk and we estimate the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6 we will use the multi-resolvent local laws from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Multi-resolvent local laws are natural generalisations of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) and they assert that longer products  G1B1G2 · · · Gk−1Bk−1Gk  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)  12  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  of resolvents Gi := G(zi) and deterministic matrices9 B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Bk−1 also become approximately determin-  istic both in average and isotropic sense in the large N limit as long as N|ℑzi| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The deterministic  approximation to the chain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) is denoted by  M(z1, B1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk−1, Bk−1, zk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  It is not simply M(z1)B1M(z2)B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' it cannot be obtained by mechanically replacing each G with M  as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) might incorrectly suggest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Instead, it is defined recursively in the length k of the chain as follows  (see [20, Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]):  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix k ∈ N and let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk ∈ C \\ R be spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' As usual, the correspond-  ing solutions to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) are denoted by M(zj), j ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, for deterministic matrices B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Bk−1 we  recursively define  M(z1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='Bk−1, zk) =  �  B1k  �−1  �  M(z1)B1M(z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5)  +  k−1  �  l=2  M(z1)⟨M(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zl)⟩M(zl, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)  �  ,  where we introduced the shorthand notation  Bmn ≡ B(zm, zn) = 1 − M(zm)⟨·⟩M(zn)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  for the stability operator acting on the space of N × N matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  It turns out that the size of M(z1, B1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , zk) in the relevant regime of small η := minj |ℑzj| is  roughly η−k+1 in the worst case, with a matching error term in the corresponding local law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This blow-up  in the small η regime comes recursively from the large norm of the inverse of the stability operator B1k  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, for a special subspace of observable matrices Bi, called regular matrices, the size of  M(z1, B1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , zk) is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For Wigner matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' for D = 0, the regular observables are  simply the traceless matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' observables B such that ⟨B⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In [15, 16, 18, 19] it was shown that  when the matrices Bi are all traceless, then M(z1, B1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , zk) hence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) are smaller by an ηk/2-factor  than for general Bi;’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The situation for deformed Wigner matrices is more complicated, since the concept of regular observ-  ables will be dependent on the precise location in the spectrum of H, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' dependent on the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' More  precisely, we will require that the trace of A tested against a deterministic energy dependent matrix has to  vanish;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this reflects the inhomogeneity introduced by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Analogously to the Wigner case, in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4  below we will show that resolvent chains (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) are much smaller when the deterministic matrices Bi are  regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Next, we give the definition of regular matrices in the chain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Using the notation A for regular  matrices, we will consider chains of resolvents and deterministic matrices of the form  ⟨G1A1 · · · GkAk⟩  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  in the averaged case, or  �  G1A1 · · · AkGk+1  �  xy  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)  in the isotropic case, with Gi := G(zi) and Ai being regular matrices according to the following Defini-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [20, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]), which generalises the earlier Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 (Regular observables – Two-point regularisation in chains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix a parameter κ > 0 and let  δ = δ(κ, ∥D∥) > 0 be small enough (see the discussion below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Consider one of the two expressions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) for some fixed length k ∈ N and bounded matrices ∥Ai∥ ≲ 1 and let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk+1 ∈ C \\ R be  spectral parameters with ℜzj ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For any j ∈ [k], we denote  1δ(zj, zj+1) := φδ(ℜzj − ℜzj+1) φδ(ℑzj) φδ(ℑzj+1)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)  9We will use the the notational convention, that the letter B denotes arbitrary (generic) matrices, while A is reserved for regular  matrices, in the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  13  where 0 ≤ φδ ≤ 1 is a smooth symmetric bump function on R satisfying φδ(x) = 1 for |x| ≤ δ/2 and  φδ(x) = 0 for |x| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here and in the following, in case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), the indices in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) are understood  cyclically modulo k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (a) For j ∈ [k], denoting sj := −sgn(ℑzjzj+1), we define the (two-point) regularisation of Aj from  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the spectral parameters (zj, zj+1) as  ˚  Azj,zj+1  j  := Aj − 1δ(zj, zj+1)⟨M(ℜzj + iℑwj)AM(ℜzj+1 + sjiℑzj+1)⟩  ⟨M(ℜzj + iℑzj)M(ℜzj+1 + sjiℑzj+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10)  (b) Moreover, we call Aj regular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (zj, zj+1) if and only if ˚  Azj,zj+1  j  = Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  As already indicated above, the two-point regularisation generalises Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5 in the sense that  ˚  Ae±iη,e±iη −→ ˚  Ae ,  and  ˚  Ae±iη,e∓iη −→ ˚  Ae ,  as  η ↓ 0 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)  with a linear speed of convergence, for e ∈ Bκ and any bounded deterministic A ∈ CN×N, where we used  that, by taking the imaginary part of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), M(z)M(z)∗ = ℑM(z)/(⟨ℑM(z)⟩ + ℑz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Moreover, we point out, that the above Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 of the regularisation is identical to [20, Defs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1] when dropping the summand with sjτ = −1 in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) of [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, for spectral  parameters zj, zj+1 satisfying 1δ(zj, zj+1) > 0 (for some δ > 0 small enough), it holds that the denomina-  tor in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10) is bounded away from zero, which shows that the linear map A �→ ˚  A is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Additionally,  we have the following Lipschitz property (see [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3]):  ˚  Az1,z2 = ˚  Aw1,w2 + O  �  |z1 − w1| + |z2 − w2|  �  I,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12)  for any z1, z2, w1, w2 ∈ C \\ R such that ℑziℑwi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' It is important that the error in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) is a constant  times the identity matrix, indicated by O(·)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Next, we give bounds on the size of M(z1, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='Ak−1, zk), the deterministic approximation to the chain  G1A1 · · · Ak−1Gk introduced in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the proof of this lemma is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let k ∈ [4] and z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk+1 ∈ C \\ R be spectral parameters with ℜzj ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Set  η := minj |ℑzj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, for bounded regular deterministic matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Ak (according to Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2),  we have the bounds  ∥M(z1, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Ak, zk+1)∥ ≲  �  1  η⌊k/2⌋  if η ≤ 1  1  ηk+1  if η > 1 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13)  |⟨M(z1, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Ak−1, zk)Ak⟩| ≲  �  1  η⌊k/2⌋−1 ∨ 1  if η ≤ 1  1  ηk  if η > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)  For the presentation of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, the main technical result underlying the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6,  we would only need (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) for k ∈ [2] from the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, the remaining bounds  covered by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 will be instrumental in several parts of our proofs (see Section 6 and Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix ǫ > 0, κ > 0, k ∈ [2], and consider z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , zk+1 ∈ C \\ R with ℜzj ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Consider  regular matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , Ak with ∥Ai∥ ≤ 1, deterministic vectors x, y with ∥x∥ + ∥y∥ ≲ 1, and set  Gi := G(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, uniformly in η := minj |ℑzj| ≥ N −1+ǫ, we have the averaged local law  ��⟨  �  G1A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Gk − M(z1, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)  �  Ak⟩  �� ≺  �  N k/2−1  √Nη  if η ≤ 1  1  Nηk+1  if η > 1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15a)  and the isotropic local law  ��⟨x,  �  G1A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Gk+1 − M(z1, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk+1)  �  y⟩  �� ≺  \uf8f1  \uf8f2  \uf8f3  N (k−1)/2  √  Nη2  if η ≤ 1  1  √  Nηk+2  if η > 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15b)  In Section 6, we will carry out the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 in the much more involved η ≤ 1 regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  For η > 1, the bound simply follows by induction on the number of resolvents in a chain by invoking the  trivial estimate ∥M(z)∥ ≲ 1/|ℑz|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The detailed argument has been carried out in [18, Appendix B] for the  case of Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Having Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 at hand, we can now prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  14  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15a) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) for k = 2 it follows that  |⟨G1A1G2A2⟩| ≺ 1 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16)  for arbitrary regular matrices A1 = ˚  Az1,z2  1  and A2 = ˚  Az2,z1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Now, using that (see [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6] for an  analogous statement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' see also (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12))  ˚  Aγi = ˚  Aγi±iη,γj±2iη + O  �  |γi − γj| + η  �  I = ˚  Aγi±iη,γj∓2iη + O  �  |γi − γj| + η  �  I ,  and analogously for (˚  A∗)γi, we obtain (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [20, Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7])  ⟨ℑG(γi + iη)˚  AγiℑG(γj + 2iη)(˚  A∗)γi⟩ ≺ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Moreover, by spectral decomposition, together with the rigidity of eigenvalues (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [2, 26]) it follows  that (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5])  N|⟨ui, ˚  Aγiuj⟩|2 ≺ (Nη)2⟨ℑG(γi + iη)˚  AγiℑG(γj + 2iη)(˚  A∗)γi⟩ ≺ (Nη)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17)  Choosing η = N −1+ξ/2 for some arbitrary small ξ > 0, we conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' DYSON BROWNIAN MOTION: PROOF OF THEOREM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8  Given the optimal a priori bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12), the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 is very similar to the analysis of the  Stochastic Eigenstate Equation (SEE) in [17, Sections 3-4] and [19, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Even if very similar to  those papers, to make the presentation clearer, here we write out the main steps of the proof and explain the  differences, but we do not write the details;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' we defer the interested reader to [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We also remark that the  proof in [17, 19] heavily relies on the analysis of SEE developed in [13] and extend in [14, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Similarly to [17, 19] we only consider the real case, the complex case is completely analogous and so  omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 dynamically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' we consider the flow  dWt = d �Bt  √  N  ,  W0 = W ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)  with �Bt a real symmetric matrix valued Brownian motion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [13, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that Wt has  a Gaussian component of size  √  t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Wt  d= W0 +  √  tU ,  with U being a GOE matrix independent of W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Denoting by λi(t) the eigenvalues of Wt (labeled in  increasing order) and by ui(t) the corresponding orthonormal eigenvectors, we will prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 for  the eigenvectors ui(T ), with T = N −1+ω, for some small fixed ω > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Since T is very small, the Gaussian  component added in the flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) can easily be removed by a standard Green function comparison (GFT)  argument as in [19, Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  By [13], it is known that the eigenvalues λi(t) and the eigenvectors ui(t) are the unique strong solution  of the following system of stochastic differential equations (SDEs):  dλi(t) = dBii(t)  √  N  + 1  N  �  j̸=i  1  λi(t) − λj(t)dt  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  dui(t) =  1  √  N  �  j̸=i  dBij(t)  λi(t) − λj(t)uj(t) −  1  2N  �  j̸=i  ui  (λi(t) − λj(t))2 dt ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)  where the matrix B(t) = (Bij(t))N  i,j=1 is a standard real symmetric Brownian motion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [13, Defi-  nition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Even if in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 we want to prove a CLT only for diagonal overlaps ⟨ui, Aui⟩, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3), it follows  that there is no closed equation for such quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For this reason, following [14, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3], we study  the evolution of the perfect matching observable (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) below) along the flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  15  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Perfect matching observable and proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We introduce the notation  pij = pij(t) = ⟨ui, Auj⟩ − δijC0 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  with A being a fixed real symmetric deterministic matrix A and C0 being a fixed constant independent of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Note that compared to [17, 19] in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) we define the diagonal pii without subtracting their expectation (see  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) above), but rather a generic constant C0 which we will choose later (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='46) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The reason  behind this choice is that in the current setting, unlike in the Wigner case [17, 19], the expectation of pii is  now i–dependent, hence the flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) below would not be satisfied if we had defined (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) with the centred  pii’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  To study moments of the pij’s we use the particle representation introduced in [13] and further developed  in [14, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='. A particle configuration, corresponding to a certain monomials of pij’s, can be encoded by a  function η : [N] → N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The image ηj = η(j) denotes the particle at the site j, and �  j ηj = n denotes  the total number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Additionally, given a particle configuration η, by ηij, with i ̸= j, we denote  a new particle configuration in which a particle at the site i moved to a new site j, if there is no particle in i  then ηij = η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We denote the set of such configuration by Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Fix a configuration η, then we define the perfect matching observable (see [14, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3]):  fλ,t,C0,C1(η) :=  N n/2  [2C1]n/2  1  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  1  M(η)E  \uf8ee  \uf8f0 �  G∈Gη  P(G)  �����λ  \uf8f9  \uf8fb ,  M(η) :=  N  �  i=1  (2ηi − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5)  with n being the total number of particles in the configuration η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The sum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) is taken over Gη, which  denotes the set of perfect matchings on the complete graph with vertex set  Vη := {(i, a) : 1 ≤ i ≤ n, 1 ≤ a ≤ 2ηi} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We also introduced the short–hand notation  P(G) :=  �  e∈E(G)  p(e),  p(e) := pi1i2 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  where e = {(i1, a1), (i2, a2)} ∈ V2  η, and E(G) denotes the edges of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) we took the  conditional expectation with respect to the entire trajectories of the eigenvalues, λ = {λ(t)}t∈[0,T ] for  some fixed 0 < T ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We also remark that the definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) differs slightly from [17, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)] and  [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)], since we now do not normalise by ⟨(A − ⟨A⟩)2⟩ but using a different constant C1 which we  will choose later in the proof (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='46) below);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this is a consequence of the fact that the diagonal overlaps  pii are not correctly centred and normalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that we did not incorporate the factor 2 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) into the  constant C1, since C1 will be chosen has a normalisation constant to compensate the size of the matrix A,  whilst the factor 2 represents the fact that diagonal overlaps, after the proper centering and normalisation  depending on A and i, would be centred Gaussian random variable of variance two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Furthermore, we  consider eigenvalues paths {λ(t)}t∈[0,T ] which lie in the event  �Ω = �Ωξ, :=  �  sup  0≤t≤T  max  i∈[N] ηf(γi(t))−1|λi(t) − γi(t)| ≤ N ξ�  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  for any ξ > 0, where ηf(γi(t)) is the local fluctuation scale defined as in [27, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In most  instances we will use this rigidity estimate in the bulk regime when ηf(γi(t)) ∼ N −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' at the edges  ηf(γi(t)) ∼ N −2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We recall that here γi(t) denote the quantiles of ρt defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The fact  that the event �Ω holds with very high probability follows by [4, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  By [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6] it follows that fλ,t is a solution of the parabolic discrete partial differential equation  (PDE):  ∂tfλ,t = B(t)fλ,t ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)  B(t)fλ,t =  �  i̸=j  cij(t)2ηi(1 + 2ηj)  �  fλ,t(ηkl) − fλ,t(η)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)  where  cij(t) :=  1  N(λi(t) − λj(t))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10)  16  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  In the remainder of this section we may often omit λ from the notation since the paths of the eigenvalues  are fixed within this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The main result of this section is the following Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, which will readily prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  For this purpose we define a version of ft(η) with centred and rescaled pii:  qλ,t(η) :=  � N  �  i=1  1  Varγi(A)ηi/2  �  N n/2  2n/2(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  1  M(η)E  \uf8ee  \uf8f0 �  G∈Gη  Q(G)  �����λ  \uf8f9  \uf8fb  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)  with ˚  Aγi denoting the regular component of A defined as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10):  ˚  Aγi := A − ⟨AℑM(γi)⟩  ⟨ℑM(γi)⟩ ,  and  Q(G) :=  �  e∈E(G)  q(e),  q(e) := ⟨ui1, ˚  Aγi1 ui2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12)  Note that the definition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) is not asymmetric for i1 ̸= i2, since in this case ⟨ui1, ˚  Aγi1 ui2⟩ =  ⟨ui1, ˚  Aγi2ui2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We now comment on the main difference between qt and ft from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' First of  all we notice the q(e)’s in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) are slightly different compared with the p(e)’s from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, we  choose the q(e)’s in such a way that the diagonal overlaps have very small expectation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' much smaller  than their fluctuations size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The price to pay for this choice is that the centering is i–dependent, hence qt  is not a solution of an equation of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We also remark that later within the proof, C0 from  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) will be chosen as  C0 = ⟨AℑM(γi0)⟩  ⟨ℑM(γi0)⟩  for some fixed i0 such that γi0 ∈ Bκ is in the bulk (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The idea behind this choice is that the  analysis of the flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) will be completely local, we can thus fix a base point i0 and ensure that  the corresponding overlap is exactly centred, then the nearby overlaps for indices |i − i0| ≤ K, for some  N–dependent K > 0, will not be exactly centred, but their expectation will be very small compared to the  size of their fluctuations:  ⟨AℑM(γi0)⟩  ⟨ℑM(γi0)⟩ − ⟨AℑM(γi)⟩  ⟨ℑM(γi)⟩ = O  �K  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  A consequence of this choice is also that the normalisation for qt and ft is different: for qt we chose a nor-  malisation that is i’s dependent, whilst for ft the normalisation C1 is i–independent and later, consistently  with the choice of C0, it will be chosen as  C1 = Varγi0(A)n/2,  which is exactly the normalisation that makes ft(η) = 1 when η is such that ηi0 = n and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For any n ∈ N there exists c(n) > 0 such that for any ǫ > 0, and for any T ≥ N −1+ǫ it  holds  sup  η |qT (η) − 1(n even)| ≲ N −c(n) ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13)  with very high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The supremum is taken over configurations η supported on bulk indices and the  implicit constant in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) depends on n and ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix n ∈ N, an index i such that γi ∈ Bκ is in the bulk, and choose a configuration  η such that ηi = n and ηj = 0 for any j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, together with a standard GFT argument  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [19, Appendix B]), readily implies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The lower bound on the variance (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15) is an explicit calculation relying on the the definition of M  from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, we use that  (i) A and hence ˚  Aγi are self-adjoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (ii) ℑM(γi) ≥ g for some g = g(κ, ∥D∥) > 0 since we are in the bulk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (iii) ⟨˚  AγiℑM(γi)⟩ = 0 by definition of the regularisation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (iv) [ℜM(γi), ℑM(γi)] = 0 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  17  Then, after writing Varγi(A) as a sum of squares and abbreviating ℑM = ℑM(γi), we find  Varγi(A) ≥  ��√  ℑM  �  A − ⟨A(ℑM)2⟩  ⟨(ℑM)2⟩  �√  ℑM  �2�  ⟨(ℑM)2⟩  ≥ g2  ��  A − ⟨A(ℑM)2⟩  ⟨(ℑM)2⟩  �2�  ⟨(ℑM)2⟩  ≥  g2  ⟨(ℑM)2⟩⟨(A − ⟨A⟩)2⟩ ,  where in the last step we used the trivial variational principle ⟨(A − ⟨A⟩)2⟩ = inft∈R⟨(A − t)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This  completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' DBM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Similarly to [17, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1] and [19, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] we introduce an equivalent particle  representation to encode moments of the pij’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, here, and previously in [17, 19], we relied on  the particle representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16) below since our arguments heavily builds on [33], which use this  latter representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Consider a particle configuration η ∈ Ωn, for some fixed n ∈ N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' η is such that �  j ηj = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We now  define the new configuration space  Λn := {x ∈ [N]2n : ni(x) is even for every i ∈ [N]  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)  where  ni(x) := |{a ∈ [2n] : xa = i}|  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15)  for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  By the correspondence  η ↔ x  ηi = ni(x)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16)  it is easy to see that these two representations are basically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The only difference is that x uniquely  determines η, but η determines only the coordinates of x as a multi-set and not its ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  From now on, given a function f defined on Ωn, we will always consider functions g on Λn ⊂ [N]2n  defined by  f(η) = f(φ(x)) = g(x),  with φ: Λn → Ωn, φ(x) = η being the projection from the x-configuration space to the η-configuration  space using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We thus defined the observable  gt(x) = gλ,t(x) := fλ,t(φ(x)) ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17)  with fλ,t from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that gt(x) is equivariant under permutation of the arguments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' it depends on  x only as a multi–set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Similarly we define  rt(x) = rλ,t(x) := qλ,t(φ(x)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18)  We remark that gt and rt are the counterpart of ft and qt, respectively, in the x-configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We can thus now write the flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) in the x–configuration space:  ∂tgt(x) = L(t)gt(x)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19)  L(t) :=  �  j̸=i  Lij(t),  Lij(t)g(x) : = cij(t)nj(x) + 1  ni(x) − 1  �  a̸=b∈[2n]  �  g(xij  ab) − g(x)  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='20)  where  xij  ab := x + δxaiδxbi(j − i)(ea + eb),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='21)  with ea ∈ R2n denoting the unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We remark that this flow is map on functions defined on Λn ⊂  [N]2n which preserves equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  For the following analysis it is convenient to define the scalar product and the natural measure on Λn:  ⟨f, g⟩Λn = ⟨f, g⟩Λn,π :=  �  x∈Λn  π(x) ¯f(x)g(x),  π(x) :=  N  �  i=1  ((ni(x) − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' )2,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='22)  as well as the norm on Lp(Λn):  ∥f∥p = ∥f∥Lp(Λn,π) :=  � �  x∈Λn  π(x)|f(x)|p  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='23)  18  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  The operator L = L(t) is symmetric with respect to the measure π and it is a negative in L2(Λn), with  associated Dirichlet form (see [32, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2]):  D(g) = ⟨g, (−L)g⟩Λn = 1  2  �  x∈Λn  π(x)  �  i̸=j  cij(t)nj(x) + 1  ni(x) − 1  �  a̸=b∈[2n]  ��g(xij  ab) − g(x)  ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Finally, by U(s, t) we denote the semigroup associated to L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' for any 0 ≤ s ≤ t it holds  ∂tU(s, t) = L(t)U(s, t),  U(s, s) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='24)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Short range approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' As a consequence of the singularity of the coefficients cij(t) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='20),  the main contribution to the flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19) comes from nearby eigenvalues, hence its analysis will be com-  pletely local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For this purpose we define the sets  J = Jκ := {i ∈ [N] : γi(0) ∈ Bκ},  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='25)  which correspond to indices with quantiles γi(0) (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)) in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Fix a point y ∈ J 2n, and an N-dependent parameter K such that 1 ≪ K ≪  √  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We remark that  y ∈ J 2n will be fixed for the rest of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Next, we define the averaging operator as a simple  multiplication operator by a “smooth” cut-off function:  Av(K, y)h(x) := Av(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' K, y)h(x),  Av(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' K, y) := 1  K  2K−1  �  j=K  1(∥x − y∥1 < j),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='26)  with ∥x − y∥1 := �2n  a=1 |xa − ya|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For notational simplicity we may often omit K, y from the notation  since they are fixed throughout the proof:  Av(x) = Av(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' K, y)h(x),  Avh(x) = Av((x))h((x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='27)  Additionally, fix an integer ℓ with 1 ≪ ℓ ≪ K, and define the short range coefficients  cS  ij(t) :=  �  cij(t)  if i, j ∈ J and |i − j| ≤ ℓ  0  otherwise,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='28)  where cij(t) is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The parameter ℓ is the length of the short range interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We now define a short–range approximation of rt, with rt defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that in the definition  of the short–range flow (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='29) below there is a slight notational difference compared to [17, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2]  and [19, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]: we now choose an initial condition h0 which depends on r0 rather than g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This  minor difference is caused by the fact that in [17, 19] the observable gt was already centred and rescaled,  while in the current case the centred and rescaled version of gt is given by rt, hence the definition in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='29)  is still conceptually the same as the one in [17, 19] (see also the paragraph above (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='45) for a more detailed  explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We point out that we make this choice to ensure that the infinite norm of the short range  approximation is always bounded by N ξ (see below (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='30)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The short range approximation ht = ht(x) is  defined as the unique solution of the parabolic equation  ∂tht(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ℓ, K, y) = S(t)ht(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ℓ, K, y)  h0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ℓ, K, y) = h0(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' K, y) : = Av(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' K, y)(r0(x) − 1(n even)),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='29)  where  S(t) :=  �  j̸=i  Sij(t),  Sij(t)h(x) := cS  ij(t)nj(x) + 1  ni(x) − 1  �  a̸=b∈[2n]  �  h(xij  ab) − h(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='30)  In the remainder of this section we may often omit K, y and ℓ from the notation, since they are fixed for  the rest of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We conclude this section defining the transition semigroup US(s, t) = US(s, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ℓ)  associated to the short range generator S(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note that ∥ht∥∞ ≤ N ξ, for any t ≥ 0 and any small ξ > 0,  since US(s, t) is a contraction and ∥h0∥∞ ≤ N ξ by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12), as a consequence of ht(x) being supported on  x ∈ J 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' L2–estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' To prove the L∞–bound in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, we first prove an L2–bound in Proposi-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 below and then use an ultracontractivity argument for the parabolic PDE (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19) (see [17, Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4]) to get an L∞–bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' To get an L2–bound we will analyse ht, the short–range version of the observ-  able gt from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17), and then we will show that ht and gt are actually close to each other using the following  finite speed of propagation (see [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4]):  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let 0 ≤ s1 ≤ s2 ≤ s1 + ℓN −1, and f be a function on Λn, then for any x ∈ Λn supported  on J it holds  ���(U(s1, s2) − US(s1, s2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ℓ))f(x)  ��� ≲ N 1+nξ s2 − s1  ℓ  ∥f∥∞ ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='31)  for any small ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The implicit constant in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) depends on n, ǫ, δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  To estimate several terms in the analysis of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='29) we will rely on the multi–resolvent local laws from  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 (in combination with the extensions in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1 in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For this purpose, for a  small ω > 2ξ > 0, we define the very high probability event (see Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  �Ω = �Ωω,ξ :=  �  ei∈Bκ,  |ℑzi|≥N −1+ω  �  n  �  k=2  �  sup  0≤t≤T  ���⟨Gt(z1)˚  A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Gt(zk)˚  Ak⟩ − 1(k = 2)⟨M(z1, ˚  A1, z2)˚  A2⟩  ��� ≤ N ξ+k/2−1  √Nη  �  ∩  �  sup  0≤t≤T  ��⟨Gt(z1)˚  A1⟩  �� ≤  N ξ  N  �  |ℑz1|  � �  �  z1,z2:e1∈Bκ,  |e1−e2|≥c1,|ℑzi|≥N−1+ω  �  sup  0≤t≤T  ��⟨(Gt(z1)B1Gt(z2)B2⟩  �� ≤ N ξ  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='32)  where ˚  A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , ˚  Ak are regular matrices defined as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 (here we used the short–hand notation  ˚  Ai = ˚  Azi,zi+1  i  ),  ⟨M(z1, ˚  A1, z2)˚  A2⟩ = ⟨M(z1)˚  A1M(z2)˚  A2⟩ + ⟨M(z1)˚  A1M(z2)⟩⟨M(z2)˚  A2M(z1)⟩  1 − ⟨M(z1)M(z2)⟩  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='33)  η := min{|ℑzi| : i ∈ [k]}, ρi,t := |ℑ⟨Mt(zi)⟩|, c1 > 0 is a fixed small constant, and B1, B2 are norm  bounded deterministic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We remark that for |e1 − e2| ≥ c1 we have the norm bound ∥M B1  12 ∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Then, by standard arguments (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='30)]), we conclude the bound (recall that �Ωξ from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  denotes the rigidity event)  max  i,j∈J |⟨ui(t), Auj(t)⟩| ≤ N ω  √  N  on �Ωω,ξ ∩ �Ωξ ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='34)  simultaneously for all i, j ∈ J and 0 ≤ t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Additionally, on �Ωω,ξ ∩ �Ωξ it also holds that  |⟨ui(t), Auj(t)⟩| ≤ N ω  �  ⟨ℑG(γi(t) + iη)AℑG(γj(t) + iη)A⟩  Nρi,tρj,t  ≲ N ω  N 1/4 ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='35)  when one among i and j is in the bulk and |i − j| ≥ cN, for some small constant c depending on c1  from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here we used that for the index in the bulk, say i, we have ρi,t ∼ 1 and for the other index  ρj,t ≳ N −1/2 as a consequence of η ≫ N −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We point out that this non optimal bound N −1/4, instead of  the optimal N −1/2, follows from the fact that the bound from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 is not optimal when one of the  two spectral parameters in close to an edge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this is exactly the same situation as in [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='31)] where  we get an analogous non optimal bound for overlaps of eigenvectors that are not in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We are now ready to prove the main technical proposition of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Note the additional term  KN −1/2 in the error E in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='37) compared to [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] and [19, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this is a  consequence of the fact the pii’s in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) are not correctly centred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We stress that the base point y in  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 is fixed throughout the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  20  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For any parameters satisfying N −1 ≪ η ≪ T1 ≪ ℓN −1 ≪ KN −1 ≪ N −1/2, and any  small ǫ, ξ > 0 it holds  ∥hT1(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ℓ, K, y)∥2 ≲ Kn/2E,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='36)  with  E := N nξ  �N ǫℓ  K  + NT1  ℓ  + Nη  ℓ  +  N ǫ  √Nη +  1  √  K  + K  √  N  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='37)  uniformly in particle configurations y such that ya = i0, for any a ∈ [2n] and i0 ∈ J , and eigenvalue  trajectory λ in the high probability event �Ωξ ∩ �Ωω,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof of this proposition is very similar to the one of [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] and [19, Proposi-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4], we thus only explain the main differences here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In the following by the star over � we denote that  the summation runs over two n-tuples of fully distinct indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The key idea in this proof is that in order to  rely on the multi–resolvent local laws (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='32) we replace the operator S(t) in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='29) with the new operator  A(t) :=  ∗  �  i,j∈[N]n  Aij(t),  Aij(t)h(x) := 1  η  � n  �  r=1  aS  ir,jr(t)  �  ∗  �  a,b∈[2n]n  (h(xij  ab) − h(x)),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='38)  where  aij = aij(t) :=  η  N((λi(t) − λj(t))2 + η2),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='39)  and aS  ij are their short range version defined as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='28) with cij(t) replaced with aij(t), and  xij  ab := x +  � n  �  r=1  δxar irδxbr ir  �  n  �  r=1  (jr − ir)(ear + ebr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='40)  The main idea behind this replacement is that infinitesimally S(t) averages only in one direction at a time,  whilst A(t) averages in all direction simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This is expressed by the fact that xij  ab from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='21)  changes two entries of x per time, instead xij  ab changes all the coordinates of x at the same time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' let  i := (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , in), j := (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , jn) ∈ [N]n, with {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , in} ∩ {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , jn} = ∅, then xij  ab ̸= x if and  only if for all r ∈ [n] it holds that xar = xbr = ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Technically, the replacement of S(t) by A(t) can be  performed at the level of Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6 of [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let S(t), A(t) be defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='30) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='38), respectively, and let µ  denote the uniform measure on Λn for which A(t) is reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then there exists a constant C(n) > 0  such that  ⟨h, S(t)h⟩Λn,π ≤ C(n)⟨h, A(t)h⟩Λn,µ ≤ 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='41)  for any h ∈ L2(Λn), on the very high probability event �Ωξ ∩ �Ωω,ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We start noticing the fact that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='29) it follows  ∂t∥ht∥2  2 = 2⟨ht, S(t)ht⟩Λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='42)  Then, combining this with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='41), and using that xij  ab = x unless xar = xbr = ir for all r ∈ [n], we  conclude that  ∂t∥ht∥2  2 ≤ C(n)⟨ht, A(t)ht⟩Λn,µ  = C(n)  2η  �  x∈Λn  ∗  �  i,j∈[N]n  � n  �  r=1  aS  irjr(t)  �  ∗  �  a,b∈[2n]n  ht(x)  �  ht(xij  ab) − ht(x)  �  � n  �  r=1  δxar irδxbr ir  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='43)  Then, proceeding as in the proof of [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5] (see also [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='40)]), we conclude that  ∂t∥ht∥2  2 ≤ −C1(n)  2η  ⟨ht⟩2  2 + C3(n)  η  E2Kn,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='44)  which implies ∥hT1∥2  2 ≤ C(n)E2Kn, by a simple Gronwall inequality, using that T1 ≫ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We point out that to go from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='43) to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='44) the proof is completely analogous to [17, Proof of Propo-  sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5], with the only exception being the proof of [17, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='41), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='43)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We thus now explain how  to obtain the analog of [17, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='41), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='43)] in the current case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The fact that we now have the  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  21  bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='35) rather than the stronger bound N −1/3 as in [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='31)] does not cause any difference in  the final estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We thus focus on the main new difficulty in the current analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' that in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='29) we  choose the initial condition depending on r0 rather than g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We recall that the difference between r0 and g0  is that r0 is defined in such a way all the eigenvector overlaps are precisely centred and normalised in an  i–dependent way, whilst for g0 we can choose the i–independent constant C0, C1 so that only the overlap  corresponding to a certain base point i0 is exactly centred and normalised, whilst the nearby overlaps are  centred and normalised only modulo a negligible error K/N (see also the paragraph below (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) for a  detailed explanation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This additional difficulty requires that to prove the analog of [17, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='41), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='43)]  we need to estimate the error produced by this mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Using that the function f(x) ≡ 1(n even) is in the kernel of L(t), for any fixed x ∈ Γ, and for any fixed  i, a, b, we conclude (recall the notation from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='27))  ht(xij  ab)  = US(0, t)  �  (Avr0)(xij  ab) − (Av1(n even))(xij  ab)  �  = Av(xij  ab)  �  US(0, t)r0(xij  ab) − 1(n even)  �  + O  �N ǫ+nξℓ  K  �  =  �  Av(x) + O  � ℓ  K  �� �  U(0, t)r0(xij  ab) − 1(n even) + O  �N 1+nξt  ℓ  ��  + O  �N ǫ+nξℓ  K  �  = Av(x)  �  gt(xij  ab) − 1(n even)  �  + O  �N ǫ+nξℓ  K  + N 1+nξt  ℓ  + N ξK  √  N  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='45)  where in the definition of gt from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17) we chose  C0 := ⟨AℑM(γi0)⟩  ⟨ℑM(γi0)⟩ ,  C1 := Varγi0 (A),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='46)  and the error terms are uniform in x ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here i0 is the index defined below (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The first three  inequalities are completely analogous to [17, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='41)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We now explain how to obtain the last inequality  at the price of the additional negligible error KN −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Recall the definition of rt from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18), then  we note that for any x supported in the bulk it holds  ∥r0(x) − g0(x)∥∞ ≲ N ξK  √  N  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='47)  for C0, C1 chosen as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='47) together with U(0, t)g0 = gt we prove the last equality in  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We point out that to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='47) we used that  ˚  Aγi0 − ˚  Aγir = O(|γi0 − γir|)I = O(KN −1)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In particular, we used this approximation to compensate the mismatch that only the diagonal overlaps  corresponding to the index i0 are properly centred and normalised in the definition of g0, whilst for nearby  indices we use this approximation to replace the approximate centering C0 and normalisation C1 from  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='46) with the correct one, which is the one in the definition of r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Then proceeding as in the proof of [19, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='41)] we conclude the analog of [17, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='43)]:  ∗  �  j  � n  �  r=1  aS  irjr(t)  �  �  qt(xij  ab) − 1(n even)  �  =  �  j  � n  �  r=1  airjr(t)  � \uf8eb  \uf8ed  N n/2  Varγi0 (A)n/22n/2(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  �  G∈Gηj  P(G) − 1(n even)  \uf8f6  \uf8f8  + O  �N nξ  Nη + N 1+nξη  ℓ  + N ξK  √  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='48)  22  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  Given (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='48), the remaining part of the proof is completely analogous to [17, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='44)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='51)] except for  the slightly different computation  ⟨ℑG(λir1 + iη)˚  Aγi0 ℑG(λir2 + iη)˚  Aγi0 ⟩  Varγi0 (A)  = −  1  4Varγi0 (A)  �  σ,τ∈{+,−}  ⟨G(λir1 + σiη)˚  Aγi0 G(λir2 + τiη)˚  Aγi0 ⟩  = −  1  4Varγi0 (A)  �  σ,τ∈{+,−}  �  G(λir1 + iση)˚  Aγir1 +iση,γir2 +iτηG(λir2 + iτη)˚  Aγir2 +iτη,γir1 +iση�  + O  �  K  N 2η3/2 + K2  N 2η +  1  N√η  �  = ⟨ℑM(γi0)⟩2 + O  �  K  N 2η3/2 + K2  N 2η +  1  √Nη  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='49)  which replaces [17, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='47)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here ˚  Aγir1 ±iη,γir2 ±iη is defined as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We also point out  that in the second equality we used the approximation (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12))  ˚  Aγi0 = ˚  Aγir1 ±iη,γir2 ±iη + O  �  |γir1 − γi0| + |γir2 − γi0| + η  �  I = ˚  Aγir1 ,γir2 + O  �K  N + η  �  I,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='50)  together with (here we present the estimate only for one representative term, the other being analogous)  ⟨G(λir1 + iη)G(λir2 + iη)˚  Aγi0 ⟩ = ⟨G(λir1 + iη)G(λir2 + iη)˚  Aγir2 +iη,γir2 +iη⟩ + O  �  1 + K  Nη  �  = ⟨M(γir2 + iη, ˚  Aγir2 +iη,γir1 +iη, γir1 + iη)⟩ + O  �  1 +  1  Nη3/2 + K  Nη  �  = O  �  1 +  1  Nη3/2 + K  Nη  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='51)  which follows by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='32) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 to estimate the deterministic term, together with the integral repre-  sentation from [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1] (see also (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15) later), to bound the error terms arising from the replacement  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Additionally, in the third equality we used the local for two resolvents from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='32):  − 1  4  �  σ,τ∈{+,−}  �  G(λir1 + iση)˚  Aγir1 +iση,γir2 +iτηG(λir2 + iτη)˚  Aγir2 +iτη,γir1 +iση�  = −1  4  �  σ,τ∈{+,−}  �  M(γir1 + iση, ˚  Aγir1 +iση,γir2 +iτη, γir2 + iτη)˚  Aγir2 +iτη,γir1 +iση�  + O  �  1  √Nη  �  = ⟨ℑM(γi0)⟩2Varγi0(A) + O  �  1  √Nη + K  N  �  ,  with the deterministic term defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='33), and in the second equality we used the approximation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='50)  again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We remark that here we presented this slightly different (compared to [17, 19]) computation only  for chain of length k = 2, the computation for longer chains is completely analogous and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This  concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  We conclude this section with the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Combining the L2–bound on ht from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 and the finite speed of prop-  agation estimates in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 we can enhance this L2–bound to an L∞–bound completely analogously  to the proof of [17, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 ] presented in [17, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  23  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' PROOF OF PROPOSITION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4  Our strategy for proving Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 (in the much more involved η ≤ 1 regime) is to derive a system  of master inequalities (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) for the errors in the local laws by cumulant expansion, then use  an iterative scheme to gradually improve their estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The cumulant expansion naturally introduces  longer resolvent chains, potentially leading to an uncontrollable hierarchy, so our master inequalities are  complemented by a set of reduction inequalities (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) to estimate longer chain in terms of shorter  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We have used a similar strategy in [18, 19] for Wigner matrices, but now, analogously to [20], dealing  with non-Hermitian i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' matrices, many new error terms due to several adjustments of the z-dependent two-  point regularisations need to be handled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' By the strong analogy to [20], our proof of the master inequalities  formulated in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 and given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 will be rather short and focus on the main differences  between [20] and the current setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  As the basic control quantities, analogously to [18, 20], in the sequel of the proof, we introduce the  normalised differences  Ψav  k (zk, Ak) := Nηk/2|⟨G1A1 · · · GkAk − M(z1, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)Ak⟩| ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)  Ψiso  k (zk+1, Ak, x, y) :=  �  Nηk+1  ���  �  G1A1 · · · AkGk+1 − M(z1, A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Ak, zk+1)  �  xy  ���  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  for k ∈ N, where we used the short hand notations  Gi := G(zi) ,  η := min  i  |ℑzi| ,  zk := (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk) ,  Ak := (A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Ak) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  The deterministic matrices ∥Ai∥ ≤ 1, i ∈ [k], are assumed to be regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Ai = ˚  Azi,zi+1, see Defini-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) and the deterministic counterparts used in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) are given recursively in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  For convenience, we extend the above definitions to k = 0 by  Ψav  0 (z) := Nη|⟨G(z) − M(z)⟩| ,  Ψiso  0 (z, x, y) :=  �  Nη  ���  G(z) − M(z)  �  xy  ��  and observe that  Ψav  0 + Ψiso  0  ≺ 1  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)  is the usual single-resolvent local law from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1), where here and in the following the arguments of Ψav/iso  k  shall occasionally be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We remark that the index k counts the number of regular matrices in the  sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Throughout the entire argument, let ǫ > 0 and κ > 0 be arbitrary but fixed, and let  D(ǫ,κ) :=  �  z ∈ C : ℜz ∈ Bκ , N 100 ≥ |ℑz| ≥ N −1+ǫ�  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  be the spectral domain, where the κ-bulk Bκ has been introduced in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Strictly speaking, we would need  to define an entire (finite) family of slightly enlarged spectral domains along which the above mentioned  iterative scheme for proving Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Since this has been carried out in detail in [20]  (see, in particular, [20, Figure 2]), we will neglect this technicality and henceforth assume all bounds on  Ψav/iso  k  to be uniform on D(ǫ,κ) in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1 (Uniform bounds in the spectral domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let ǫ > 0 and κ > 0 as above and let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We  say that the bounds  ��⟨G(z1)B1 · · · G(zk)Bk − M(z1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)Bk⟩  �� ≺ Eav ,  ���  �  G(z1)B1 · · · BkG(zk+1) − M(z1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Bk, zk+1)  �  xy  ��� ≺ Eiso  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5)  hold (ǫ, κ)-uniformly (or simply uniformly)for some deterministic control parameters Eav/iso = Eav/iso(N, η),  depending only on N and η := mini |ℑzi|, if the implicit constant in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) are uniform in bounded deter-  ministic matrices ∥Bj∥ ≤ 1, deterministic vectors ∥x∥, ∥y∥ ≤ 1, and admissible spectral parameters  zj ∈ D(ǫ,κ) satisfying 1 ≥ η := minj |ℑzj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Moreover, we may allow for additional restrictions on the deterministic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For example, we may  talk about uniformity under the additional assumption that some (or all) of the matrices are regular (in the  sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  24  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  Note that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) is stated for a fixed choice of spectral parameters zj in the left hand side, but it is in  fact equivalent to an apparently stronger statement, when the same bound holds with a supremum over the  spectral parameters (with the same constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' While one implication is trivial, the other direction follows  from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5) by a standard grid argument (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', the discussion after [18, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We can now formulate Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, in the language of our basic control quantities Ψav/iso  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 (Estimates on Ψav/iso  1  and Ψav/iso  2  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For any ǫ > 0 and κ > 0 we have  Ψav  1 + Ψiso  1  ≺ 1  and  Ψav  2 + Ψiso  2  ≺  �  Nη  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  (ǫ, κ)-uniformly in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The η ≥ 1 case was already explained right after Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The more  critical η ≤ 1 case immediately follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Master inequalities and reduction lemma: Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We now state the relevant part of  a non-linear infinite hierarchy of coupled master inequalities for Ψav  k and Ψiso  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In fact, for our purposes, it  is sufficient to have only the inequalities for k ∈ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Slightly simplified versions of this master inequalities  will be used in Appendix A for general k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 is given in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 (Master inequalities, see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Assume that for some deterministic  control parameters ψav/iso  j  we have that  Ψav/iso  j  ≺ ψav/iso  j  ,  j ∈ [4] ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  holds uniformly in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then we have  Ψav  1 ≺ 1 + ψav  1  Nη + ψiso  1  + (ψav  2 )1/2  (Nη)1/2  + (ψiso  2 )1/2  (Nη)1/4 ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8a)  Ψiso  1  ≺ 1 + ψiso  1  + ψav  1  (Nη)1/2  + (ψiso  2 )1/2  (Nη)1/4 ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8b)  Ψav  2 ≺ 1 + (ψav  1 )2 + (ψiso  1 )2 + ψav  2  Nη  + ψiso  2  + (ψav  4 )1/2  (Nη)1/2  + (ψiso  3 )1/2 + (ψiso  4 )1/2  (Nη)1/4  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8c)  Ψiso  2  ≺ 1 + ψiso  1  + ψav  1 ψiso  1  + (ψiso  1 )2  Nη  + ψiso  2  + (ψiso  1 ψiso  3 )1/2  (Nη)1/2  + (ψiso  3 )1/2 +(ψiso  4 )1/2  (Nη)1/4  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8d)  again uniformly in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  As shown in the above proposition, resolvent chains of length k = 1, 2 are estimated by resolvent chains  up to length 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In order to avoid the indicated infinite hierarchy of master inequalities with higher and  higher k indices, we will need the following reduction lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 (Reduction inequalities, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' As in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), assume that Ψav/iso  j  ≺ ψav/iso  j  holds for 1 ≤ j ≤ 4 uniformly in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then we have  Ψav  4 ≺ (Nη)2 + (ψav  2 )2 ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)  uniformly in regular matrices, and  Ψiso  3  ≺ Nη  �  1 + ψiso  2  √Nη  � �  1 + ψav  2  Nη  �1/2  ,  Ψiso  4  ≺ (Nη)3/2  �  1 + ψiso  2  √Nη  � �  1 + ψav  2  Nη  �  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10)  again uniformly in regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This is completely analogous to [20, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9] and hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The principle idea is to write  out the lhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10) by spectral decomposition and tacitly employ a Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This  leaves us with shortened chains, where certain resolvents G are replaced with absolute values |G|, which  can be handled by means of a suitable integral representation [20, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]  □  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  25  Now the estimates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) follow by combining Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 in an iterative scheme,  which has been carried out in detail in [20, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This completes the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Proof of the master inequalities in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 is very similar to  the proof of the master inequalities in [20, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Therefore, we shall only elaborate on (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8a) as  a showcase in some detail and briefly discuss (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8b)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8d) afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  First, we notice that [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] also holds for deformed Wigner matrices (see Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In order to formulate it, recall the definition of the second order renormalisation, denoted by underline,  from [20, Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For a function f(W) of the Wigner matrix W, we define  Wf(W) := Wf(W) − �E  ��  W(∂�  W f)(W)  �  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)  where ∂�  W denotes the directional derivative in the direction of �  W, which is a GUE/GOE matrix (depending  on the symmetry class of W) that is independent of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The expectation is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the matrix �  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Note that, if W itself a GUE/GOE matrix, then EWf(W) = 0, while for W with general single entry  distributions, this expectation is independent of the first two moments of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In other words, the underline  renormalises the product Wf(W) to second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We note that �E�  WR�  W = ⟨R⟩ and furthermore, that the directional derivative of the resolvent is given by  ∂�  W G = −G�  WG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For example, in the special case and f(W) = (W + D − z)−1 = G, we thus have  WG = WG + ⟨G⟩G  by definition of the underline in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Under the assumption (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), for any regular matrix A = ˚  A we have that  ⟨(G − M)˚  A⟩ = −⟨WG˚  A′⟩ + O≺ (Eav  1 ) ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12)  for some other regular matrix A′ = ˚  A′, which linearly depends on A (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18) for an explicit formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Here G = G(z) and η := |ℑz|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For the error term we used the shorthand notation  Eav  1  :=  1  Nη1/2  �  1 + ψav  1  Nη  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13)  By simple complex conjugation of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12), we may henceforth assume that z = e + iη with η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The  representation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) will be verified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Now, using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) we compute the even moments of ⟨(G−M)˚  A⟩  as  E |⟨(G − M)A⟩|2p =  ��−E⟨WGA′⟩⟨(G − M)A⟩p−1⟨(G − M)∗A∗⟩p�� + O≺  �  (Eav  1 )2p�  and then apply a cumulant expansion to the first summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The term corresponding to the second cumulant  (also called the Gaussian term) is bounded from above by (a p-dependent constant times)  E  �|⟨GGA′GA⟩| + |⟨G∗GA′G∗A∗⟩|  N 2  |⟨(G − M)A⟩|2p−2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)  The main technical tool to estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) is the following contour integral representation for the square of  resolvent (see [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This is given by  G(z)2 =  1  2πi  �  Γ  G(ζ)  (ζ − z)2 dζ ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15)  where the contour Γ = Γ(z) is the boundary of a half band J × [i˜η, i∞) which is parametrised counter-  clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Here the closed interval J is chosen as Bκ′ for a suitable κ′ ∈ (0, κ) and hence contains e = ℜz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  the parameter ˜η is chosen to be smaller than η/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Applying the integral identity (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='15) to the product GG in the term ⟨GGA′GA⟩ gives us that  |⟨GGA′GA⟩| ≲  ������  �  Γ  ⟨G(ζ)A′GA⟩  (ζ − z)2  dζ  ������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16)  Now, the parameter κ′ in the definition of J = Bκ′ is chosen in such a way that the distance between z  and the vertical parts of the contour Γ is greater than δ > 0 from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 (see also [20, Figure 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Hence, for ζ on the vertical parts of Γ, every matrix is considered regular w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (z, ζ), which allows us to  26  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  use the upper bounds for products of two resolvents and two regular matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Therefore, estimating this  contribution with the aid of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3, we find that  |⟨GGA′GA⟩| ≺  �  1 + ψav  2  Nη  �  +  �  J  |⟨G(x + i˜η)A′G(e + iη)A⟩|  (x − e)2 + η2  dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In this integral over J, the horizontal part of Γ, we decompose A and A′ in accordance to the spectral  parameters of the adjacent resolvents and use the following regularity property:  A = ˚  Ae+iη,e+iη = ˚  Ae+iη,x+i˜η + O(|x − e| + η)I ,  A′ =  ˚  (A′)  e+iη,e+iη =  ˚  (A′)  x+i˜η,e+iη + O(|x − e| + η)I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Now the integral over J is represented as a sum of four integrals: one of them contains two regular matrices,  the rest have at least one identity matrix with a small error factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For the first one we use the same estimates  as for the integral over the vertical part of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For the other terms, thanks to the identity matrix, we use a  resolvent identity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' for G(x + i˜η)IG(e + iη) and note that the  �  |x − e| + η  �  error improves the original  1/η-blow up of  �  J  1  (x−e)2+η2 dx to an only | log η|-divergent singularity, which is incorporated into ‘≺’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  For the term ⟨G∗GA′G∗A∗⟩ from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14), we use a similar strategy: After an application of the Ward  identity G∗G = ℑG/η, we decompose the deterministic matrices A, A′ according to the spectral parameters  of their neighbouring resolvents in the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This argument gives us that each of terms ⟨GGA′GA⟩ and  ⟨G∗GA′G∗A∗⟩ is stochastically dominated by  1  η  �  1 + ψav  2  Nη  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Contributions stemming from higher order cumulants in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) are estimated exactly in the same way as  in [20, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8a) is concluded by applying Young inequalities to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) (see [20,  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Finally we show that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) holds:  Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof of this representation is simpler than the proof of its analogue [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2]  because in the current setting all terms in [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] containing the particular chiral symmetry matrix  E− are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In the same way as in [20] we arrive at the identity  ⟨(G − M)A⟩ = −⟨WGX [A] M⟩ + ⟨G − M⟩⟨(G − M)X[A]M⟩ ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17)  where we introduced the bounded linear operator X[B] :=  �  1 − ⟨M · M⟩  �−1[B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Indeed, boundedness  follows from the explicit formula  X[B] = B + ⟨MBM⟩  1 − ⟨M 2⟩  by means of the lower bound  |1 − ⟨M 2⟩| = |(1 − ⟨MM ∗⟩) − 2i⟨MℑM⟩| ≥ 2⟨ℑM⟩2 ≳ 1 ,  obtained by taking the imaginary part of the MDE (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), in combination with ∥M∥ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Next, completely analogously to [20, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4], we find the decomposition  X[A]M =  �  X[A]M  �◦ + O(η)I = ˚  A′ + O(η)I ,  A′ := X[A]M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18)  Plugging this into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='17), we thus infer  ⟨(G − M)A⟩ = −⟨WGA′⟩ + ⟨G − M⟩⟨(G − M)˚  A′⟩ +  �  −⟨WG⟩ + ⟨G − M⟩2�  O(η),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19)  The second term in the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='19) is obviously bounded by ψav  1 /(N 2η3/2) and in the third term we use  the usual local law (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) to estimate it by O≺  �  N −1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Combining these information gives (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  Notice that the above arguments leading to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8a) are completely identical to the ones required in the  proof of the analogous master inequality in [20, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8] with one minor but key modification:  Every term involving the chiral symmetry matrix E− in [20] is simply absent and hence, with the notation  of [20], all sums over signs �  σ=± · · · collapse to a single summand with E+ ≡ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' With this recipe, the  proofs of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8b)–(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8d) are completely analogous to the ones given in [20, Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5 and Appendix E]  and hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  27  APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ADDITIONAL TECHNICAL LEMMAS  In this appendix, we prove several technical lemmas underlying the proofs our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Bounds on averaged multi-resolvent chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8, especially controlling  the event �Ω in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='32), we need to extend the key estimate  ���  G(z1)˚  A1G(z2)˚  A2  ��� ≺ 1 for ℜz1, ℜz2 ∈ Bκ  from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16), which underlies the proof of the ETH in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6, in two directions: First, we consider  chains with an arbitrary number k of resolvents in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, but we will only need a quite weak sub-  optimal bound which makes its proof quite direct and short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Second, in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, we no longer restrict  ℜz1, ℜz2 ∈ Bκ to the bulk, but assume that |ℜz1 − ℜz2| + |ℑz1| + |ℑz2| ≥ ν for some N-independent  constant ν > 0 and additionally allow for arbitrary (non-regular) matrices A1, A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This second extension  requires Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7 on the deformation D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' the boundedness of M(z) also for ℜz /∈ Bκ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this is  a slightly stronger requirement than just the boundedness of D assumed in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Both extensions  are relevant for constructing the high probability event �Ω in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='32) for the DBM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proofs of  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 are simple extensions and slight adjustments of the arguments leading to  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix ǫ > 0, κ > 0, k ∈ N, and consider z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , zk ∈ C\\R with ℜzj ∈ Bκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Consider regular  matrices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' , Ak with ∥Ai∥ ≤ 1, deterministic vectors x, y with ∥x∥ + ∥y∥ ≲ 1, and set Gi := G(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Define  Gk := �G1A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Ak−1 �GkAk,  �Gj ∈ {Gj, |Gj|} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)  Then, uniformly in η := minj |ℑzj| ≥ N −1+ǫ, we have  ��⟨Gk⟩  �� ≺ N k/2−1  √Nη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' To keep the notation simple we only consider the case when Gk = G1A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Ak−1GkAk, the general  case is completely analogous and so omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We split the proof into three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In Step (i) we first prove  the slightly weaker bound  ��⟨Gk⟩  �� ≺ N k/2−1 for any k ≥ 3 and a similar bound in isotropic sense;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' then,  using Step (i) as an input, we will prove the better estimate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) for k = 3, 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' finally, similarly to Step (i),  we prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) for any k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Step (i): Similarly to the proof of the reduction inequalities in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 (see [20, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9]) we readily  see that for k = 2j:  ⟨(GA)2j⟩ ≲ N  �  ⟨|G|A(GA)j/2−1|G|A(G∗A)j/2−1⟩2  j even,  ⟨|G|A(GA)(j−1)/2|G|A(G∗A)(j−1)/2⟩⟨|G|A(GA)(j−3)/2|G|A(G∗A)(j−3)/2⟩  j odd,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)  and for k = 2j − 1:  ⟨(GA)2j−1⟩ ≲ ⟨|G|A(GA)j−2|G|A(G∗A)j−2⟩1/2⟨|G|A(GA)j−1|G|A(G∗A)j−1⟩1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  We proceed by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' First we use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) for j = 2, together with ⟨GAGA⟩ ≺ 1 from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4,  to get the bound ⟨G4⟩ ≺ N, and then use this bound as an input to obtain ⟨G3⟩ ≺ N 1/2 using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then  proceeding exactly in the same way we prove that if ⟨Gk⟩ ≺ N k/2−1 holds then the same bound holds for  chains of length k + 1 and k + 2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Similarly, in the isotropic chains we prove ⟨x, Gky⟩ ≺ N (k−1)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  this concludes Step (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Step (ii): Given the bounds ⟨Gk⟩ ≺ N k/2−1, ⟨x, Gky⟩ ≺ N (k−1)/2, the estimate in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) for k = 3, 4  immediately follows by writing the equation for Gk, performing cumulant expansion and using the corre-  sponding bounds on M(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This was done in [18, Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5], hence  we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Step (iii): The proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) for k ≥ 5 proceeds by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' We first show that it holds for k = 5, 6 and  then we prove that if it holds for k − 2 and k − 1 then it holds for k and k + 1 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' To keep the notation  short with a slight abuse of notation we denote (GA)k = G1A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Ak−1Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  By Step (ii), it follows that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) holds for k = 3, 4, and for k = 2 we have ⟨GAGA⟩ ≺ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then by  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) we immediately conclude that the same bound is true for k = 6, which together with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) also imply  the desired bound for k = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The key point is that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) splits a longer k-chain (k even) into a product of  shorter chains of length k1, k2 with k1 + k2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' As long as k ≥ 5, at least one of the shorter chain has  28  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  already length at least three, so we gain the factor (Nη)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In fact chains of length k = 2, from which  we do not gain any extra factor, |⟨GAGA⟩| ≺ 1, appear only once when we apply (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) for k = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' But in  this case the other factor is a chain of length four with a gain of a (Nη)−1/2 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Similarly, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4) splits  the long k chain (k odd) into the square root of two chains of length k − 1, and k + 1, and for k ≥ 5 we  have the (Nη)−1/2 factor from both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The induction step then readily follows by using again (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)–(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4)  as explained above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In fact, in most steps of the induction we gain more than one  factor (Nη)−1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' this would allow us to improve the bound (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2), but for the purpose of the present paper  the suboptimal estimate (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  We now turn to the second extension of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='16) allowing for arbitrary spectral parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' not neces-  sarily in the bulk, but separated by a safe distance ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fix ǫ, ν > 0 and let the deformation D ∈ CN×N satisfy Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Let z1, z2 ∈ C \\ R  be spectral parameters with ∆ := |ℜz1 − ℜz2| + |ℑz1| + |ℑz2| ≥ ν > 0 and B1, B2 ∈ CN×N bounded  deterministic matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Then, uniformly in η := min  �  |ℑz1|, |ℑz2|  �  ≥ N −1+ǫ, it holds that  ���  G(z1)B1G(z2)B2  ��� ≺ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof is very similar to that of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, relying on a system of master inequalities  (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) complemented by the reduction inequalities (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4), we just comment on the minor  differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Recall that the naive size in averaged sense for a chain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) with k resolvents and arbitrary deterministic  matrices in between is of order η−k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' generically this is the size of the corresponding deterministic term  in the usual multi-resolvent local law (see [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5] with a = 0 for the case of Wigner matrices)  |⟨G1B1 · · · GkBk − M(z1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)Bk⟩| ≺  1  Nηk  ���  �  G1B1 · · · BkGk+1 − M(z1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Bk, zk+1)  �  xy  ��� ≺  1  √Nη ηk  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6)  with the customary short hand notations  Gi := G(zi) ,  η := min  i  |ℑzi| ,  zk := (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk) ,  Bk := (B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Bk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  In the following, we will consider every deterministic matrix Bj together with its neighbouring resolvents,  GjBjGj+1, which we will call the unit of Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For every distinct10 unit GjBjGj+1 in the initial resolvent  chain with a regular Bj, the naive size of M gets reduced by an η-factor, yielding the bound η−⌊k/2⌋+1  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) when all matrices are regular, with an (almost) matching improvement in the error (for odd k,  the error is bigger by a factor η−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, if the spectral parameters zj and zj+1 are far away (as  quantified by the parameter ν > 0 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) below), then any matrix Bj in the chain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' GjBjGj+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  behaves as if it were regular because the corresponding stability operator has no singular direction (see  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In fact, when Bj is the identity, then this effect is seen even easier by the resolvent identity  GjGj+1 = (Gj − Gj+1)/(zj − zj+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Therefore, when mimicking the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, instead  of counting well separated regular matrices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' their corresponding units are distinct), we need to count  distinct units within the chain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3) for which the corresponding spectral parameters are far away;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' their  overall effects are the same – modulo the difference, that now the errors for odd k do not get increased by  η−1/2 when compared to the M-bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  To be more precise, in our new setup, where all consecutive spectral parameters are separated by a safe  distance ν > 0, we introduce the modified11 normalised differences  �Ψav  k (zk, Bk) := Nη⌊k/2⌋|⟨G1B1 · · · GkBk − M(z1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)Bk⟩| ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7)  �Ψiso  k (zk+1, Bk, x, y) :=  �  Nη η⌊k/2⌋ ���  �  G1B1 · · · BkGk+1 − M(z1, B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', Bk, zk+1)  �  xy  ���  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)  for k ∈ N as a new set of basic control quantities (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The deterministic counterparts  used in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) are again given recursively in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, and contrarily to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2), the  10This means that sharing a resolvent is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' But in the averaged case, one of the k resolvents can be “reused” in this  counting of units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  11Notice that for odd k, the η-power in the prefactor is slightly different from those in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  29  deterministic matrices ∥Bj∥ ≤ 1, j ∈ [k], are not assumed to be regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This “lack of regularity” is  compensated by the requirement that consecutive spectral parameters zj, zj+1 of the unit of Bj satisfy  ∆j := |ℜzj − ℜzj+1| + |ℑzj| + |ℑzj+1| ≥ ν > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9)  Just as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, in case of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), the indices in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) are understood cyclically modulo k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Chains  satisfying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) for all j ∈ [k] are called “good”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Hence, in a “good” chain one can potentially gain a factor  η from every unit GjBjGj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Therefore, analogous to the regularity requirement for all deterministic  matrices in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1, the normalised differences �Ψav/iso  k  in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8) will only be used for  “good” chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  As already indicated above, the analogy between our new setup and the setup of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4 is not  perfect due to the following reason: For k = 1 the error bounds in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) improve by √η for B1 being a  regular matrix, but for ∆1 ≥ ν > 0, the improvement is by a full power of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12 This discrepancy causes  slightly different η-powers for odd k in all estimates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  We now claim that for “good” chains, the requirement (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9) for all j ∈ [k] reduces the naive sizes of the  errors in the usual multi-resolvent local laws (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) at least by a factor η⌈k/2⌉ for k = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Previously, in the  proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, these sizes got reduced by a factor √η for every matrix Bj which was regular in  the sense of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Now, compared to this regularity gain, the main effect for our new, say, ν-gain  for “good” chains is that for every j ∈ [k], the inverse of the stability operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) (explicitly given in  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) below) is bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  ∥B−1  j,j+1[R]∥ ≲ ∥R∥  for all  j ∈ [k] ,  R ∈ CN×N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10)  Armed with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10), completely analogously to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, one then starts a proof of the mas-  ter inequalities (similar to those in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3): First, one establishes suitable underlined lemmas  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5 and [20, Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='8, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='9]), where now no splittings of observables into  singular and regular parts (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='18) and [20, Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='35)]) are necessary, since the bounded ma-  trices Bj are arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Afterwards, the proof proceeds by cumulant expansion (see (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)), where resolvent  chains of length k are estimated by resolvent chains of length up to 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This indicated infinite hierarchy  is handled by suitable reduction inequalities, completely analogously to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, along this  procedure, we also create non-“good" chains, but a direct inspection shows that there are always sufficiently  many "good" chains providing the necessary improvements, exactly as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For  example, in the Gaussian term appearing in the cumulant expansion of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7) for k = 2, analogous to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14),  we encounter the term  |⟨G2B2G1B1G2G1B1G2B2⟩|  N 2  ≺ |⟨G2B2G1B1G2B1G2B2⟩| + |⟨G2B2G1B1G1B1G2B2⟩|  N 2  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11)  where we used a resolvent identity for G2G1 and that ∆ = |ℜz1 −ℜz2|+|ℑz1|+|ℑz2| ≥ ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Now, the  two chains on the rhs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11) cannot be cast in the form (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7), since the indices of spectral parameters  are not alternating (these chains are not “good").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, after application of a reduction inequality (see  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)), we find that  |⟨G2B2G1B1G2B1G2B2⟩| ≺ N  �  ⟨|G2|B2|G1|B∗  2⟩⟨|G1|B1|G2|B∗  1⟩⟨|G2|B1|G2|B∗  1⟩⟨|G2|B2|G2|B∗  2⟩  �1/2  and analogously for the second summand in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Estimating the two shorter chains involving only G2  by 1/η via a trivial Schwarz inequality, this yields that  |⟨G2B2G1B1G2G1B1G2B2⟩|  N 2  ≺  �  ⟨|G2|B2|G1|B∗  2⟩⟨|G1|B1|G2|B∗  1⟩  �1/2  Nη  ,  where the remaining shorter chains are again “good” and can hence eventually be cast in terms of �Ψav  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' A  similar mechanism works for any other term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' This completes the discussion of the discrepancies between  the current setup and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Notice that this argument always assumes that we have a single resolvent local law and that M’s are  bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' At potential cusps in the scDos ρ we do not have a single resolvent local law (see the discussion  below (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1)) and the estimates on ∥M(z)∥ for ℜz close to edges (and cusps) of ρ may deteriorate for  general deformation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' However, these two phenomena are simply excluded by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='7 on the  12For the averaged case, this improvement is really artificial, since the ∆1 ≥ ν-requirement means that η ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  30  GAUSSIAN FLUCTUATIONS IN THE EQUIPARTITION PRINCIPLE FOR WIGNER MATRICES  deformation D (see also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' In particular, this assumption allows us to show exactly same  estimates on M(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk) as given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3, which serve as an input in the proof sketched above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  To conclude, similarly to Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='4, our method again shows that  ����  G(z1)B1G(z2) − M(z1, B1, z2)  �  B2  ��� ≺  1  (Nη)1/2  which, together with the corresponding bound  ��⟨M(z1, B1, z2)B2⟩  �� ≲ 1, immediately yields the desired  bound and completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' The proof is completely analogous to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2 from [20], hence  we only show how the three main technical aspects of the latter should be adjusted to our setup of deformed  Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Recursive Relations: The principal idea is to derive several different recursive relations for M(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=', zk)  (which itself is given by one of those in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='1) by a so-called meta argument [21, 20], which can  then be employed to prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3 iteratively in the number of spectral parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' These recursive  relations are identical to those in [20, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2] when dropping the σ = −1 terms in [20, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2) and  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='3)] and writing the N × N identity instead of E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Stability Operator: The inverse of the stability operator (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='6) can be expressed in the following explicit form  B−1  12 [R] = R +  ⟨R⟩  1 − ⟨M1M2⟩M1M2 = R +  1  β12  ⟨R⟩M1M2 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12)  where β12 := 1 − ⟨M1M2⟩ is the only non-trivial eigenvalue of B12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Completely analogously to [20,  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='5 (b)], it holds that  |β12| ≳  �  |ℜz1 − ℜz2| + |ℑz1| + |ℑz2|  �  ∧ 1 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13)  which, in combination with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12), in particular implies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) for k = 1 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14) for k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Longer Chains: In order to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) for k = 3, similarly to [20] we verify at first (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='13) for k = 2 in  the case when exactly one observable is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' For this purpose we again use the recursive relation of the  form [20, Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='2)]:  M(z1, A1, z2, A2, z3) = M(z1, X12[A1]M2A2, z2) + M(z1, X12[A1]M2, z3)⟨M(z2, A2, z3)⟩ ,  where we denoted the linear operator Xmn as  Xmn[R] :=  �  1 − ⟨Mm · Mn⟩  �−1[R]  for  R ∈ CN×N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)  Now, similarly to the arguments around [20, Equation (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='14)], we observe a balancing cancellation in the  last term, which comes from the continuity with respect to one of spectral parameters of the regular part of  a deterministic matrix when another spectral parameter is fixed (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  □  REFERENCES  [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
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+page_content=' Equipartition principle for Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Forum of Mathematics, Sigma 9, E44 (2021)  [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Bialas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Spiechowicz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Łuczka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Quantum analogue of energy equipartition theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Journal of Physics A: Mathematical  and Theoretical 52, 15LT01 (2019)  [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
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+page_content=' Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Annales de l’Institut Henri  Poincaré - Probabilités et Statistiques 56, 2822–2867 (2020)  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
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+page_content=' Cipolloni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' arXiv: 2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='10694  (2022)  [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Benigni, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Lopatto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Fluctuations in local Quantum Unique Ergodicity for generalized Wigner matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
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+page_content=' Local deformed semicircle law and complete delocalization for Wigner matrices with random potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='  Journal of Mathematical Physics 54, 103504 (2013)  [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Marcinek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' High dimensional normality of noisy eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Thesis, Harvard University (2020)  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Marcinek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' High dimensional normality of noisy eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' Communications in Mathematical Physics 395, 1007–  1096 (2022)  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
+page_content=' ˘Snirel’man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE4T4oBgHgl3EQfoA2t/content/2301.05181v1.pdf'}
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+ML-FEED: Machine Learning Framework for
+Efficient Exploit Detection (Extended version)
+Tanujay Saha
+Electrical and Computer Engineering
+Princeton University
+NJ, USA
+tsaha@princeton.edu
+Tamjid Al Rahat
+Electrical and Computer Engineering
+University of California Los Angeles
+CA, USA
+tamjid@ucla.edu
+Najwa Aaraj
+Cryptography Research Center
+Technology Innovation Institute
+Abu Dhabi, UAE
+najwa@tii.ae
+Yuan Tian
+Electrical and Computer Engineering
+University of California Los Angeles
+CA, USA
+yuant@ucla.edu
+Niraj K. Jha
+Electrical and Computer Engineering
+Princeton University
+NJ, USA
+jha@princeton.edu
+Abstract—Machine learning (ML)-based methods have re-
+cently become attractive for detecting security vulnerability
+exploits. Unfortunately, state-of-the-art ML models like long
+short-term memories (LSTMs) and transformers incur significant
+computation overheads. This overhead makes it infeasible to
+deploy them in real-time environments. We propose a novel ML-
+based exploit detection model, ML-FEED, that enables highly
+efficient inference without sacrificing performance. We develop
+a novel automated technique to extract vulnerability patterns
+from the Common Weakness Enumeration (CWE) and Common
+Vulnerabilities and Exposures (CVE) databases. This feature
+enables ML-FEED to be aware of the latest cyber weaknesses.
+Second, it is not based on the traditional approach of classifying
+sequences of application programming interface (API) calls into
+exploit categories. Such traditional methods that process entire
+sequences incur huge computational overheads. Instead, ML-
+FEED operates at a finer granularity and predicts the exploits
+triggered by every API call of the program trace. Then, it uses
+a state table to update the states of these potential exploits
+and track the progress of potential exploit chains. ML-FEED
+also employs a feature engineering approach that uses natural
+language processing-based word embeddings, frequency vectors,
+and one-hot encoding to detect semantically-similar instruction
+calls. Then, it updates the states of the predicted exploit categories
+and triggers an alarm when a vulnerability fingerprint executes.
+Our experiments show that ML-FEED is 72.9× and 75, 828.9×
+faster than state-of-the-art lightweight LSTM and transformer
+models, respectively. We trained and tested ML-FEED on 79
+real-world exploit categories. It predicts categories of exploit
+in real-time with 98.2% precision, 97.4% recall, and 97.8% F1
+score. These results also outperform the LSTM and transformer
+baselines. In addition, we evaluated ML-FEED on the attack
+traces of CVE vulnerability exploits in three popular Java
+libraries and detected all three reported critical vulnerabilities
+in them.
+I. INTRODUCTION
+Digital platforms worldwide have been facing increasingly
+sophisticated malicious campaigns directed at them. Most
+cyber-attacks are executed by exploiting system vulnerabili-
+ties. Vulnerabilities are weaknesses in a system that hackers
+can exploit them in multiple ways. Vulnerability scanning
+involves detecting such potential weaknesses. This requires
+administrator-level access to the system resources. Exploit
+detection requires more sophistication than vulnerability detec-
+tion. This is because system administrators often do not grant
+permission to exploit detection software to access internal
+resources.
+Researchers have devoted significant efforts towards ad-
+dressing the challenge of detecting vulnerabilities and exploits,
+developing popular methods like fuzzing, symbolic analysis,
+and rule-based testing. Although these methods are quite
+popular, they do have some drawbacks. A significant limitation
+is that most such methods require manual definitions of attack
+signatures and patterns. Manual description of rules limits the
+efficiency of these methods when dealing with large code-
+bases [1]. Moreover, these traditional vulnerability detection
+techniques suffer from a large number of false positives,
+performance issues, and inability to identify the vulnerability
+type [2]. Researchers have overcome these drawbacks by
+integrating machine learning (ML), more specifically deep
+learning, into vulnerability detection techniques [3].
+Using ML requires minimal manual effort. Thus, it expe-
+dites the process. State-of-the-art ML methods, like long-short
+term memories (LSTMs) and transformers, classify application
+programming interface (API) call sequences obtained from
+program execution traces as benign or malignant. They can
+even predict the exploit type. However, these models incur
+huge computation overheads. This diminishes their usefulness.
+This article presents ML-FEED (Machine Learning Frame-
+work for Efficient Exploit Detection) that combines ML and
+non-ML methods to achieve highly efficient exploit classifi-
+cation of execution traces. Next, we discuss the drawbacks
+of state-of-the-art techniques and how ML-FEED overcomes
+them. Transformers analyze the entire sequence of instruction
+calls at every timestep during inference, resulting in huge
+computation overheads. LSTMs use representations of the
+sequence history to reason about the entire sequence at every
+arXiv:2301.04314v1  [cs.CR]  11 Jan 2023
+
+timestep. Although these representations help them capture
+semantic information in the sequence, handling these represen-
+tations incurs large computation overheads. Therefore, unlike
+traditional ML-based sequence models, ML-FEED does not
+analyze the entire sequence of instruction calls at once. It first
+predicts the exploits that an instruction call may trigger. Then,
+it uses this prediction to update the states of the predicted
+exploits in a state table. This state table keeps track of the
+current execution state of every exploit in our threat model.
+When the state of one or more exploits in the table represents
+a vulnerability fingerprint, it raises the alarm for a potential
+attack. Our experiments show that ML-FEED is faster than
+the state-of-the-art lightweight LSTM (transformer) model by
+72.9× (75, 828.9×) and 4.5× (2500.6×) more compact.
+Besides designing a lightweight model for efficient in-
+ference, we also design a general framework for extracting
+threat intelligence from vulnerability databases like Common
+Weakness Enumeration (CWE) and Common Vulnerabilities
+and Exposures (CVE). We use this extracted intelligence to
+construct an exploit sequences database, which we further use
+to train our ML model. This methodology enables us to update
+our ML model with the latest exploits, thus mitigating the
+impact of a lack of sufficient data consisting of heterogeneous
+vulnerabilities.
+The novelties of our approach are as follows.
+1) We propose a hybrid system comprising ML classifiers
+and state tables to classify exploit chains efficiently.
+2) We analyze individual API calls instead of entire API
+call sequences. This analysis at a finer granularity enables
+ML-FEED to demonstrate much higher efficiency than
+state-of-the-art sequence models.
+3) We propose a methodology to automatically extract intel-
+ligence from vulnerability databases and create an exploit
+sequence database to update ML-based exploit detection
+systems with the latest exploits.
+The article is organized as follows. Section II provides the
+background information necessary for understanding the rest
+of the article. Section III describes the methodology for exploit
+classification. Section IV details the results of our experiments.
+Section V discusses related work in this area. Section VI
+concludes the article.
+II. BACKGROUND
+This section discusses background material for program
+representations and ML models.
+A. Call Graph
+A call graph is a control-flow graph representing the calling
+relationships between functions in a program. Fig. 1 shows
+an example of a call graph where each node represents a
+function, and each edge represents the calling relationship
+between the functions. For example, if a function f1 calls f2
+during the program flow, the calling relationship is represented
+by a directed edge (f1,f2).
+0
+1
+2
+3
+7
+4
+6
+5
+𝑓!
+𝑓"
+𝑓#
+𝑓$
+𝑓%
+𝑓&
+𝑓%
+𝑓%
+𝑓&
+𝑓'
+Fig. 1: Example of a call graph.
+B. System Dependence Graph
+We use a System Dependence Graph (SDG) [4] as the
+inter-procedural graph representation of programs to extract
+the vulnerable API sequences for training our model. An
+SDG G = (V, Ec, Ed) is a three-tuple, where V represents
+program statements, Ec encodes control dependency edges,
+and Ed denotes data dependency edges. For nodes X and Y
+in an SDG, Y is control-dependent on X if, during execution,
+X can directly affect whether Y is executed. Also, X is
+data-dependent on Y
+if Y
+is an assignment and X can
+access the value assigned in Y . Since these data and control
+dependencies are computed for individual program procedures
+(or functions), they do not account for inter-procedural control
+and data relationships. As a result, they cannot determine the
+transitive dependencies by default. Therefore, we augment the
+SDG with information on inter-procedural relationships. SDG
+edges also represent the direct dependence between the call
+site and the called procedure, which allows us to determine
+the transitive dependencies due to a call.
+To construct the SDG, a call graph and pointer analysis
+is simultaneously performed over the entire program. A call
+graph is a control-flow graph that represents the calling
+relationships between functions in a program. Each node
+represents a function and each edge represents the calling rela-
+tionship between the functions. Since we target object-oriented
+programs, we use the Class Hierarchy Analysis (CHA) [5] call
+graph algorithm implemented in the Wala framework. On the
+other hand, pointer analysis is a static code analysis technique
+that determines which pointers (i.e., references) can point to
+which variables or storage locations. Since we target Java-
+based program in our analyses, we use an Andersen-style
+pointer analysis, which is available in the Wala toolchain.
+An SDG creates three additional edges to handle function
+calls: (1) call edge, (2) parameter-in edge, and (3) parameter-
+out edge. A call edge connects the node at the call site to the
+entry node of the called procedure. Parameter-in edges connect
+the actual-in parameter nodes to the formal-in parameter nodes
+of the called function, and parameter-out edges connect the
+formal-out nodes to the actual-out nodes. Fig. 2 shows an
+SDG for two functions, f1 and f2, where f1 calls f2. When
+
+𝑓!
+<Entry>
+call 𝑓!
+𝑥
+<actual_in>
+𝑓!
+<Entry>
+<Cond>
+𝑥
+<formal_in>
+𝑦
+<formal_out>
+𝑦
+<actual_out>
+Param-in
+edge
+Param-out
+edge
+Call edge
+1
+2
+data-dep
+control-dep
+Fig. 2: Example of an SDG for call graph nodes 1 and 2. Here,
+an SDG node represents a statement and an edge represents
+the control or data dependency between program statements.
+f1 calls f2 during the program flow, a call edge from the call
+site in caller f1 to the entry node of callee f2 represents this
+calling relationship. Also, the arguments passed by the caller
+are represented by the actual in nodes, and the formal in
+nodes represent the arguments received by the callee. The
+returned value from the caller is represented by the formal out
+node, whereas the value received by the callee represents the
+actual out node. As shown in Fig. 2, the solid and dotted lines
+represent the control and data dependencies between the SDG
+nodes, respectively.
+C. Sequence Models
+Sequence models are ML-based models that can analyze
+data with sequential dependencies. Time-series data, video and
+audio clips, and text streams are a few examples of sequential
+data. A popular deep learning model used for sequential
+modeling is the LSTM. This is because LSTMs are capable
+of handling long-range sequential dependencies. Security an-
+alysts often use LSTMs and their variants to analyze system-
+call sequences for vulnerability detection. More sophisticated
+versions of the LSTM, like bidirectional-LSTMs [6] and CNN-
+LSTMs [7], usually achieve good performance over vanilla
+LSTM models. However, LSTM models require significant
+computation overheads due to their increased complexities.
+Therefore, we compare ML-FEED with the state-of-the-art
+lightweight LSTM model that has reasonable accuracy while
+taking minimal computation time. As shown in Fig. ??, this
+architecture is a cascade of an LSTM cell followed by a
+feedforward neural network.
+Transformers are another category of sequence analysis
+models. Transformers have become popular because they are
+even more successful at capturing long-range dependencies
+with the help of the attention mechanism. This mechanism
+considers the entire series during inference and determines
+how to prioritize the different elements of a sequence. Further-
+more, their advanced semantic knowledge representation en-
+ables sophisticated modeling of semantically-similar functions.
+Although very successful, transformers are often huge mod-
+els that incur enormous computation overheads. Transformer
+architectures that have performed well on natural language
+processing (NLP) tasks have also been successful in system-
+call sequence analysis. Some of the most popular transformer
+architectures used for detecting vulnerabilities include the
+generative pre-trained (GPT) transformer and its variants,
+BERT and BART. Unlike BERT and BART, the GPT family of
+transformers is a decoder-based model. It is easier to model
+our problem statement with a decoder model [8]. Thus, we
+compare the performance of ML-FEED with that of GPT-2.
+III. METHODOLOGY
+We describe our methodology for efficient exploit detection
+in this section. We begin by extracting vulnerable instruction
+sequences from vulnerability databases and publicly available
+vulnerability exploit programs. Then, we convert the individual
+instruction calls into labeled feature vectors. These vectors
+constitute our training data. We use our training data to design
+and train our neural network architecture for the ML-FEED
+exploit classifier.
+Instruction 
+extraction
+Feature 
+engineering
+Test 
+program
+Exploit 
+classifier
+Vulnerability 
+fingerprint 
+executed?
+Yes
+No
+White-listed 
+instruction?
+No
+Sequence 
+monitor
+Fetch next 
+instruction
+Stop 
+execution
+Yes
+Yes
+Fig. 3: Overview of our inference pipeline
+We give a high-level overview of the ML-FEED infer-
+ence pipeline in Fig. 3. First, we extract instruction calls
+sequentially from the test program during inference. If the
+current instruction call is present in the white-list of non-
+harmful functions, we proceed to analyze the next instruction.
+Otherwise, we continue to analyze the current function by
+converting it into a feature vector. The exploit classifier then
+analyzes this feature vector to predict the potential exploits
+that the current instruction call can execute. Finally, this list of
+potential exploits is provided as input to the sequence monitor.
+The sequence monitor tracks the order of the instruction
+calls and raises an alarm if the current instruction executes
+a vulnerability fingerprint. If the current instruction does not
+implement an exploit, we analyze the next instruction of the
+test program execution trace.
+
+During inference, ML-FEED raises an alarm when it detects
+the execution of a CVE/CWE vulnerability fingerprint. Prior
+ML-based methods (like LSTMs and transformers) and non-
+ML-based methods (like subgraph matching) for vulnerability
+detection are not efficient enough to be deployed in real-time.
+Non-ML-based methods incur a time overhead of 28.57× [1].
+As we show later, prior ML-based methods like lightweight
+LSTMs and transformers incur time overheads of 72.86× and
+76178.88×, respectively. We compare the efficiency of ML-
+FEED with prior methods in Section III-B7.
+A. Representation of Vulnerable Programs
+This section describes the extraction of exploit instruction
+sequences from the CVE and CWE databases. We give an
+overview of this extraction procedure in Fig. 4. First, we use
+the CWE database to define a high-level vulnerability finger-
+print. Although these fingerprints can represent the exploit’s
+high-level behavior, they are not sufficient for capturing the
+fine-grained relationships between the program entities (e.g.,
+language-specific APIs). Therefore, we use the CVE database
+to extract the low-level program entities and their relationships
+that trigger the corresponding vulnerabilities. We refer to this
+low-level representation of the vulnerability fingerprint as a
+vulnerability query. Next, we create the vulnerable program’s
+SDG representation. Finally, we perform graph search based
+on the extracted query to obtain the vulnerable program flow.
+1) CWE and CVE databases: We use both CWE and CVE
+databases to analyze vulnerabilities. The CWE database con-
+tains broad categories of system vulnerabilities and their high-
+level descriptions. For example, SQL injection vulnerabilities
+and cross-site request forgery have been assigned CWE-89 and
+CWE-352, respectively. Therefore, we use the CWE descrip-
+tions to extract the high-level semantics of the vulnerabilities
+in our threat model. However, the CWE database does not
+provide the low-level system-specific vulnerability information
+that we require for real-world exploit classification. The CVE
+database contains this low-level information. It contains the
+specific flaws and weaknesses of various systems. Therefore,
+we obtain the system-specific vulnerability information from
+the CVE database.
+For example, SQL injection vulnerability has been assigned
+a CWE, namely CWE-89. However, SQL injection (CWE-89)
+has multiple entries in the CVE database. Each entry corre-
+sponds to a unique, vulnerable software version. Some popular
+CVEs for SQL injection are CVE-2007-6602 and CVE-2017-
+11508. We utilize the high-level vulnerability semantics from
+the CWE database and the low-level vulnerability details from
+the CVE database for further analysis.
+2) Vulnerability fingerprints: We describe the process of
+extracting vulnerability fingerprints from the CWE database
+in this section. First, we analyze the CWE database to define
+the high-level vulnerability semantics. Then, we represent this
+information with the help of graphs. The nodes of these graphs
+represent entities involved in a particular vulnerability. The
+edges represent the relationships between these entities. These
+relationships must hold for the vulnerability to be triggered.
+For example, we can describe the semantics of SQL injection
+(CWE-89) using a fingerprint graph, as shown in Fig. 4,
+where a vulnerability is triggered when an SQL command is
+constructed using user-controlled (external) input and executed
+by the system.
+3) Vulnerability queries: Vulnerability fingerprints defined
+by high-level semantics are insufficient for capturing the
+fine-grained and diverse behaviors of vulnerable programs.
+For example, user-controllable external inputs may trigger a
+vulnerability (e.g., SQL injection) in databases. However, user-
+controllable information can originate from various sources,
+such as files or network streams. Similarly, different programs
+might use other APIs for executing SQL commands. To cap-
+ture this low-level information, we transform the vulnerability
+fingerprints into vulnerability queries. To construct the queries,
+we further analyze the CVEs of the corresponding CWE. We
+extract the relevant entities and their relationships that exploit
+a vulnerability from the CVE database. Then, we map the enti-
+ties to their related programming language-specific elements.
+For example, as shown in Fig. 4, we map a user-controlled
+external input from files to the corresponding APIs (e.g.,
+String BufferedReader.readLine()) for reading the input file.
+This query representation allows us to check for vulnerable
+sequences in the SDG representation of the program.
+4) Vulnerable instruction sequences:
+A vulnerable se-
+quence is a series of instructions that exploits a vulnerability.
+In this section, we describe the process of generating such
+sequences. As illustrated in Fig. 4, first, we perform pointer
+analysis and obtain the precise call graph for an input program.
+The call graph contains the calling relationships between
+functions in the program. Then, we process the call graph to
+obtain the program’s SDG. The SDG captures both control-
+and data-flow dependencies of a program.
+Finally, we search the SDG for the vulnerability queries
+that we generated before. This graph search outputs the
+sequence of basic blocks and instructions representing the
+program segment that triggers the vulnerability. We can do
+this analysis for vulnerability exploits in any programming
+language. However, we have analyzed Java-based exploits with
+the Wala code disassembler for our experiments.
+In this work, we use the SDG to extract vulnerable API
+calls from the programs in our dataset. Prior research utilizes
+other graph representations, along with the SDG, to detect
+vulnerabilities and explore various properties of the program.
+For example, SSLint [9] uses a Program Dependence Graph
+(PDG) that combines both Control Flow Graphs (CFGs) and
+Data Flow Graphs to perform intra-procedural analysis for
+detecting vulnerable properties within a function in an SSL
+program. Yamaguchi et al. [2] combine an Abstract Syntax
+Tree (AST), CFG, and PDG to introduce a new graph represen-
+tation called Code Property Graph to analyze lower-level code
+properties, such as expressions and conditions in a program.
+While all these representations have proven to be effective
+for various applications, the SDG representation is the best
+candidate for our problem. An SDG, being an inter-procedural
+representation of the PDG, combines the control flow and data
+
+String BufferedReader.readLine()
+Statement Connection.createStatement()
+boolean Statement.execute(String)
+Data flow
+Data flow
+Input 
+program
+Generate 
+vulnerability 
+queries
+Execute
+graph 
+search
+for 
+query
+SDG representation 
+of the input program
+Query for a SQL injection vulnerability
+Sequence(s) of vulnerable instructions
+CWE-based fingerprint 
+graph for SQL Injection
+Pointer
+analysis
+Call graph
+Generate SDG
+External 
+Input
+Execute
+Data flow
+Data flow
+SQL
+An attacker could exploit this
+vulnerability by uploading a file
+containing an SQL query to the
+configuration dashboard.
+Fig. 4: Extracting instruction sequences from CVE and CWE databases and source codes of exploits.
+flow relationships between the APIs in a program. This inter-
+procedural representation of vulnerable programs enables us
+to capture the holistic program execution environment.
+However, SDG construction can be imprecise for certain
+program features (e.g., reflected function calls) and can gen-
+erate sequences that may not occur at run-time in the given
+program. Fortunately, this is an advantage for the generality
+of our training model. In other words, since an imprecise (i.e.,
+overapproximated) SDG for certain program components can
+represent a program path that may not occur at run-time, it
+allows our model to learn those paths as well, since they could
+occur at run-time in other programs with different inputs.
+B. Feature Engineering
+We need to convert the instruction (or API) calls of
+execution traces into feature vectors that can be analyzed
+by ML models. Unfortunately, most prior research ignores
+relevant information in the API calls during feature extraction.
+Some omitted discriminative information includes API call
+arguments, API output types, API categories, and API analysis
+scope. Table I compares the feature representation of ML-
+FEED with prior research.
+TABLE I: Feature representation comparison
+Model
+1
+2
+3
+4
+5
+Roy et al. [10]
+
+
+
+
+
+Russel et al. [11]
+
+
+
+
+
+MaMaDroid [12]
+
+
+
+
+
+VulDeePecker [6]
+
+
+
+
+
+µ-VulDeePecker [13]
+
+
+
+
+
+Yamaguchi et al. [14]
+
+
+
+
+
+SHARKS [15]
+
+
+
+
+
+GRAVITAS [16]
+
+
+
+
+
+ML-FEED
+
+
+
+
+
+The various columns in Table I are as follows: (1) API
+name, (2) API category, (3) API analysis scope, (4) API
+package and class, and (5) API input/output type.
+TABLE II: An overview of feature representation
+Feature
+name
+Feature
+encoding
+Dimension
+Examples
+API name
+NLP
+70
+readLine()
+API category
+One-hot
+9
+invokespecial
+API
+analysis
+scope
+One-hot
+2
+Application
+API package
+One-hot
+22
+Ljava/io/
+BufferedReader
+API
+input/
+output type
+Frequency
+vector
+24
+Ljava/sql/ Result-
+Set
+There are two reasons for the omission of various features in
+prior research. First, converting all the information into vectors
+is often a non-trivial task. Second, more information in the
+feature vector increases vector size. Higher-dimensional fea-
+ture vectors require more sophisticated ML models. However,
+including this information leads to better performance [10].
+For example, ignoring the API call parameters for a read
+operation can make the feature vectors of all read operation-
+based API calls similar. However, the read operation may be
+malicious if the input comes from the user, whereas it is benign
+if it is system-generated. Therefore, we should not ignore this
+information in order to avoid misrepresenting API calls. In our
+experiments, we do not overlook any such information. Table
+II shows the details of the feature representations that we use in
+ML-FEED. It uses a hybrid feature engineering approach that
+combines the benefits of NLP embeddings, one-hot encoding,
+and frequency vectors.
+1) NLP embeddings: One of the many challenges of vulner-
+ability exploit detection is detecting semantically-similar APIs.
+For example, if an API f1 triggers an exploit, then probably,
+a semantically-identical process f2 may also trigger the same
+exploit. Hence, it is essential to capture the behavioral seman-
+tics of the APIs. We observe that the API names are good
+indicators of behavioral semantics. For example, the APIs for
+opening a new file in Java are newFile() and createFile().
+This observation inspires us to use NLP-based embeddings
+
+Stringdata=
+/*Readdatausinganoutboundtcpconnection*/
+socket=newSocket("host.example.org",123);
+/*readinputfromsocket*
+readerInputStream=new InputStreamReader(socket.getInputStream());
+readerBuffered=new BufferedReader(readerInputStream);
+/* PoTENTIALFLAw:Readdatausinganoutboundtcpconnection*/
+data=readerBufferedreadLine(:
+[..]
+dbConnection=getDBConnection()：
+sqlstatement=dbConnection.createstatement();
+/* PoTENTIALFLAw:dataflowsinto SQLstatementusedin execute)
+whichcouldresultinSQLInjection*/
+[]TCVE-2021-40129["17 = invokevirtual <BufferedReader,readLine()>14"],
+["conditional branch(eg,toiindex=34)14,4"]
+["invokevirtual<close()>14"]
+["82 = phi17,17,3,3",
+"87=invokestatic<io,getDBConnection(）>"]
+["89=invokeinterface<Connection,createStatement()>87"],
+["go = new<StringBuilder>"],
+["invokespecial<StringBuilder,<init>()>90"],
+["94 = invokevirtual <StringBuilder, append(Ljava/lang/String;)> 90,92"],
+["101=invokevirtual<StringBuilder,toString(;>9g"],
+["103 = invokeinterface <Statement,execute(Ljava/lang/String;)> 89,101"]Start
+ml
+n1
+-
+-
+-
+S1
+0
+A1
+C1
+J
+-
+-
+c1
+-
+-
+-
+-
+B
+足
+-
+1
+-
+1
+-
+A2
+el
+-
+1
+-
+-
+A3
+M
+-
+2
+P1
+Data Edge
+A4
+Control Edge
+Startto encode the API names.
+We use a pre-trained lightweight word2vec embedding
+model, called Fasttext [17], to compute the embedding of
+the API name. Word2vec is an NLP model that learns word
+associations from a large text corpus. A trained word2vec
+model is capable of detecting synonymous words. It outputs
+vector representations of words such that similar words are
+closer in the vector space. These vector representations of
+words are known as word embeddings. Fasttext is trained on
+the text corpus of Wikipedia and returns 10-dimensional word
+embeddings. In our dataset, no API name has more than seven
+words. Hence, we concatenate the seven embeddings to obtain
+a 70-dimensional feature vector for the API name. If an API
+name has less than seven words, we fill in the missing values
+with zeros.
+2) One-hot encoding: We have many categorical features
+in our dataset. For example, the API analysis scope has two
+categories: Application and Primordial. We experimented with
+label encoding and one-hot encoding to encode the categorical
+features.
+Label encoding assigns a unique integer value to every
+possible feature category. Unfortunately, it suffers from spu-
+rious correlations. To avoid this problem, we use the one-
+hot encoding method. For categorical features with n possible
+types, a one-hot encoder creates an n-dimensional binary
+vector. Every category is assigned a unique index in the vector.
+For example, to encode category i via one-hot encoding, only
+the ith element of the vector is assigned a 1 and the remaining
+elements are assigned 0’s. We demonstrate label encoding and
+one-hot encoding of all the categories of API types in Table
+III.
+TABLE III: Example feature encodings of API types
+Feature
+Label
+encoding
+One-hot encoding
+binaryop
+1
+1
+0
+0
+0
+0
+0
+0
+0
+0
+conversion
+2
+0
+1
+0
+0
+0
+0
+0
+0
+0
+getCaught
+Exception
+3
+0
+0
+1
+0
+0
+0
+0
+0
+0
+getstatic
+4
+0
+0
+0
+1
+0
+0
+0
+0
+0
+invoke
+interface
+5
+0
+0
+0
+0
+1
+0
+0
+0
+0
+invoke
+special
+6
+0
+0
+0
+0
+0
+1
+0
+0
+0
+invoke static
+7
+0
+0
+0
+0
+0
+0
+1
+0
+0
+invoke
+virtual
+8
+0
+0
+0
+0
+0
+0
+0
+1
+0
+phi
+9
+0
+0
+0
+0
+0
+0
+0
+0
+1
+3) Frequency vector: Frequency vectors contain the fre-
+quencies of various elements in the API call. We use them
+to encode the API input/output (I/O) types, which denote the
+frequency of every I/O class among the API call parameters.
+We present an example of frequency vectors in Table IV. For
+conciseness, we only show 4-dimensional frequency vectors
+in the table. As shown in Table II, we have 24 unique API
+input/output types. Therefore, we have a 24-dimensional fre-
+quency vector representing the API call input types. Likewise,
+we have another 24-dimensional frequency vector representing
+the API output types.
+TABLE IV: Example frequency vector-based encodings of API
+call parameters of Java APIs
+API name
+Input/Output types
+String
+Level
+Throwable
+File
+log
+1
+1
+1
+0
+getString
+1
+0
+0
+0
+4) Combination of encodings: We concatenate the hybrid
+encodings of various elements to construct the final feature
+vector for an instruction call. We show the dimension of every
+feature in Table II. We have two feature vectors corresponding
+to the last row in the table – one for the API inputs and
+another for the API outputs. We have one feature vector for
+all the other rows. Thus, the concatenated feature vector of
+the instruction call has a dimension of 151.
+Next, we describe how our model, ML-FEED, classifies
+exploits efficiently. ML-FEED is a cascade of a classifier and
+a dynamic state table. Unlike most prior research, ML-FEED
+does not analyze entire sequences of instruction calls at once.
+It looks into the problem at a finer granularity and analyzes one
+instruction call at a time. This strategy eliminates the necessity
+of using sophisticated sequence models, resulting in reduced
+computation overhead. The ML-FEED classifier module takes
+the 151-dimensional feature vector of an instruction call as
+input and outputs the potential exploits that the instruction
+call might trigger.
+We show the pipeline for training the ML-FEED exploit
+classifier in Fig. ??. The training procedure consists of the
+following steps. First, we split the dataset of vulnerable
+instruction sequences into a training set with 85% of the data
+and a test set with the remaining 15%. We use the training set
+to extract individual instructions from the sequences and label
+them with the exploits that they can potentially execute. Then,
+we convert these instructions into labeled feature vectors. We
+describe the process of obtaining such vectors from the dataset
+of instruction sequences in Section III-B5. Next, we use the
+dataset of labeled feature vectors to design the neural network
+for our classifier. We use 3-fold cross-validation to select the
+optimal design of our neural network. We elaborate on this
+process in Section III-B6. Finally, we train the optimal neural
+network architecture with the training dataset to get our final
+model.
+5) Transforming the dataset: As mentioned in Section
+III-A, we convert the exploits in the CWE database to vul-
+nerable instruction sequences. We need to extract the features
+and labels of the individual instruction calls from the dataset of
+instruction sequences. We describe this information extraction
+procedure next. First, we obtain the feature vectors of all the
+instruction calls via the feature engineering method discussed
+in Section III-B. Then, we generate the labels for all the feature
+vectors. The labels contain exploits that the corresponding in-
+struction calls can trigger. A single instruction call can trigger
+multiple exploits. The exploits are generally independent of
+each other. Thus, we need a multi-label classifier that predicts
+all the exploits that a given instruction can execute. We design
+
+our classifier output to be an n-dimensional binary vector
+to enable this. We assign the values corresponding to the
+triggered exploits in this vector as 1, whereas the rest are
+assigned 0. For example, if instruction call A executes exploits
+1 and 23, only the first and 23rd elements of the n-dimensional
+output vector will be 1.
+6) Classifier model architecture: The classifier is a fully-
+connected feedforward neural network. We need to design a
+lightweight neural network that achieves high accuracy. The
+neural network architecture has the following constraint – the
+number of neurons in the input and output layers is pre-
+determined by the input and output dimensions. Therefore,
+we only need to determine the number of hidden layers and
+the number of neurons in each hidden layer. Furthermore, we
+target a lightweight model to reduce its latency. Hence, we start
+with one hidden layer. We observe that architectures with one
+hidden layer do not produce good results. Thus, we expand
+our search to two hidden layers. Two hidden layers achieve
+much better results. Finally, we reduce the number of neurons
+in the hidden layers to achieve the most lightweight model
+with comparable accuracy. Table V describes our chosen
+architecture. The output layer has 79 neurons because the
+output of the classifier is a 79-dimensional binary vector that
+denotes which of the 79 exploit categories are executed at a
+given time.
+TABLE V: The model architecture of the ML-FEED classifier
+Number of neurons
+Input layer
+151
+Hidden layer 1
+150
+Hidden layer 2
+100
+Output layer
+79
+We train this classifier model on the complete training
+dataset and evaluate it on the test dataset.
+7) Sequence monitor: Cyber-attackers can exploit a vul-
+nerability by executing a set of malicious instructions in a
+particular sequence. However, running the same set of instruc-
+tions in a different order may not trigger the vulnerability.
+Thus, we need to track the series of program statements
+and instruction calls to detect exploits. Traditional sequence
+models, namely LSTMs and transformers, can track the order
+implicitly. However, this leads to high computation overheads.
+Therefore, we propose a state table-based approach to track
+sequences of program instructions.
+First, we discuss a naive pattern matching approach that
+can be implemented with a state table and its drawbacks.
+Then, we show how we can overcome these drawbacks with
+ML-FEED. Every row in the state table corresponds to a
+unique exploit. We can initialize the exploit states with the
+first instruction call in the respective exploit fingerprints in
+the naive approach. When we encounter any instruction call
+in the current state table during program execution, we update
+the corresponding states with the following instruction call of
+the exploit fingerprint. If the encountered instruction call was
+the last one in the fingerprint(s), we would raise an alarm
+indicating that the API call has executed the corresponding
+exploit(s). Since we analyze the API call parameters like the
+API name, input types, and API package name, this method is
+also applicable to obfuscated codebases. This naive approach
+is similar to the subgraph matching approaches used for
+vulnerability detection [1].
+Although the naive method is much more lightweight than
+sequence models, it has two drawbacks that limit its applica-
+bility in real-world scenarios. First, the naive pattern matching
+approach cannot detect semantically-similar instruction calls.
+As a result, the attacker may avoid detection by executing
+different APIs with the same functionality as the instruction
+in the fingerprint. Second, checking the existence of every in-
+struction in the state table is a time-intensive task. This timing
+overhead reduces the usefulness of naive pattern matching in
+real-time scenarios [1].
+Our solution, ML-FEED, addresses these drawbacks of
+naive pattern matching. Instead of storing the instructions in
+the state table, we store the feature vectors of the instruction
+calls. Then, we compare the feature vectors of the executed
+APIs with the entries in the state table. We use cosine
+similarity to measure the similarity between them. This method
+enables us to capture semantic similarities between APIs. If
+the cosine similarity is above a threshold value, we update
+the state table with the feature vector of the following API in
+the exploit chain. We experimentally observed that a threshold
+value of 0.9 enables the model to achieve the highest F1 score
+and accuracy.
+To address the second drawback of considerable time over-
+heads, we do not compare the feature vector of the executed
+API with all the states in the state table. Instead, we compare
+it with only the states of the exploit categories predicted by the
+ML-FEED classifier. Our experiments show that this method
+requires nearly 10× fewer comparisons per instruction call.
+8) Instruction white-list: To further reduce inference time
+overheads, ML-FEED also uses a pre-defined white-list of
+instructions that are not relevant to the sensitive API calls.
+The instructions in the white-list are those that do not have
+any control or data dependencies on the exploit chain [1].
+For example, Java uses the getCaughtException API to handle
+exception calls during program execution. The presence of
+this API call has no impact on the success or failure of
+triggering an exploit in our threat model. Therefore, We
+can safely discard frequently used white-listed APIs without
+losing sensitive data and control dependency information.
+Furthermore, this ignorance of non-sensitive API calls does
+not affect ML-FEED’s ability to detect potential exploits.
+9) Inference pipeline: We have designed ML-FEED to
+efficiently detect exploits without compromising accuracy. We
+demonstrate our end-to-end inference pipeline in Fig. 5. As
+shown in the figure, we input the executed instruction into
+our framework at run-time. We proceed to the next executed
+instruction if the executed instruction is present in our white-
+list of non-sensitive API calls. Otherwise, we analyze the
+instruction to determine if it triggers an exploit. First, we
+obtain the feature vector for the executed instruction. The ML-
+FEED exploit classifier processes this feature vector. Next,
+
+White-listed
+instruction?
+Feature Engineering
+Fetch next
+instruction
+NLP embedding 
+ 
+One-hot encoding 
+ 
+Frequency vector
+Exploit classifier
+Potential exploits
+#Potential exploits = 0?
+No
+Yes
+No
+Extract states of
+potential exploits
+Yes
+Cosine
+similarity
+Similarity >
+threshold ?
+Update state
+table
+Exploit  
+signature 
+executed?
+No
+Stop execution
+Yes
+Yes
+No
+Executed
+instruction
+Sequence monitor
+State table
+Exploit
+State
+1
+( -0.43, 0.95, ... , 0)
+2
+( 0.21, -0.67, ..., 1)
+Fig. 5: The ML-FEED inference pipeline
+the classifier outputs the list of potential exploits that the
+executed instruction might trigger. If this list is empty, we
+terminate our analysis of the current instruction and proceed
+to the next executed instruction. Otherwise, we input the list
+of potential exploits to the ML-FEED sequence monitor. The
+sequence monitor has a dynamic state table that stores the
+states of all the exploits in our threat model. From the state
+table, we extract the exploit states of all the potential exploits
+predicted by the classifier. Next, we compute the cosine
+similarities between the states of the extracted exploits with
+the feature vector of the executed instruction. As mentioned in
+Section III-B7, we can successfully detect semantically-similar
+instructions by making this comparison. Finally, we update
+the exploit states similar to the feature vector of the executed
+instruction with the feature vectors of the following instruction
+in the corresponding exploit fingerprints. However, if the state
+table indicates that the executed instruction has implemented a
+potential vulnerability fingerprint, we stop the execution and
+raise an alarm. Otherwise, we proceed to analyze the next
+instruction.
+IV. EXPERIMENTAL SETUP AND EVALUATION
+This section describes our experimental setup, provides
+implementation details, and discusses evaluation results.
+A. Experimental Setup
+In what follows, we discuss our vulnerability database
+construction and the experimental setup for the evaluation.
+1) Vulnerability database: We train our model with 23
+unique CWEs from the CWE dataset. Table VI shows the
+complete list of the CWE vulnerabilities that we consider.
+As shown in the table, we evaluate multiple exploits and
+payloads for each CWE, resulting in an analysis of 79 exploits.
+Based on the vulnerability database, we construct an exploit
+database from the Juliet Test Suite [18] for Java. containing
+code examples for 112 CWEs. Given a code example from
+this suite, we first construct the Java bytecode using the javac
+compiler. We then use Wala to perform pointer analysis on
+the bytecode and build the call graphs. Finally, we create the
+SDG using pointer analysis and call graphs. A node in the
+SDG represents an instruction and an edge represents control
+or data flow between instructions. Although our method can
+analyze exploits in any programming language, we focus on
+Java-based exploits for our evaluation.
+We represent every exploit with code snippets in order to
+cover all the features of a CWE. We generate 226,207 code
+snippets to represent 79 exploits across 23 CWEs. We also
+generate 1,101,388 benign code snippets to represent non-
+vulnerable code executions. Thus, the total size of our database
+is 1,327,595 instruction sequences. We split these sequences
+into training and test set with a train-test split of 85%-15%.
+2) Exploit database: We construct our exploit database
+from the Juliet Test Suite [18] for Java, containing code
+examples for 112 CWEs. Given a code example from this
+suite, we first construct the Java bytecode using the javac
+compiler. We then use Wala to perform pointer analysis on
+the bytecode and build the call graphs.
+
+TABLE VI: Details of CWEs considered under our threat
+model
+CWE id
+Description
+Number of
+exploits
+CWE-15
+External control of system or
+configuration settings
+2
+CWE-23
+Relative path traversal
+1
+CWE-78
+OS command injection
+2
+CWE-81
+Cross-site scripting
+1
+CWE-89
+SQL injection
+10
+CWE-90
+LDAP injection
+1
+CWE-129
+Improper validation of array index
+14
+CWE-134
+Uncontrolled format string
+3
+CWE-190
+Integer overflow
+3
+CWE-191
+Integer underflow
+5
+CWE-197
+Numeric truncation error
+6
+CWE-226
+Sensitive information uncleared before
+release
+1
+CWE-256
+Plaintext storage of password
+1
+CWE-259
+Hard-coded password
+1
+CWE-319
+Sensitive information in clear text
+2
+CWE-369
+Division by zero
+10
+CWE-400
+Resource exhaustion
+6
+CWE-470
+Unsafe reflection
+1
+CWE-506
+Embedded malicious code
+1
+CWE-570
+Expression always false
+1
+CWE-606
+Unchecked loop condition
+3
+CWE-643
+Xpath injection
+1
+CWE-789
+Uncontrolled memory allocation
+3
+Finally, we create the SDG using pointer analysis and call
+graphs. A node in the SDG represents an instruction, and
+an edge represents control or data flow between instructions.
+Although our method can analyze exploits in any programming
+language, we focus on Java-based exploits for our evaluation.
+3) Parameters: We use the most lightweight state-of-the-
+art LSTM and transformer models for comparison to ML-
+FEED. The hidden state of the LSTM cell is a 6-dimensional
+vector. The lightweight GPT-2 transfomer that we use in our
+experiments has 117 million parameters and 12 decoder layers
+with 12 attention heads in each self-attention layer. The GPT-2
+model processes a series of 1024 API calls at any given time
+instance.
+4) Machine configuration:
+We ran experiments on a
+Lenovo ThinkPad laptop with an Intel Core i7-8665U CPU.
+The CPU has four cores with a maximum operating frequency
+of 4.2 GHz for each core. The processor base frequency is
+1.90GHz.
+B. Experimental Results
+This section provides experimental results obtained by
+running ML-FEED and state-of-the-art ML-based sequence
+models. We report the performance metrics of our model on
+every CWE in our threat model in Table VII. We also denote
+the CWEs that belong to the OWASP top-10 vulnerabilities.
+The non-profit organization, OWASP, ranks the most critical
+web application security risks based on their frequency of
+occurrence, severity, and magnitude of the potential impact
+on the system. Security analysts frequently use the OWASP
+top 10 vulnerability list as a guideline.
+1) Metrics: We define the performance metrics in terms of
+true positives (TP), true negatives (TN), false positives (FP),
+and false negatives (FN). TP and TN refer to the number
+of correctly classified positive and negative labels. FP and
+FN refer to the number of incorrect positive and negative
+predictions, respectively. These values are often represented
+in the form of a confusion matrix. We use these metrics to
+compute the accuracy, precision, recall, and F1-score of the
+models.
+1) Accuracy: Accuracy denotes the overall performance of
+the classifier. Although it is a good evaluation metric, it
+suffers from biased results when evaluating imbalanced
+datasets.
+Accuracy =
+TP + TN
+TP + TN + FP + FN
+2) Precision: Precision measures what fraction of the posi-
+tive predictions of the model is correct. We can compute
+the precision value from the confusion matrix of a binary
+classification problem with the following formula.
+Precision =
+TP
+TP + FP
+3) Recall: Recall measures what fraction of the true positive
+labels in a binary dataset are predicted as positive. The
+formula for computing the recall value from the confusion
+matrix of a binary classification problem is as follows.
+Recall =
+TP
+TP + FN
+4) F1 score: The F1 score is a statistical measure calculated
+as the harmonic mean of precision and recall values. It
+is a popular metric for evaluating binary classification
+models, especially with an imbalanced dataset.
+F1 = 2 × precision × recall
+precision + recall
+We report the performance metrics of our model on every
+CWE in our threat model in Table VII. We show the results
+of only one exploit per CWE in Table VII. We also denote the
+CWEs that belong to the OWASP top-10 vulnerabilities.
+In Table VII, we observe that the accuracies of all the
+models are quite high in comparison to other metrics. This
+is due to an imbalanced dataset. Thus, precision, recall, and
+F1 scores are more reliable metrics for evaluating our model
+on this dataset than accuracy. We observe that ML-FEED
+achieves the best F1 scores for most CWEs. The transformer
+performs better than ML-FEED for a few CWEs. However, we
+observe later in Table IX that the transformer incurs orders of
+magnitude higher computation overhead than ML-FEED. This
+makes ML-FEED more practical in real-world scenarios.
+2) Error analysis:
+Here, we analyze the shortcomings
+of ML-FEED. ML-FEED outputs false positives and false
+negatives for some of the CWEs. We observe in Table VII
+that all the state-of-the-art ML models perform poorly on
+CWE-319. CWE-319 vulnerabilities arise due to the storage
+
+TABLE VII: Performance of ML models. All values depict percentages (%). The best performances are denoted in bold.
+LSTM
+Transformer
+ML-Feed
+CWE id
+OWASP
+top-10
+Acc.
+Prec.
+Rec.
+F1
+Acc.
+Prec.
+Rec.
+F1
+Acc.
+Prec.
+Rec.
+F1
+CWE-15
+
+99.5
+93.8
+98.9
+96.2
+99.6
+96.3
+99.0
+97.6
+99.8
+97.5
+99.6
+98.6
+CWE-23
+
+96.4
+93.6
+94.9
+96.4
+93.6
+94.9
+96.4
+93.6
+94.9
+96.4
+93.6
+94.9
+CWE-78
+
+99.9
+91.2
+73.8
+81.6
+99.9
+91.4
+76.2
+83.1
+99.9
+91.9
+80.9
+86.0
+CWE-81
+
+99.9
+91.9
+80.9
+86.0
+99.9
+92.1
+83.3
+87.5
+99.9
+92.5
+88.1
+90.2
+CWE-89
+
+99.2
+93.8
+99.2
+96.4
+99.5
+96.3
+99.4
+97.8
+99.7
+97.5
+99.7
+98.6
+CWE-90
+
+99.1
+92.9
+69.0
+79.2
+99.9
+95.2
+72.7
+82.5
+99.9
+96.4
+86.2
+91.0
+CWE-129
+
+99.8
+93.8
+93.9
+93.9
+99.9
+96.2
+96.2
+96.2
+99.9
+97.4
+98.0
+97.7
+CWE-134
+
+99.9
+93.3
+52.5
+67.2
+99.9
+95.6
+61.4
+74.8
+99.9
+95.6
+72.9
+82.7
+CWE-190
+
+99.5
+93.8
+98.9
+96.3
+99.7
+96.2
+99.3
+97.7
+99.8
+97.6
+99.7
+98.6
+CWE-191
+
+99.9
+93.2
+79.9
+86.1
+99.9
+95.8
+92.5
+94.1
+99.9
+97.4
+93.0
+95.2
+CWE-197
+
+99.9
+92.6
+41.7
+57.5
+99.9
+92.6
+50.0
+64.9
+99.9
+96.3
+74.3
+83.9
+CWE-226
+
+99.9
+93.4
+85.4
+89.2
+99.9
+96.0
+91.9
+93.9
+99.9
+97.3
+95.6
+96.5
+CWE-256
+
+99.9
+93.4
+85.4
+89.2
+99.9
+96.0
+91.9
+93.9
+99.9
+97.3
+95.6
+96.5
+CWE-259
+
+99.9
+93.2
+83.6
+88.2
+99.9
+95.9
+82.1
+88.5
+99.9
+97.3
+92.3
+94.7
+CWE-319
+
+99.9
+50.0
+2.1
+4.0
+99.9
+50.0
+3.8
+7.1
+99.9
+50.0
+4.3
+8.0
+CWE-369
+
+99.6
+93.8
+98.7
+96.2
+99.7
+96.2
+99.2
+97.7
+99.8
+97.5
+99.5
+98.5
+CWE-400
+
+99.9
+88.9
+20.5
+33.3
+99.9
+88.9
+32.0
+47.1
+99.9
+88.9
+26.7
+41.0
+CWE-470
+
+99.9
+92.7
+65.4
+76.7
+99.9
+94.5
+72.2
+81.9
+99.9
+96.4
+80.3
+87.6
+CWE-506
+
+99.9
+93.5
+72.5
+81.6
+99.9
+95.6
+88.6
+92.0
+99.9
+97.1
+85.3
+90.8
+CWE-570
+
+99.9
+75.0
+8.1
+14.6
+99.9
+95.6
+88.6
+92.0
+99.9
+97.1
+85.3
+90.8
+CWE-606
+
+99.9
+85.7
+75.0
+80.0
+99.9
+86.8
+82.5
+84.6
+99.9
+87.2
+85.0
+86.1
+CWE-643
+
+99.9
+92.7
+65.4
+76.7
+99.9
+94.5
+72.2
+81.9
+99.9
+96.4
+80.3
+87.6
+CWE-789
+
+99.9
+93.6
+55.1
+69.4
+99.9
+95.2
+76.9
+85.1
+99.9
+96.8
+79.2
+87.1
+of sensitive information in cleartext. One must manually check
+the repository for sensitive content or train separate NLP
+models to catch sensitive data in cleartext to detect this
+vulnerability. We aim to tackle this problem in the future by
+cascading a sensitivity-predicting NLP model with ML-FEED.
+In Table VII, we also observe that ML-based methods have
+a relatively low recall for detecting CWE-400 vulnerabilities.
+CWE-400 is a denial-of-service (DoS) attack. DoS attacks are
+difficult to prevent because they are highly dependent on the
+system architecture. Thus, system-specific intrusion detection
+systems are more efficient in detecting DoS attacks. However,
+the high precision scores achieved by ML-based methods in
+Table VII can help improve the efficiency of IDS [15, 19].
+We can use the precision, recall, and F1 score metrics
+to evaluate only binary classification problems. However,
+our exploit classification task is a multi-label classification
+problem with 79 labels. For a multi-label classification task
+with n labels, we have n confusion matrices. Precision, recall,
+and F1 scores cannot handle multiple confusion matrices by
+design. However, we can compute these metrics for each of
+the n confusion matrices and average them to obtain the
+mean precision, recall, and F1 scores. We show the mean per-
+formance metrics of ML-FEED and state-of-the-art sequence
+models across all 79 labels in Table VIII. In Table VII, we
+observed that accuracy is not a reliable performance metric
+for our dataset. Thus, we do not report the average accuracy
+in Table VIII.
+TABLE VIII: Performance of ML models on 79 exploits. All
+values depict percentages (%).
+Model
+Precision
+Recall
+F1
+LSTM
+96.4
+93.6
+94.9
+Transformer
+97.6
+96.1
+96.8
+ML-FEED
+98.2
+97.4
+97.8
+3) Computation overhead: We demonstrate the computa-
+tion overheads of each model in terms of their inference time
+and model size in Table IX. Higher overheads degrade the
+applicability of the models in real-world scenarios.
+TABLE IX: The inference time and model size of ML models
+Model
+Inference
+time (µs)
+Number of
+parameters
+Model size
+LSTM
+475.81
+213,116
+2.5 MB
+Transformer
+495162.72
+117M
+523 MB
+ML-FEED
+6.53
+46,788
+970.5 KB
+Table IX shows that ML-FEED is 72.9× and 75828.9×
+faster than the lightweight LSTM and GPT-2 models, respec-
+tively. It also has 4.5× and 2500.6× fewer parameters than
+the LSTM and GPT-2 models, respectively.
+In Fig. 6, we observe that the LSTM is more efficient
+than the transformer. The transformer is inefficient because it
+analyzes a sequence of the last 1024 API calls for inference at
+every time step. However, the LSTM analyzes only two fixed-
+length vectors at every time step: the current API call and
+the hidden state. Fig. 6 also shows that ML-FEED is more
+efficient than the LSTM. This is because it does not maintain
+and update a hidden state vector at every time step. Instead,
+ML-FEED uses a state-table to keep track of the sequence of
+API calls.
+The LSTM analyzes the current API call along with its
+hidden state. The hidden state is a fixed length vector that
+encapsulates the history of the sequence. On the contrary, the
+transformer analyzes a sequence of the last 1024 API calls for
+inference at every time-step. This increases the effectiveness
+of the transformer over LSTMs. However, this makes the
+transformers inefficient. Like the LSTM, ML-FEED analyzes
+only the current API call. However, it is more efficient than the
+LSTM because it does not maintain and update a hidden state
+
+TABLE X: Vulnerability detector feature comparison
+Model
+Run-
+time
+Rule-
+based
+ML-
+based
+Efficient
+Constraint solving [20]
+
+
+
+
+Modular checking [21]
+
+
+
+
+VulDeePecker [6]
+
+
+
+
+µ-VulDeePecker [13]
+
+
+
+
+Yamaguchi et al. [14]
+
+
+
+
+SHARKS [15]
+
+
+
+
+GRAVITAS [16]
+
+
+
+
+MLASA [? ]
+
+
+
+
+GPT-2 [8]
+
+
+
+
+SySevVR [22]
+
+
+
+
+Russell et al. [11]
+
+
+
+
+LSTM [23]
+
+
+
+
+ML-FEED
+
+
+
+
+vector at every time-step. Instead, ML-FEED uses a state-table
+to keep track of the sequence of API calls. We demonstrate
+the efficiency of ML-FEED in Fig. 6.
+loge
+Fig. 6: Comparison of the efficiency of ML-FEED with state-
+of-the-art ML-based sequence models
+C. Comparison
+Table X shows that ML-FEED is the only framework that
+satisfies all the features in the table. ML-FEED can perform
+run-time analysis effectively for all exploits in its training set.
+The rule-based fingerprint detector in the sequence monitor
+makes it more efficient than other state-of-the-art ML methods.
+The exploit classifier in the inference pipeline enables pattern-
+based exploit detection to detect a wide range of exploits.
+D. Evaluation with Real Attack Traces
+To demonstrate the practical usage of our model, we use
+ML-FEED to evaluate three popular open-source libraries with
+existing CVEs. We choose open-source libraries because our
+model requires access to the source code to analyze potential
+vulnerabilities. We evaluate ML-FEED on the attack traces
+of CVE-2021-25646, CVE-2021-39139, and CVE-2020-1958.
+ML-FEED successfully identifies all three vulnerabilities. We
+describe the detected vulnerabilities next.
+1) Remote code execution: ML-FEED identifies two remote
+code execution (RCE) vulnerabilities in the popular Apache
+Druid and XStream library. Apache Druid is a popular library
+that provides real-time database support for powering modern
+analytics applications. On the other hand, XStream allows
+users to serialize objects to XML and back again. How-
+ever, while evaluating Apache Druid, ML-FEED detects an
+RCE vulnerability that allows a malicious attacker to execute
+JavaScript codes embedded in various types of requests. An
+authenticated user can send a specially-crafted request, regard-
+less of server configuration, which forces Druid to run user-
+provided JavaScript code for that request. This vulnerability
+is reported in CVE-2021-25646, with a high severity score of
+8.8. ML-FEED identifies another RCE vulnerability that may
+allow a remote attacker to load and execute arbitrary code from
+a remote host just by manipulating the processed input stream.
+This vulnerability has been reported in CVE-2021-39139, with
+a high severity score of 8.5.
+2) Lightweight directory access protocol (LDAP) injection:
+ML-FEED also identifies an LDAP injection vulnerability in
+the Apache Druid library. This vulnerability allows Druid
+API callers to bypass the credentialsValidator.userSearch filter
+barrier that determines if a valid LDAP user is allowed to
+authenticate with Druid. Callers can also retrieve any LDAP
+attribute of users from the LDAP server, as long as that
+information is visible to the Druid server. This information
+disclosure does not require the caller to be a valid LDAP user.
+This vulnerability is reported in CVE-2020-1958 [24] , with a
+medium severity score of 6.5.
+V. RELATED WORK
+Vulnerability exploit detection is a fundamental challenge in
+software security, for which researchers have proposed many
+notable solutions. The two broad categories of classical vul-
+nerability detection approaches are rule-based and similarity-
+based methods. Rule-based methods involve manually crafting
+the rules of vulnerabilities [20, 21, 25, 26]. These rules
+attempt to encompass both the signatures and behaviors of
+the exploits [1]. The vulnerabilities and exploit payloads are
+detected by naive pattern matching against the rules. As
+discussed in Section III, naive pattern matching suffers from
+generalization limitations and considerable time overheads [1].
+ML-FEED overcomes these limitations by using ML and
+reducing the search space for vulnerabilities. Similarity-based
+approaches rely on code sharing and code cloning between
+applications. The assumption is that the exposure is replicated
+if a vulnerable code is cloned. These approaches aim to
+find similarities with vulnerable codebases. Similarity-based
+vulnerability detection techniques are effective only when
+codebases are cloned into the testing project.
+Vulnerability detection methods have undergone a radical
+transformation with the advent of ML [15, 16, 19, 27]. ML-
+based methods perform much better than rule-based meth-
+ods. Although ML can accurately learn the underlying pat-
+terns of vulnerabilities, manual effort is still required to
+characterize the vulnerable programs. VulDeePecker [6] and
+
+ML-FEED
+LSTM
+Transformer
+20
+15
+10
+5
+0
+Inference time
+Number of parametersµVulDeePecker [13] are among the first fully-automated ML
+models to detect and classify vulnerabilities successfully.
+Sequence models like transformers and LSTMs have found
+success in modeling exploit chains and detecting exploits.
+However, they suffer from degraded performance in capturing
+long-range dependencies in the attack sequences. Transformers
+are better at capturing long-range dependencies than RNNs.
+Hence, they have found various applications in cybersecurity
+analysis [28]. However, transformers incur large computation
+overheads, limiting their application in real-world scenarios.
+Recent methods like ATLAS [29] and ALchemist [30] have
+extended the applications of sequence-based ML analysis to
+domains like advanced persistent threat analysis and audit
+report analysis-based attack investigation. Malware detection
+is another domain in which ML has been highly success-
+ful [31, 32, 33]. ML has also shown promising results in
+detecting attacks on networks [34], operating systems [35], and
+web applications [36]. Despite the rapid progress of ML meth-
+ods, they suffer from a lack of explainability. ML algorithms
+generally work as black boxes. Hence, security analysts cannot
+easily interpret their predictions. This lack of transparency
+diminishes the trust analysts place in ML models. Fortunately,
+researchers have recently developed explainable ML models
+for cybersecurity [37, 38, 39, 40] that can potentially increase
+the confidence that can be placed in ML models.
+VI. CONCLUSION
+In this article, we presented ML-FEED, a lightweight ML-
+based exploit detection framework that is orders of magnitude
+faster than state-of-the-art methods while achieving better
+performance. It analyzes sequences of API calls and raises
+an alarm when an exploit is executed. It can also classify
+the exploit into CWE categories. We also proposed an auto-
+mated method to extract intelligence from the CWE and CVE
+vulnerability databases, thus mitigating the need for human
+intervention. Finally, we used ML-FEED to analyze three
+public libraries and found the existing CVEs in all of them.
+Finally, ML-FEED was successful in detecting existing CVEs
+in three public libraries.
+ACKNOWLEDGMENT
+We thank Bhishma Dedhia for his help with implementing
+the GPT-2 transformer model.
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+page_content='ML-FEED: Machine Learning Framework for  Efficient Exploit Detection (Extended version)  Tanujay Saha  Electrical and Computer Engineering  Princeton University  NJ, USA  tsaha@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='edu  Tamjid Al Rahat  Electrical and Computer Engineering  University of California Los Angeles  CA, USA  tamjid@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='edu  Najwa Aaraj  Cryptography Research Center  Technology Innovation Institute  Abu Dhabi, UAE  najwa@tii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='ae  Yuan Tian  Electrical and Computer Engineering  University of California Los Angeles  CA, USA  yuant@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='edu  Niraj K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Jha  Electrical and Computer Engineering  Princeton University  NJ, USA  jha@princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='edu  Abstract—Machine learning (ML)-based methods have re-  cently become attractive for detecting security vulnerability  exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Unfortunately, state-of-the-art ML models like long  short-term memories (LSTMs) and transformers incur significant  computation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This overhead makes it infeasible to  deploy them in real-time environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We propose a novel ML-  based exploit detection model, ML-FEED, that enables highly  efficient inference without sacrificing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We develop  a novel automated technique to extract vulnerability patterns  from the Common Weakness Enumeration (CWE) and Common  Vulnerabilities and Exposures (CVE) databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This feature  enables ML-FEED to be aware of the latest cyber weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Second, it is not based on the traditional approach of classifying  sequences of application programming interface (API) calls into  exploit categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Such traditional methods that process entire  sequences incur huge computational overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Instead, ML-  FEED operates at a finer granularity and predicts the exploits  triggered by every API call of the program trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, it uses  a state table to update the states of these potential exploits  and track the progress of potential exploit chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML-FEED  also employs a feature engineering approach that uses natural  language processing-based word embeddings, frequency vectors,  and one-hot encoding to detect semantically-similar instruction  calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, it updates the states of the predicted exploit categories  and triggers an alarm when a vulnerability fingerprint executes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Our experiments show that ML-FEED is 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9× and 75, 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9×  faster than state-of-the-art lightweight LSTM and transformer  models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We trained and tested ML-FEED on 79  real-world exploit categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It predicts categories of exploit  in real-time with 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2% precision, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='4% recall, and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='8% F1  score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' These results also outperform the LSTM and transformer  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' In addition, we evaluated ML-FEED on the attack  traces of CVE vulnerability exploits in three popular Java  libraries and detected all three reported critical vulnerabilities  in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' INTRODUCTION  Digital platforms worldwide have been facing increasingly  sophisticated malicious campaigns directed at them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Most  cyber-attacks are executed by exploiting system vulnerabili-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Vulnerabilities are weaknesses in a system that hackers  can exploit them in multiple ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Vulnerability scanning  involves detecting such potential weaknesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This requires  administrator-level access to the system resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Exploit  detection requires more sophistication than vulnerability detec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This is because system administrators often do not grant  permission to exploit detection software to access internal  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Researchers have devoted significant efforts towards ad-  dressing the challenge of detecting vulnerabilities and exploits,  developing popular methods like fuzzing, symbolic analysis,  and rule-based testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Although these methods are quite  popular, they do have some drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A significant limitation  is that most such methods require manual definitions of attack  signatures and patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Manual description of rules limits the  efficiency of these methods when dealing with large code-  bases [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Moreover, these traditional vulnerability detection  techniques suffer from a large number of false positives,  performance issues, and inability to identify the vulnerability  type [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Researchers have overcome these drawbacks by  integrating machine learning (ML), more specifically deep  learning, into vulnerability detection techniques [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Using ML requires minimal manual effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, it expe-  dites the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' State-of-the-art ML methods, like long-short  term memories (LSTMs) and transformers, classify application  programming interface (API) call sequences obtained from  program execution traces as benign or malignant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' They can  even predict the exploit type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, these models incur  huge computation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This diminishes their usefulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  This article presents ML-FEED (Machine Learning Frame-  work for Efficient Exploit Detection) that combines ML and  non-ML methods to achieve highly efficient exploit classifi-  cation of execution traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Next, we discuss the drawbacks  of state-of-the-art techniques and how ML-FEED overcomes  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Transformers analyze the entire sequence of instruction  calls at every timestep during inference, resulting in huge  computation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' LSTMs use representations of the  sequence history to reason about the entire sequence at every  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='04314v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CR]  11 Jan 2023  timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Although these representations help them capture  semantic information in the sequence, handling these represen-  tations incurs large computation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore, unlike  traditional ML-based sequence models, ML-FEED does not  analyze the entire sequence of instruction calls at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It first  predicts the exploits that an instruction call may trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then,  it uses this prediction to update the states of the predicted  exploits in a state table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This state table keeps track of the  current execution state of every exploit in our threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  When the state of one or more exploits in the table represents  a vulnerability fingerprint, it raises the alarm for a potential  attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Our experiments show that ML-FEED is faster than  the state-of-the-art lightweight LSTM (transformer) model by  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9× (75, 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9×) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5× (2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6×) more compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Besides designing a lightweight model for efficient in-  ference, we also design a general framework for extracting  threat intelligence from vulnerability databases like Common  Weakness Enumeration (CWE) and Common Vulnerabilities  and Exposures (CVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We use this extracted intelligence to  construct an exploit sequences database, which we further use  to train our ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This methodology enables us to update  our ML model with the latest exploits, thus mitigating the  impact of a lack of sufficient data consisting of heterogeneous  vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The novelties of our approach are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  1) We propose a hybrid system comprising ML classifiers  and state tables to classify exploit chains efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  2) We analyze individual API calls instead of entire API  call sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This analysis at a finer granularity enables  ML-FEED to demonstrate much higher efficiency than  state-of-the-art sequence models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  3) We propose a methodology to automatically extract intel-  ligence from vulnerability databases and create an exploit  sequence database to update ML-based exploit detection  systems with the latest exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Section II provides the  background information necessary for understanding the rest  of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Section III describes the methodology for exploit  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Section IV details the results of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Section V discusses related work in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Section VI  concludes the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' BACKGROUND  This section discusses background material for program  representations and ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Call Graph  A call graph is a control-flow graph representing the calling  relationships between functions in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 1 shows  an example of a call graph where each node represents a  function, and each edge represents the calling relationship  between the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For example, if a function f1 calls f2  during the program flow, the calling relationship is represented  by a directed edge (f1,f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  0  1  2  3  7  4  6  5  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  𝑓"  𝑓#  𝑓$  𝑓%  𝑓&  𝑓%  𝑓%  𝑓&  𝑓\'  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 1: Example of a call graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' System Dependence Graph  We use a System Dependence Graph (SDG) [4] as the  inter-procedural graph representation of programs to extract  the vulnerable API sequences for training our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' An  SDG G = (V, Ec, Ed) is a three-tuple, where V represents  program statements, Ec encodes control dependency edges,  and Ed denotes data dependency edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For nodes X and Y  in an SDG, Y is control-dependent on X if, during execution,  X can directly affect whether Y is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Also, X is  data-dependent on Y  if Y  is an assignment and X can  access the value assigned in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Since these data and control  dependencies are computed for individual program procedures  (or functions), they do not account for inter-procedural control  and data relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As a result, they cannot determine the  transitive dependencies by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore, we augment the  SDG with information on inter-procedural relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' SDG  edges also represent the direct dependence between the call  site and the called procedure, which allows us to determine  the transitive dependencies due to a call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  To construct the SDG, a call graph and pointer analysis  is simultaneously performed over the entire program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A call  graph is a control-flow graph that represents the calling  relationships between functions in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Each node  represents a function and each edge represents the calling rela-  tionship between the functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Since we target object-oriented  programs, we use the Class Hierarchy Analysis (CHA) [5] call  graph algorithm implemented in the Wala framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' On the  other hand, pointer analysis is a static code analysis technique  that determines which pointers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=', references) can point to  which variables or storage locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Since we target Java-  based program in our analyses, we use an Andersen-style  pointer analysis, which is available in the Wala toolchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  An SDG creates three additional edges to handle function  calls: (1) call edge, (2) parameter-in edge, and (3) parameter-  out edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A call edge connects the node at the call site to the  entry node of the called procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Parameter-in edges connect  the actual-in parameter nodes to the formal-in parameter nodes  of the called function, and parameter-out edges connect the  formal-out nodes to the actual-out nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 2 shows an  SDG for two functions, f1 and f2, where f1 calls f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' When  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  <Entry>  call 𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  𝑥  <actual_in>  𝑓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  <Entry>  <Cond>  𝑥  <formal_in>  𝑦  <formal_out>  𝑦  <actual_out>  Param-in  edge  Param-out  edge  Call edge  1  2  data-dep  control-dep  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 2: Example of an SDG for call graph nodes 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Here,  an SDG node represents a statement and an edge represents  the control or data dependency between program statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  f1 calls f2 during the program flow, a call edge from the call  site in caller f1 to the entry node of callee f2 represents this  calling relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Also, the arguments passed by the caller  are represented by the actual in nodes, and the formal in  nodes represent the arguments received by the callee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The  returned value from the caller is represented by the formal out  node, whereas the value received by the callee represents the  actual out node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 2, the solid and dotted lines  represent the control and data dependencies between the SDG  nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Sequence Models  Sequence models are ML-based models that can analyze  data with sequential dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Time-series data, video and  audio clips, and text streams are a few examples of sequential  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A popular deep learning model used for sequential  modeling is the LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This is because LSTMs are capable  of handling long-range sequential dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Security an-  alysts often use LSTMs and their variants to analyze system-  call sequences for vulnerability detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' More sophisticated  versions of the LSTM, like bidirectional-LSTMs [6] and CNN-  LSTMs [7], usually achieve good performance over vanilla  LSTM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, LSTM models require significant  computation overheads due to their increased complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Therefore, we compare ML-FEED with the state-of-the-art  lightweight LSTM model that has reasonable accuracy while  taking minimal computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=', this  architecture is a cascade of an LSTM cell followed by a  feedforward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Transformers are another category of sequence analysis  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Transformers have become popular because they are  even more successful at capturing long-range dependencies  with the help of the attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This mechanism  considers the entire series during inference and determines  how to prioritize the different elements of a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Further-  more, their advanced semantic knowledge representation en-  ables sophisticated modeling of semantically-similar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Although very successful, transformers are often huge mod-  els that incur enormous computation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Transformer  architectures that have performed well on natural language  processing (NLP) tasks have also been successful in system-  call sequence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Some of the most popular transformer  architectures used for detecting vulnerabilities include the  generative pre-trained (GPT) transformer and its variants,  BERT and BART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Unlike BERT and BART, the GPT family of  transformers is a decoder-based model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It is easier to model  our problem statement with a decoder model [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, we  compare the performance of ML-FEED with that of GPT-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' METHODOLOGY  We describe our methodology for efficient exploit detection  in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We begin by extracting vulnerable instruction  sequences from vulnerability databases and publicly available  vulnerability exploit programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, we convert the individual  instruction calls into labeled feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' These vectors  constitute our training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We use our training data to design  and train our neural network architecture for the ML-FEED  exploit classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Instruction  extraction  Feature  engineering  Test  program  Exploit  classifier  Vulnerability  fingerprint  executed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Yes  No  White-listed  instruction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  No  Sequence  monitor  Fetch next  instruction  Stop  execution  Yes  Yes  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 3: Overview of our inference pipeline  We give a high-level overview of the ML-FEED infer-  ence pipeline in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, we extract instruction calls  sequentially from the test program during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' If the  current instruction call is present in the white-list of non-  harmful functions, we proceed to analyze the next instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Otherwise, we continue to analyze the current function by  converting it into a feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The exploit classifier then  analyzes this feature vector to predict the potential exploits  that the current instruction call can execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Finally, this list of  potential exploits is provided as input to the sequence monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The sequence monitor tracks the order of the instruction  calls and raises an alarm if the current instruction executes  a vulnerability fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' If the current instruction does not  implement an exploit, we analyze the next instruction of the  test program execution trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  During inference, ML-FEED raises an alarm when it detects  the execution of a CVE/CWE vulnerability fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Prior  ML-based methods (like LSTMs and transformers) and non-  ML-based methods (like subgraph matching) for vulnerability  detection are not efficient enough to be deployed in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Non-ML-based methods incur a time overhead of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='57× [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  As we show later, prior ML-based methods like lightweight  LSTMs and transformers incur time overheads of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='86× and  76178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='88×, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We compare the efficiency of ML-  FEED with prior methods in Section III-B7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Representation of Vulnerable Programs  This section describes the extraction of exploit instruction  sequences from the CVE and CWE databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We give an  overview of this extraction procedure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, we use  the CWE database to define a high-level vulnerability finger-  print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Although these fingerprints can represent the exploit’s  high-level behavior, they are not sufficient for capturing the  fine-grained relationships between the program entities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=',  language-specific APIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore, we use the CVE database  to extract the low-level program entities and their relationships  that trigger the corresponding vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We refer to this  low-level representation of the vulnerability fingerprint as a  vulnerability query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Next, we create the vulnerable program’s  SDG representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Finally, we perform graph search based  on the extracted query to obtain the vulnerable program flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  1) CWE and CVE databases: We use both CWE and CVE  databases to analyze vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The CWE database con-  tains broad categories of system vulnerabilities and their high-  level descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For example, SQL injection vulnerabilities  and cross-site request forgery have been assigned CWE-89 and  CWE-352, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore, we use the CWE descrip-  tions to extract the high-level semantics of the vulnerabilities  in our threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, the CWE database does not  provide the low-level system-specific vulnerability information  that we require for real-world exploit classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The CVE  database contains this low-level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It contains the  specific flaws and weaknesses of various systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore,  we obtain the system-specific vulnerability information from  the CVE database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, SQL injection vulnerability has been assigned  a CWE, namely CWE-89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, SQL injection (CWE-89)  has multiple entries in the CVE database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Each entry corre-  sponds to a unique, vulnerable software version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Some popular  CVEs for SQL injection are CVE-2007-6602 and CVE-2017-  11508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We utilize the high-level vulnerability semantics from  the CWE database and the low-level vulnerability details from  the CVE database for further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  2) Vulnerability fingerprints: We describe the process of  extracting vulnerability fingerprints from the CWE database  in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, we analyze the CWE database to define  the high-level vulnerability semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, we represent this  information with the help of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The nodes of these graphs  represent entities involved in a particular vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The  edges represent the relationships between these entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' These  relationships must hold for the vulnerability to be triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, we can describe the semantics of SQL injection  (CWE-89) using a fingerprint graph, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 4,  where a vulnerability is triggered when an SQL command is  constructed using user-controlled (external) input and executed  by the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  3) Vulnerability queries: Vulnerability fingerprints defined  by high-level semantics are insufficient for capturing the  fine-grained and diverse behaviors of vulnerable programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, user-controllable external inputs may trigger a  vulnerability (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=', SQL injection) in databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, user-  controllable information can originate from various sources,  such as files or network streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Similarly, different programs  might use other APIs for executing SQL commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' To cap-  ture this low-level information, we transform the vulnerability  fingerprints into vulnerability queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' To construct the queries,  we further analyze the CVEs of the corresponding CWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We  extract the relevant entities and their relationships that exploit  a vulnerability from the CVE database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, we map the enti-  ties to their related programming language-specific elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 4, we map a user-controlled  external input from files to the corresponding APIs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=',  String BufferedReader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='readLine()) for reading the input file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  This query representation allows us to check for vulnerable  sequences in the SDG representation of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  4) Vulnerable instruction sequences:  A vulnerable se-  quence is a series of instructions that exploits a vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  In this section, we describe the process of generating such  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 4, first, we perform pointer  analysis and obtain the precise call graph for an input program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The call graph contains the calling relationships between  functions in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, we process the call graph to  obtain the program’s SDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The SDG captures both control-  and data-flow dependencies of a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Finally, we search the SDG for the vulnerability queries  that we generated before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This graph search outputs the  sequence of basic blocks and instructions representing the  program segment that triggers the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We can do  this analysis for vulnerability exploits in any programming  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, we have analyzed Java-based exploits with  the Wala code disassembler for our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  In this work, we use the SDG to extract vulnerable API  calls from the programs in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Prior research utilizes  other graph representations, along with the SDG, to detect  vulnerabilities and explore various properties of the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, SSLint [9] uses a Program Dependence Graph  (PDG) that combines both Control Flow Graphs (CFGs) and  Data Flow Graphs to perform intra-procedural analysis for  detecting vulnerable properties within a function in an SSL  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Yamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' [2] combine an Abstract Syntax  Tree (AST), CFG, and PDG to introduce a new graph represen-  tation called Code Property Graph to analyze lower-level code  properties, such as expressions and conditions in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  While all these representations have proven to be effective  for various applications, the SDG representation is the best  candidate for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' An SDG, being an inter-procedural  representation of the PDG, combines the control flow and data  String BufferedReader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='readLine()  Statement Connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='createStatement()  boolean Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='execute(String)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Data flow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Data flow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='program  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Generate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='vulnerability  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='queries  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Execute  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='graph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='search  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='query  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='SDG representation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='of the input program  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Query for a SQL injection vulnerability  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Sequence(s) of vulnerable instructions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-based fingerprint  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='graph for SQL Injection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Pointer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='analysis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Call graph  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Generate SDG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='External  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Execute  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Data flow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Data flow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='SQL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='An attacker could exploit this  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='vulnerability by uploading a file  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='containing an SQL query to the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='configuration dashboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 4: Extracting instruction sequences from CVE and CWE databases and source codes of exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  flow relationships between the APIs in a program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This inter-  procedural representation of vulnerable programs enables us  to capture the holistic program execution environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  However, SDG construction can be imprecise for certain  program features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=', reflected function calls) and can gen-  erate sequences that may not occur at run-time in the given  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Fortunately, this is an advantage for the generality  of our training model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' In other words, since an imprecise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=',  overapproximated) SDG for certain program components can  represent a program path that may not occur at run-time, it  allows our model to learn those paths as well, since they could  occur at run-time in other programs with different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Feature Engineering  We need to convert the instruction (or API) calls of  execution traces into feature vectors that can be analyzed  by ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Unfortunately, most prior research ignores  relevant information in the API calls during feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Some omitted discriminative information includes API call  arguments, API output types, API categories, and API analysis  scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Table I compares the feature representation of ML-  FEED with prior research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  TABLE I: Feature representation comparison  Model  1  2  3  4  5  Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' [10]  \x17  \x13  \x17  \x17  \x13  Russel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' [11]  \x13  \x17  \x17  \x13  \x13  MaMaDroid [12]  \x13  \x13  \x17  \x13  \x13  VulDeePecker [6]  \x13  \x17  \x17  \x17  \x13  µ-VulDeePecker [13]  \x13  \x17  \x17  \x17  \x13  Yamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' [14]  \x13  \x17  \x17  \x13  \x13  SHARKS [15]  \x17  \x13  \x17  \x13  \x17  GRAVITAS [16]  \x17  \x13  \x17  \x13  \x17  ML-FEED  \x13  \x13  \x13  \x13  \x13  The various columns in Table I are as follows: (1) API  name, (2) API category, (3) API analysis scope, (4) API  package and class, and (5) API input/output type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  TABLE II: An overview of feature representation  Feature  name  Feature  encoding  Dimension  Examples  API name  NLP  70  readLine()  API category  One-hot  9  invokespecial  API  analysis  scope  One-hot  2  Application  API package  One-hot  22  Ljava/io/  BufferedReader  API  input/  output type  Frequency  vector  24  Ljava/sql/ Result-  Set  There are two reasons for the omission of various features in  prior research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, converting all the information into vectors  is often a non-trivial task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Second, more information in the  feature vector increases vector size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Higher-dimensional fea-  ture vectors require more sophisticated ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However,  including this information leads to better performance [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, ignoring the API call parameters for a read  operation can make the feature vectors of all read operation-  based API calls similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, the read operation may be  malicious if the input comes from the user, whereas it is benign  if it is system-generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore, we should not ignore this  information in order to avoid misrepresenting API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' In our  experiments, we do not overlook any such information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Table  II shows the details of the feature representations that we use in  ML-FEED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It uses a hybrid feature engineering approach that  combines the benefits of NLP embeddings, one-hot encoding,  and frequency vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  1) NLP embeddings: One of the many challenges of vulner-  ability exploit detection is detecting semantically-similar APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, if an API f1 triggers an exploit, then probably,  a semantically-identical process f2 may also trigger the same  exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Hence, it is essential to capture the behavioral seman-  tics of the APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We observe that the API names are good  indicators of behavioral semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For example, the APIs for  opening a new file in Java are newFile() and createFile().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  This observation inspires us to use NLP-based embeddings  Stringdata=  /*Readdatausinganoutboundtcpconnection*/  socket=newSocket("host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='org",123);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  /*readinputfromsocket*  readerInputStream=new InputStreamReader(socket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='getInputStream());' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  readerBuffered=new BufferedReader(readerInputStream);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  /* PoTENTIALFLAw:Readdatausinganoutboundtcpconnection*/  data=readerBufferedreadLine(:  [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='.]  dbConnection=getDBConnection()：  sqlstatement=dbConnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='createstatement();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  /* PoTENTIALFLAw:dataflowsinto SQLstatementusedin execute)  whichcouldresultinSQLInjection*/  []TCVE-2021-40129["17 = invokevirtual <BufferedReader,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='readLine()>14"],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ["conditional branch(eg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='toiindex=34)14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='4"]  ["invokevirtual<close()>14"]  ["82 = phi17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
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+page_content='  "87=invokestatic<io,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='getDBConnection(）>"]  ["89=invokeinterface<Connection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='createStatement()>87"],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ["go = new<StringBuilder>"],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ["invokespecial<StringBuilder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='<init>()>90"],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ["94 = invokevirtual <StringBuilder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' append(Ljava/lang/String;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=')> 90,92"],  ["101=invokevirtual<StringBuilder,toString(;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='>9g"],  ["103 = invokeinterface <Statement,execute(Ljava/lang/String;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=')> 89,101"]Start  ml  n1 S1  0  A1  C1  J c1 B  足 1 1 A2  el 1 A3  M 2  P1  Data Edge  A4  Control Edge  Startto encode the API names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  We use a pre-trained lightweight word2vec embedding  model, called Fasttext [17], to compute the embedding of  the API name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Word2vec is an NLP model that learns word  associations from a large text corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A trained word2vec  model is capable of detecting synonymous words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It outputs  vector representations of words such that similar words are  closer in the vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' These vector representations of  words are known as word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Fasttext is trained on  the text corpus of Wikipedia and returns 10-dimensional word  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' In our dataset, no API name has more than seven  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Hence, we concatenate the seven embeddings to obtain  a 70-dimensional feature vector for the API name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' If an API  name has less than seven words, we fill in the missing values  with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  2) One-hot encoding: We have many categorical features  in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For example, the API analysis scope has two  categories: Application and Primordial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We experimented with  label encoding and one-hot encoding to encode the categorical  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Label encoding assigns a unique integer value to every  possible feature category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Unfortunately, it suffers from spu-  rious correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' To avoid this problem, we use the one-  hot encoding method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For categorical features with n possible  types, a one-hot encoder creates an n-dimensional binary  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Every category is assigned a unique index in the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, to encode category i via one-hot encoding, only  the ith element of the vector is assigned a 1 and the remaining  elements are assigned 0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We demonstrate label encoding and  one-hot encoding of all the categories of API types in Table  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='TABLE III: Example feature encodings of API types  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
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+page_content='3) Frequency vector: Frequency vectors contain the fre-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='quencies of various elements in the API call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We use them  to encode the API input/output (I/O) types, which denote the  frequency of every I/O class among the API call parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  We present an example of frequency vectors in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For  conciseness, we only show 4-dimensional frequency vectors  in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As shown in Table II, we have 24 unique API  input/output types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore, we have a 24-dimensional fre-  quency vector representing the API call input types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Likewise,  we have another 24-dimensional frequency vector representing  the API output types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  TABLE IV: Example frequency vector-based encodings of API  call parameters of Java APIs  API name  Input/Output types  String  Level  Throwable  File  log  1  1  1  0  getString  1  0  0  0  4) Combination of encodings: We concatenate the hybrid  encodings of various elements to construct the final feature  vector for an instruction call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We show the dimension of every  feature in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We have two feature vectors corresponding  to the last row in the table – one for the API inputs and  another for the API outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We have one feature vector for  all the other rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, the concatenated feature vector of  the instruction call has a dimension of 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Next, we describe how our model, ML-FEED, classifies  exploits efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML-FEED is a cascade of a classifier and  a dynamic state table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Unlike most prior research, ML-FEED  does not analyze entire sequences of instruction calls at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  It looks into the problem at a finer granularity and analyzes one  instruction call at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This strategy eliminates the necessity  of using sophisticated sequence models, resulting in reduced  computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The ML-FEED classifier module takes  the 151-dimensional feature vector of an instruction call as  input and outputs the potential exploits that the instruction  call might trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  We show the pipeline for training the ML-FEED exploit  classifier in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='. The training procedure consists of the  following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, we split the dataset of vulnerable  instruction sequences into a training set with 85% of the data  and a test set with the remaining 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We use the training set  to extract individual instructions from the sequences and label  them with the exploits that they can potentially execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then,  we convert these instructions into labeled feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We  describe the process of obtaining such vectors from the dataset  of instruction sequences in Section III-B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Next, we use the  dataset of labeled feature vectors to design the neural network  for our classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We use 3-fold cross-validation to select the  optimal design of our neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We elaborate on this  process in Section III-B6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Finally, we train the optimal neural  network architecture with the training dataset to get our final  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  5) Transforming the dataset: As mentioned in Section  III-A, we convert the exploits in the CWE database to vul-  nerable instruction sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We need to extract the features  and labels of the individual instruction calls from the dataset of  instruction sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We describe this information extraction  procedure next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, we obtain the feature vectors of all the  instruction calls via the feature engineering method discussed  in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, we generate the labels for all the feature  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The labels contain exploits that the corresponding in-  struction calls can trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A single instruction call can trigger  multiple exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The exploits are generally independent of  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, we need a multi-label classifier that predicts  all the exploits that a given instruction can execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We design  our classifier output to be an n-dimensional binary vector  to enable this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We assign the values corresponding to the  triggered exploits in this vector as 1, whereas the rest are  assigned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For example, if instruction call A executes exploits  1 and 23, only the first and 23rd elements of the n-dimensional  output vector will be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  6) Classifier model architecture: The classifier is a fully-  connected feedforward neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We need to design a  lightweight neural network that achieves high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The  neural network architecture has the following constraint – the  number of neurons in the input and output layers is pre-  determined by the input and output dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore,  we only need to determine the number of hidden layers and  the number of neurons in each hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Furthermore, we  target a lightweight model to reduce its latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Hence, we start  with one hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We observe that architectures with one  hidden layer do not produce good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, we expand  our search to two hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Two hidden layers achieve  much better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Finally, we reduce the number of neurons  in the hidden layers to achieve the most lightweight model  with comparable accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Table V describes our chosen  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The output layer has 79 neurons because the  output of the classifier is a 79-dimensional binary vector that  denotes which of the 79 exploit categories are executed at a  given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  TABLE V: The model architecture of the ML-FEED classifier  Number of neurons  Input layer  151  Hidden layer 1  150  Hidden layer 2  100  Output layer  79  We train this classifier model on the complete training  dataset and evaluate it on the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  7) Sequence monitor: Cyber-attackers can exploit a vul-  nerability by executing a set of malicious instructions in a  particular sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, running the same set of instruc-  tions in a different order may not trigger the vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Thus, we need to track the series of program statements  and instruction calls to detect exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Traditional sequence  models, namely LSTMs and transformers, can track the order  implicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, this leads to high computation overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Therefore, we propose a state table-based approach to track  sequences of program instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  First, we discuss a naive pattern matching approach that  can be implemented with a state table and its drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Then, we show how we can overcome these drawbacks with  ML-FEED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Every row in the state table corresponds to a  unique exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We can initialize the exploit states with the  first instruction call in the respective exploit fingerprints in  the naive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' When we encounter any instruction call  in the current state table during program execution, we update  the corresponding states with the following instruction call of  the exploit fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' If the encountered instruction call was  the last one in the fingerprint(s), we would raise an alarm  indicating that the API call has executed the corresponding  exploit(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Since we analyze the API call parameters like the  API name, input types, and API package name, this method is  also applicable to obfuscated codebases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This naive approach  is similar to the subgraph matching approaches used for  vulnerability detection [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Although the naive method is much more lightweight than  sequence models, it has two drawbacks that limit its applica-  bility in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, the naive pattern matching  approach cannot detect semantically-similar instruction calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  As a result, the attacker may avoid detection by executing  different APIs with the same functionality as the instruction  in the fingerprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Second, checking the existence of every in-  struction in the state table is a time-intensive task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This timing  overhead reduces the usefulness of naive pattern matching in  real-time scenarios [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Our solution, ML-FEED, addresses these drawbacks of  naive pattern matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Instead of storing the instructions in  the state table, we store the feature vectors of the instruction  calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Then, we compare the feature vectors of the executed  APIs with the entries in the state table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We use cosine  similarity to measure the similarity between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This method  enables us to capture semantic similarities between APIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' If  the cosine similarity is above a threshold value, we update  the state table with the feature vector of the following API in  the exploit chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We experimentally observed that a threshold  value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9 enables the model to achieve the highest F1 score  and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  To address the second drawback of considerable time over-  heads, we do not compare the feature vector of the executed  API with all the states in the state table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Instead, we compare  it with only the states of the exploit categories predicted by the  ML-FEED classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Our experiments show that this method  requires nearly 10× fewer comparisons per instruction call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  8) Instruction white-list: To further reduce inference time  overheads, ML-FEED also uses a pre-defined white-list of  instructions that are not relevant to the sensitive API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The instructions in the white-list are those that do not have  any control or data dependencies on the exploit chain [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  For example, Java uses the getCaughtException API to handle  exception calls during program execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The presence of  this API call has no impact on the success or failure of  triggering an exploit in our threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Therefore, We  can safely discard frequently used white-listed APIs without  losing sensitive data and control dependency information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Furthermore, this ignorance of non-sensitive API calls does  not affect ML-FEED’s ability to detect potential exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  9) Inference pipeline: We have designed ML-FEED to  efficiently detect exploits without compromising accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We  demonstrate our end-to-end inference pipeline in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As  shown in the figure, we input the executed instruction into  our framework at run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We proceed to the next executed  instruction if the executed instruction is present in our white-  list of non-sensitive API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Otherwise, we analyze the  instruction to determine if it triggers an exploit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' First, we  obtain the feature vector for the executed instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The ML-  FEED exploit classifier processes this feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Next,  White-listed  instruction?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Feature Engineering  Fetch next  instruction  NLP embedding  One-hot encoding  Frequency vector  Exploit classifier  Potential exploits  #Potential exploits = 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  No  Yes  No  Extract states of  potential exploits  Yes  Cosine  similarity  Similarity >  threshold ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Update state  table  Exploit  signature  executed?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  No  Stop execution  Yes  Yes  No  Executed  instruction  Sequence monitor  State table  Exploit  State  1  ( -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='43, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='95, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' , 0)  2  ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='21, -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='67, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=', 1)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 5: The ML-FEED inference pipeline  the classifier outputs the list of potential exploits that the  executed instruction might trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' If this list is empty, we  terminate our analysis of the current instruction and proceed  to the next executed instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Otherwise, we input the list  of potential exploits to the ML-FEED sequence monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The  sequence monitor has a dynamic state table that stores the  states of all the exploits in our threat model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' From the state  table, we extract the exploit states of all the potential exploits  predicted by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Next, we compute the cosine  similarities between the states of the extracted exploits with  the feature vector of the executed instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As mentioned in  Section III-B7, we can successfully detect semantically-similar  instructions by making this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Finally, we update  the exploit states similar to the feature vector of the executed  instruction with the feature vectors of the following instruction  in the corresponding exploit fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, if the state  table indicates that the executed instruction has implemented a  potential vulnerability fingerprint, we stop the execution and  raise an alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Otherwise, we proceed to analyze the next  instruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' EXPERIMENTAL SETUP AND EVALUATION  This section describes our experimental setup, provides  implementation details, and discusses evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Experimental Setup  In what follows, we discuss our vulnerability database  construction and the experimental setup for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  1) Vulnerability database: We train our model with 23  unique CWEs from the CWE dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Table VI shows the  complete list of the CWE vulnerabilities that we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  As shown in the table, we evaluate multiple exploits and  payloads for each CWE, resulting in an analysis of 79 exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Based on the vulnerability database, we construct an exploit  database from the Juliet Test Suite [18] for Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' containing  code examples for 112 CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Given a code example from  this suite, we first construct the Java bytecode using the javac  compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We then use Wala to perform pointer analysis on  the bytecode and build the call graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Finally, we create the  SDG using pointer analysis and call graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A node in the  SDG represents an instruction and an edge represents control  or data flow between instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Although our method can  analyze exploits in any programming language, we focus on  Java-based exploits for our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  We represent every exploit with code snippets in order to  cover all the features of a CWE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We generate 226,207 code  snippets to represent 79 exploits across 23 CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We also  generate 1,101,388 benign code snippets to represent non-  vulnerable code executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, the total size of our database  is 1,327,595 instruction sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We split these sequences  into training and test set with a train-test split of 85%-15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  2) Exploit database: We construct our exploit database  from the Juliet Test Suite [18] for Java, containing code  examples for 112 CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Given a code example from this  suite, we first construct the Java bytecode using the javac  compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We then use Wala to perform pointer analysis on  the bytecode and build the call graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='TABLE VI: Details of CWEs considered under our threat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE id  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Description  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Number of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='exploits  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='External control of system or  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='configuration settings  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-23  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Relative path traversal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-78  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='OS command injection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-81  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Cross-site scripting  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-89  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='SQL injection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-90  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='LDAP injection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-129  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Improper validation of array index  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-134  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Uncontrolled format string  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-190  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Integer overflow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-191  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Integer underflow  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-197  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Numeric truncation error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-226  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Sensitive information uncleared before  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='release  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-256  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Plaintext storage of password  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-259  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Hard-coded password  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-319  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Sensitive information in clear text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-369  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Division by zero  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Resource exhaustion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-470  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Unsafe reflection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-506  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Embedded malicious code  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-570  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Expression always false  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-606  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Unchecked loop condition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-643  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Xpath injection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='CWE-789  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Uncontrolled memory allocation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' we create the SDG using pointer analysis and call  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' A node in the SDG represents an instruction, and  an edge represents control or data flow between instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Although our method can analyze exploits in any programming  language, we focus on Java-based exploits for our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  3) Parameters: We use the most lightweight state-of-the-  art LSTM and transformer models for comparison to ML-  FEED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The hidden state of the LSTM cell is a 6-dimensional  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The lightweight GPT-2 transfomer that we use in our  experiments has 117 million parameters and 12 decoder layers  with 12 attention heads in each self-attention layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The GPT-2  model processes a series of 1024 API calls at any given time  instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  4) Machine configuration:  We ran experiments on a  Lenovo ThinkPad laptop with an Intel Core i7-8665U CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The CPU has four cores with a maximum operating frequency  of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2 GHz for each core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The processor base frequency is  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='90GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Experimental Results  This section provides experimental results obtained by  running ML-FEED and state-of-the-art ML-based sequence  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We report the performance metrics of our model on  every CWE in our threat model in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We also denote  the CWEs that belong to the OWASP top-10 vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The non-profit organization, OWASP, ranks the most critical  web application security risks based on their frequency of  occurrence, severity, and magnitude of the potential impact  on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Security analysts frequently use the OWASP  top 10 vulnerability list as a guideline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  1) Metrics: We define the performance metrics in terms of  true positives (TP), true negatives (TN), false positives (FP),  and false negatives (FN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' TP and TN refer to the number  of correctly classified positive and negative labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' FP and  FN refer to the number of incorrect positive and negative  predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' These values are often represented  in the form of a confusion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We use these metrics to  compute the accuracy, precision, recall, and F1-score of the  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  1) Accuracy: Accuracy denotes the overall performance of  the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Although it is a good evaluation metric, it  suffers from biased results when evaluating imbalanced  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Accuracy =  TP + TN  TP + TN + FP + FN  2) Precision: Precision measures what fraction of the posi-  tive predictions of the model is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We can compute  the precision value from the confusion matrix of a binary  classification problem with the following formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Precision =  TP  TP + FP  3) Recall: Recall measures what fraction of the true positive  labels in a binary dataset are predicted as positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The  formula for computing the recall value from the confusion  matrix of a binary classification problem is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Recall =  TP  TP + FN  4) F1 score: The F1 score is a statistical measure calculated  as the harmonic mean of precision and recall values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It  is a popular metric for evaluating binary classification  models, especially with an imbalanced dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  F1 = 2 × precision × recall  precision + recall  We report the performance metrics of our model on every  CWE in our threat model in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We show the results  of only one exploit per CWE in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We also denote the  CWEs that belong to the OWASP top-10 vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  In Table VII, we observe that the accuracies of all the  models are quite high in comparison to other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This  is due to an imbalanced dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, precision, recall, and  F1 scores are more reliable metrics for evaluating our model  on this dataset than accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We observe that ML-FEED  achieves the best F1 scores for most CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The transformer  performs better than ML-FEED for a few CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, we  observe later in Table IX that the transformer incurs orders of  magnitude higher computation overhead than ML-FEED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This  makes ML-FEED more practical in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  2) Error analysis:  Here, we analyze the shortcomings  of ML-FEED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML-FEED outputs false positives and false  negatives for some of the CWEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We observe in Table VII  that all the state-of-the-art ML models perform poorly on  CWE-319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' CWE-319 vulnerabilities arise due to the storage  TABLE VII: Performance of ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' All values depict percentages (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The best performances are denoted in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  LSTM  Transformer  ML-Feed  CWE id  OWASP  top-10  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  F1  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Prec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
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+page_content='0  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='8  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='0  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  CWE-643  \x13  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='7  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='4  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='4  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='3  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6  CWE-789  \x17  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='4  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='8  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  of sensitive information in cleartext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' One must manually check  the repository for sensitive content or train separate NLP  models to catch sensitive data in cleartext to detect this  vulnerability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We aim to tackle this problem in the future by  cascading a sensitivity-predicting NLP model with ML-FEED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  In Table VII, we also observe that ML-based methods have  a relatively low recall for detecting CWE-400 vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  CWE-400 is a denial-of-service (DoS) attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' DoS attacks are  difficult to prevent because they are highly dependent on the  system architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, system-specific intrusion detection  systems are more efficient in detecting DoS attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However,  the high precision scores achieved by ML-based methods in  Table VII can help improve the efficiency of IDS [15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  We can use the precision, recall, and F1 score metrics  to evaluate only binary classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However,  our exploit classification task is a multi-label classification  problem with 79 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' For a multi-label classification task  with n labels, we have n confusion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Precision, recall,  and F1 scores cannot handle multiple confusion matrices by  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, we can compute these metrics for each of  the n confusion matrices and average them to obtain the  mean precision, recall, and F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We show the mean per-  formance metrics of ML-FEED and state-of-the-art sequence  models across all 79 labels in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' In Table VII, we  observed that accuracy is not a reliable performance metric  for our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Thus, we do not report the average accuracy  in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  TABLE VIII: Performance of ML models on 79 exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' All  values depict percentages (%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Model  Precision  Recall  F1  LSTM  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='4  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9  Transformer  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='1  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='8  ML-FEED  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='2  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='4  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='8  3) Computation overhead: We demonstrate the computa-  tion overheads of each model in terms of their inference time  and model size in Table IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Higher overheads degrade the  applicability of the models in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  TABLE IX: The inference time and model size of ML models  Model  Inference  time (µs)  Number of  parameters  Model size  LSTM  475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='81  213,116  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5 MB  Transformer  495162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='72  117M  523 MB  ML-FEED  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='53  46,788  970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5 KB  Table IX shows that ML-FEED is 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9× and 75828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='9×  faster than the lightweight LSTM and GPT-2 models, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It also has 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5× and 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='6× fewer parameters than  the LSTM and GPT-2 models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 6, we observe that the LSTM is more efficient  than the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The transformer is inefficient because it  analyzes a sequence of the last 1024 API calls for inference at  every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, the LSTM analyzes only two fixed-  length vectors at every time step: the current API call and  the hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 6 also shows that ML-FEED is more  efficient than the LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This is because it does not maintain  and update a hidden state vector at every time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Instead,  ML-FEED uses a state-table to keep track of the sequence of  API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The LSTM analyzes the current API call along with its  hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The hidden state is a fixed length vector that  encapsulates the history of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' On the contrary, the  transformer analyzes a sequence of the last 1024 API calls for  inference at every time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This increases the effectiveness  of the transformer over LSTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, this makes the  transformers inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Like the LSTM, ML-FEED analyzes  only the current API call.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, it is more efficient than the  LSTM because it does not maintain and update a hidden state  TABLE X: Vulnerability detector feature comparison  Model  Run-  time  Rule-  based  ML-  based  Efficient  Constraint solving [20]  \x13  \x13  \x17  \x13  Modular checking [21]  \x13  \x17  \x17  \x13  VulDeePecker [6]  \x17  \x17  \x13  \x17  µ-VulDeePecker [13]  \x17  \x17  \x13  \x17  Yamaguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' [14]  \x17  \x17  \x13  \x13  SHARKS [15]  \x13  \x17  \x13  \x17  GRAVITAS [16]  \x13  \x17  \x13  \x17  MLASA [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ]  \x13  \x17  \x13  \x17  GPT-2 [8]  \x17  \x17  \x13  \x17  SySevVR [22]  \x17  \x17  \x13  \x17  Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' [11]  \x17  \x17  \x13  \x13  LSTM [23]  \x17  \x17  \x13  \x17  ML-FEED  \x13  \x13  \x13  \x13  vector at every time-step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Instead, ML-FEED uses a state-table  to keep track of the sequence of API calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We demonstrate  the efficiency of ML-FEED in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  loge  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' 6: Comparison of the efficiency of ML-FEED with state-  of-the-art ML-based sequence models  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Comparison  Table X shows that ML-FEED is the only framework that  satisfies all the features in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML-FEED can perform  run-time analysis effectively for all exploits in its training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The rule-based fingerprint detector in the sequence monitor  makes it more efficient than other state-of-the-art ML methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  The exploit classifier in the inference pipeline enables pattern-  based exploit detection to detect a wide range of exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Evaluation with Real Attack Traces  To demonstrate the practical usage of our model, we use  ML-FEED to evaluate three popular open-source libraries with  existing CVEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We choose open-source libraries because our  model requires access to the source code to analyze potential  vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We evaluate ML-FEED on the attack traces  of CVE-2021-25646, CVE-2021-39139, and CVE-2020-1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ML-FEED successfully identifies all three vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We  describe the detected vulnerabilities next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  1) Remote code execution: ML-FEED identifies two remote  code execution (RCE) vulnerabilities in the popular Apache  Druid and XStream library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Apache Druid is a popular library  that provides real-time database support for powering modern  analytics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' On the other hand, XStream allows  users to serialize objects to XML and back again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' How-  ever, while evaluating Apache Druid, ML-FEED detects an  RCE vulnerability that allows a malicious attacker to execute  JavaScript codes embedded in various types of requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' An  authenticated user can send a specially-crafted request, regard-  less of server configuration, which forces Druid to run user-  provided JavaScript code for that request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This vulnerability  is reported in CVE-2021-25646, with a high severity score of  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML-FEED identifies another RCE vulnerability that may  allow a remote attacker to load and execute arbitrary code from  a remote host just by manipulating the processed input stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  This vulnerability has been reported in CVE-2021-39139, with  a high severity score of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  2) Lightweight directory access protocol (LDAP) injection:  ML-FEED also identifies an LDAP injection vulnerability in  the Apache Druid library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This vulnerability allows Druid  API callers to bypass the credentialsValidator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='userSearch filter  barrier that determines if a valid LDAP user is allowed to  authenticate with Druid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Callers can also retrieve any LDAP  attribute of users from the LDAP server, as long as that  information is visible to the Druid server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This information  disclosure does not require the caller to be a valid LDAP user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  This vulnerability is reported in CVE-2020-1958 [24] , with a  medium severity score of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' RELATED WORK  Vulnerability exploit detection is a fundamental challenge in  software security, for which researchers have proposed many  notable solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The two broad categories of classical vul-  nerability detection approaches are rule-based and similarity-  based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Rule-based methods involve manually crafting  the rules of vulnerabilities [20, 21, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' These rules  attempt to encompass both the signatures and behaviors of  the exploits [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The vulnerabilities and exploit payloads are  detected by naive pattern matching against the rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' As  discussed in Section III, naive pattern matching suffers from  generalization limitations and considerable time overheads [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ML-FEED overcomes these limitations by using ML and  reducing the search space for vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Similarity-based  approaches rely on code sharing and code cloning between  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' The assumption is that the exposure is replicated  if a vulnerable code is cloned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' These approaches aim to  find similarities with vulnerable codebases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Similarity-based  vulnerability detection techniques are effective only when  codebases are cloned into the testing project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Vulnerability detection methods have undergone a radical  transformation with the advent of ML [15, 16, 19, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML-  based methods perform much better than rule-based meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Although ML can accurately learn the underlying pat-  terns of vulnerabilities, manual effort is still required to  characterize the vulnerable programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' VulDeePecker [6] and  ML-FEED  LSTM  Transformer  20  15  10  5  0  Inference time  Number of parametersµVulDeePecker [13] are among the first fully-automated ML  models to detect and classify vulnerabilities successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Sequence models like transformers and LSTMs have found  success in modeling exploit chains and detecting exploits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  However, they suffer from degraded performance in capturing  long-range dependencies in the attack sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Transformers  are better at capturing long-range dependencies than RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Hence, they have found various applications in cybersecurity  analysis [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' However, transformers incur large computation  overheads, limiting their application in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Recent methods like ATLAS [29] and ALchemist [30] have  extended the applications of sequence-based ML analysis to  domains like advanced persistent threat analysis and audit  report analysis-based attack investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Malware detection  is another domain in which ML has been highly success-  ful [31, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML has also shown promising results in  detecting attacks on networks [34], operating systems [35], and  web applications [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Despite the rapid progress of ML meth-  ods, they suffer from a lack of explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' ML algorithms  generally work as black boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Hence, security analysts cannot  easily interpret their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' This lack of transparency  diminishes the trust analysts place in ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Fortunately,  researchers have recently developed explainable ML models  for cybersecurity [37, 38, 39, 40] that can potentially increase  the confidence that can be placed in ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' CONCLUSION  In this article, we presented ML-FEED, a lightweight ML-  based exploit detection framework that is orders of magnitude  faster than state-of-the-art methods while achieving better  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It analyzes sequences of API calls and raises  an alarm when an exploit is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' It can also classify  the exploit into CWE categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' We also proposed an auto-  mated method to extract intelligence from the CWE and CVE  vulnerability databases, thus mitigating the need for human  intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content=' Finally, we used ML-FEED to analyze three  public libraries and found the existing CVEs in all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  Finally, ML-FEED was successful in detecting existing CVEs  in three public libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
+page_content='  ACKNOWLEDGMENT  We thank Bhishma Dedhia for his help with implementing  the GPT-2 transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
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+page_content=' Tyson, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfGgkb/content/2301.04314v1.pdf'}
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+arXiv:2301.05201v1  [math.FA]  12 Jan 2023
+ON WEAK-TYPE (1, 1) FOR AVERAGING TYPE
+OPERATORS
+SERGI BAENA-MIRET∗∗ AND MAR´IA J. CARRO∗
+Abstract. It is known that, due to the fact that L1,∞ is not a Banach
+space, if (Tj)j is a sequence of bounded operators so that
+Tj : L1 −→ L1,∞,
+with norm less than or equal to ||Tj|| and �
+j ||Tj|| < ∞, nothing can be
+said about the operator T = �
+j Tj. This is the origin of many difficult and
+open problems. However, if we assume that
+Tj : L1(u) −→ L1,∞(u),
+∀u ∈ A1,
+with norm less than or equal to ϕ(||u||A1)||Tj||, where ϕ is a nondecreasing
+function and A1 the Muckenhoupt class of weights, then we prove that,
+essentially,
+T : L1(u) −→ L1,∞(u),
+∀u ∈ A1.
+We shall see that this is the case of many interesting problems in Harmonic
+Analysis.
+1. Introduction
+Let {Tθ}θ be a family of operators indexed in a probability measure space
+such that
+(1.1)
+Tθ : L1(Rn) −→ L1,∞(Rn)
+with norm less than or equal to a uniform constant C. What can we say about
+the boundedness of the average operator
+TAf(x) =
+�
+Tθf(x)dP(θ),
+x ∈ Rn,
+whenever is well defined? The following trivial example shows that, at first
+sight, nothing of interest can be concluded: for 0 < θ < 1, set
+Tθf(x) =
+� 1
+0 f(y)dy
+|x − θ| ,
+x ∈ (0, 1),
+so clearly Tθ satisfies (1.1), but
+TAf(x) =
+� 1
+0
+Tθf(x)dθ ≡ ∞,
+∀x ∈ (0, 1).
+2020 Mathematics Subject Classification. 42A99, 46E30, 47B34, 42A38.
+Key words and phrases. Muckenhoupt weights, restricted weak-type extrapolation, average
+operators, Fourier multiplier operators, the Bochner-Riesz operator.
+The authors were partially supported by grant PID2020-113048GB-I00 funded by
+MCIN/AEI/ 10.13039/501100011033.
+E-mail addresses:
+∗mjcarro@ucm.es, ∗∗sergibaena@ub.edu.
+1
+
+2
+On weak-type (1, 1) for averaging type operators
+However, things change completely, and this is one of the main goals of this
+paper, if we assume that
+Tθ : L1(u) −→ L1,∞(u),
+∀u ∈ A1,
+where A1 is the class of Muckenhoupt weights defined as follows: we say that
+u ∈ A1 if u is a nonnegative locally integrable function (called weight) so that
+there exists a positive constant C such that
+Mu(x) ≤ Cu(x),
+a.e. x ∈ Rn,
+where M is the Hardy-Littlewood maximal operator defined by
+Mf(x) = sup
+Q∋x
+1
+|Q|
+�
+Q
+|f(y)| dy,
+f ∈ L1
+loc(Rn),
+with the supremum being taken over all cubes Q ⊆ Rn containing x ∈ Rn. We
+denote by ∥u∥A1 the least constant C satisfying such inequality. Besides, it is
+wellknown ([5, 29]) that
+M : L1(u) −→ L1,∞(u)
+⇐⇒
+u ∈ A1,
+with ||M||L1(u)→L1,∞(u) ≤ C||u||A1.
+Let us start with a very simple and motivating example. Let m be a bounded
+variation function on R that is right-continuous and normalized by the condition
+m(−∞) = 0. Then,
+m(ξ) =
+� ξ
+−∞
+dm(t) =
+�
+R
+χ(−∞,ξ)(t) dm(t) =
+�
+R
+χ(t,∞)(ξ) dm(t),
+∀ξ ∈ R,
+where dm is the Lebesgue-Stieltjes measure associated with m and it is a finite
+measure. Hence, if we consider the Fourier multiplier operator
+Tmf(x) =
+�
+R
+m(ξ) ˆf(ξ)e2πixξdξ,
+x ∈ R,
+for every Schwartz function f, where
+ˆf(ξ) =
+�
+R
+f(x)e−2πixξdx,
+ξ ∈ R,
+is the Fourier transform of the function f, a formal computation shows that
+Tmf(x) =
+�
+R
+Htf(x)dm(t),
+∀x ∈ R,
+where
+Htf(x) = Tχ(t,∞)f(x) =
+� ∞
+t
+ˆf(ξ)e2πixξdξ,
+x ∈ R.
+Now, Ht is essentially a Hilbert transform operator (recall that Hf = Tmf with
+m(ξ) = −i sgn ξ) because
+χ(t,∞)(ξ) = sgn(ξ − t) + 1
+2
+,
+∀ξ ∈ R.
+Thus, since
+Ht : Lp(R) −→ Lp(R),
+∀p > 1,
+
+3
+we have, using the Minkowski’s integral inequality and the density of the
+Schwartz functions on Lp(R), that every right-continuous bounded variation
+function is a Fourier multiplier on Lp(R) for every p > 1. However, even though
+we also have
+Ht : L1(R) −→ L1,∞(R),
+we cannot deduce (at least not immediately) that the same boundedness holds
+for Tm due to the lack of the Minkowski’s integral inequality for the space
+L1,∞(R).
+The main theorem of this paper will show that since
+Ht : L1(u) −→ L1,∞(u),
+ϕ(||u||A1),
+∀u ∈ A1,
+with ϕ being a nondecreasing function on [1, ∞) and independent of t ∈ R, then
+for every measurable set E ⊆ Rn,
+||TmχE||L1,∞(u) ≤ C(m)ϕ(C2||u||A1)(1 + log ||u||A1)u(E),
+∀u ∈ A1.
+The result will be proved using an extended version of the Rubio de Francia’s
+extrapolation theorem which deals with the theory of Muckenhoupt weights
+(see Theorem 1.1). Other interesting applications will be given in Section 4.
+Let us now recall (see [5, 29]) that for p > 1,
+M : Lp(v) −→ Lp(v)
+⇐⇒
+v ∈ Ap,
+where this class of weights is defined by the condition
+∥v∥Ap = sup
+Q⊆Rn
+� 1
+|Q|
+�
+Q
+v(x) dx
+�� 1
+|Q|
+�
+Q
+v(x)
+1
+1−p dx
+�p−1
+< ∞,
+and, given a weight v, Lp(v) is the Lebesgue space defined as the set of mea-
+surable functions f such that
+||f||Lp(v) =
+��
+Rn |f(x)|pv(x) dx
+� 1
+p
+< ∞.
+Indeed, (see [18]) for every p ≥ 1,
+M : Lp(v) −→ Lp,∞(v)
+⇐⇒
+v ∈ Ap,
+with Lp,∞(v) being the weak Lebesgue space defined as the set of measurable
+functions f so that
+∥f∥Lp,∞(v) = sup
+y>0
+yλv
+f(y)
+1
+p < ∞.
+Here, λv
+f is the distribution function of f with respect to v defined by
+λv
+f(y) = v
+��
+x ∈ Rn : |f(x)| > y
+��
+,
+y > 0.
+(Here we are using the standard notation v(E) =
+�
+E v(x) dx for every measur-
+able set E ⊆ Rn. If v = 1, we shall write λf and |E|. See [3] for more details
+about this topic.)
+An important result for our purpose concerning Ap weights is the extrapola-
+tion theorem of Rubio de Francia [32, 33] (see also [15, 17, 19, 20, 21]) which,
+nowadays, can be formulated as follows:
+
+4
+On weak-type (1, 1) for averaging type operators
+Theorem 1.1 ([17]). Let (f, g) be a pair of measurable functions such that for
+some 1 ≤ p0 < ∞,
+||g||Lp0(v) ≤ ϕ(∥v∥Ap0 )||f||Lp0(v),
+∀v ∈ Ap0,
+with ϕ being a nondecreasing function on [1, ∞). Then, for every 1 < p < ∞,
+||g||Lp(v) ≤ C1ϕ
+�
+C2∥v∥
+max
+�
+1, p0−1
+p−1
+�
+Ap
+�
+||f||Lp(v),
+∀v ∈ Ap,
+with C1 and C2 being two positive constants independent of all parameters in-
+volved.
+We have to emphasize here that although p0 can be 1, it is not possible, in
+general, to extrapolate till the endpoint p = 1 (take just T = M ◦ M or see,
+for instance, [30] where a counterexample is given in the case of commutators).
+However, in the recent papers [9, 12], a Rubio de Francia extrapolation theory
+for operators satisfying a weighted restricted weak-type boundedness for the
+class of weights �Ap (slightly bigger than the class Ap) has been developed. The
+main advantage of this new class of weights is that allows to obtain boundedness
+estimates at the endpoint p = 1.
+Definition 1.2. We define
+�Ap =
+�
+v ∈ L1
+loc(Rn) : ∃ h ∈ L1
+loc(Rn) and ∃ u ∈ A1 with v = (Mh)1−pu
+�
+,
+endowed with the norm
+∥v∥ �
+Ap = inf
+�
+∥u∥
+1
+p
+A1 : v = (Mh)1−pu
+�
+.
+Clearly, �A1 = A1, while for 1 < p < ∞, Ap ⊊ �Ap.
+It holds that (see [9, 14, 24]) for every 1 ≤ p < ∞ and every v ∈ �Ap,
+M : Lp,1(v) −→ Lp,∞(v),
+∥M∥Lp,1(v)−→Lp,∞(v) ≤ C∥v∥ �
+Ap,
+where the Lorentz space Lp,1(v) is defined as the set of measurable functions f
+such that
+∥f∥Lp,1(v) = p
+� ∞
+0
+λv
+f(y)
+1
+p dy < ∞.
+Then, the restricted weak-type Rubio de Francia extrapolation result proved
+in [9] can be stated as follows:
+Theorem 1.3 ([9]). Let 1 < p0 < ∞ and let T be an operator such that
+T : Lp0,1(v) −→ Lp0,∞(v),
+ϕ(∥v∥ �
+Ap0),
+∀v ∈ �Ap0,
+where ϕ is a positive nondecreasing function on [1, ∞). Then, T is of weighted
+restricted weak-type (1, 1) for every weight in A1; that is, for any measurable
+set E ⊆ Rn, there exists a constant C > 0 independent of E such that
+(1.2)
+∥TχE∥L1,∞(u) ≤ C∥u∥
+1− 1
+p0
+A1
+ϕ
+�
+∥u∥
+1
+p0
+A1
+�
+u(E),
+∀u ∈ A1.
+
+5
+For simplicity, whenever an operator T satisfies that for every measurable
+set E,
+∥TχE∥L1,∞(u) ≤ Cuu(E),
+we shall denote it by
+T : L1
+R(u) −→ L1,∞(u),
+Cu.
+Remark 1.4. The complete result that T is of weighted weak-type (1, 1) (i.e.,
+that the estimate in (1.2) holds for every f ∈ L1(u)) is, in general, false (see
+[9]). However, under certain mild condition in the operator T (see Section 2.2)
+the weighted weak-type (1, 1) boundedness can be proved.
+Remark 1.5. We should emphasize here that our operators do not need to be
+sublinear. However, if T is sublinear, it was proved in [34] that
+T : L1
+R(u) −→ L1,∞(u)
+is equivalent to have the boundedness on the space
+B∗(u) =
+�
+f :
+� ∞
+0
+λu
+f(t)
+�
+1 + log ||f||1
+λu
+f(t)
+�
+dt < ∞
+�
+,
+which can be endowed with a quasi-norm.
+Our main goal will be consequence of the fact that the converse of Theorem
+1.3 is also true, and hence
+T : L1
+R(u) −→ L1,∞(u), ∀u ∈ A1 ⇐⇒ T : Lp0,1(v) −→ Lp0,∞(v), ∀v ∈ �Ap0.
+Indeed, if p′ is the conjugate exponent of p > 1 (that is, 1
+p + 1
+p′ = 1) our main
+theorem reads as follows:
+Theorem 1.6. Let (f, g) be a pair of measurable functions such that
+||g||L1,∞(u) ≤ ϕ(∥u∥A1)||f||L1(u),
+∀u ∈ A1,
+with ϕ being a nondecreasing function on [1, ∞). Then, for every 1 < p < ∞,
+||g||Lp,∞(v) ≤ Φ(∥v∥ �
+Ap)||f||Lp,1(v),
+∀v ∈ �Ap,
+where
+Φ(r) = C1ϕ(C2rp)rp−1(1 + log r)
+2
+p′ ,
+r ≥ 1,
+with C1 and C2 being two positive constants independent of all parameters in-
+volved.
+As a consequence we obtain the following corollary:
+Corollary 1.7. Let c = (cj)j ∈ ℓ1 and let {Tj}j be such that
+Tj : L1(u) −→ L1,∞(u),
+ϕ(||u||A1),
+∀u ∈ A1,
+where ϕ is a positive nondecreasing function on [1, ∞). Then, for every u ∈ A1,
+�
+j
+cjTj : L1
+R(u) −→ L1,∞(u),
+C1||c||ℓ1ϕ(C2||u||A1)(1 + log ||u||A1).
+
+6
+On weak-type (1, 1) for averaging type operators
+As usual, we shall use the symbol A ≲ B to indicate that there exists a
+universal positive constant C, independent of all important parameters, such
+that A ≤ CB. When A ≲ B and B ≲ A, we will write A ≈ B.
+The paper is organized as follows. In Section 2, we will see some previous
+notions, the necessary definitions and some technical results which shall be used
+later on. Indeed, there we will prove Lemma 2.5 which will be essential in the
+proof of the main result given in Section 3. Further, Section 4 contains our
+main examples and applications. Finally, we also include a last section related
+with similar results in the context of limited extrapolation.
+2. Preliminary notions and some technical results
+2.1. A1 weights. Let us start by recalling some wellknown facts of the class
+A1:
+i) ([16, Theorem 7.7]) A weight u belongs to A1 if and only if there exists
+h ∈ L1
+loc(Rn) and K such that K, K−1 ∈ L∞(Rn) satisfying that, for some
+0 < µ < 1,
+u(x) = K(x)(Mh(x))µ,
+a.e. x ∈ Rn,
+where L∞(Rn) consists of all measurable functions f such that
+||f||∞ := ||f||L∞(Rn) = ess sup f < ∞.
+ii) ([12, Lemma 2.12]) For every h ∈ L1
+loc(Rn), every u ∈ A1 and 0 < µ < 1,
+then (Mh)µu1−µ ∈ A1 with
+(2.1)
+����(Mh)µu1−µ
+����
+A1
+≲ ∥u∥A1
+1 − µ .
+iii) ([31, Lemma 5.1]) If t = 1 +
+1
+2n+1∥u∥A1 , then
+(2.2)
+ut ∈ A1
+and
+||ut||A1 ≲ ||u||A1.
+2.2. (ε, δ)-atomic operators. As mentioned above, in general, the following
+implication does not hold for every u ∈ A1:
+T : L1
+R(u) −→ L1,∞(u)
+=⇒
+T : L1(u) −→ L1,∞(u),
+even if T is a sublinear operator. However, it was proved in [9, Theorem 3.5]
+that for a quite big class of operators the above implication is true.
+Definition 2.1. Given δ > 0, a function a ∈ L1(Rn) is called a δ-atom if it
+satisfies the following properties:
+(i)
+�
+Rn a(x) dx = 0, and
+(ii) there exists a cube Q ⊆ Rn such that |Q| ≤ δ and supp a ⊆ Q.
+Definition 2.2. (a) A sublinear operator T is called (ε, δ)-atomic if, for every
+ε > 0, there exists δ > 0 satisfying that
+∥Ta∥L1(Rn)+L∞(Rn) ≤ ε∥a∥1,
+for every δ-atom a.
+
+7
+(b) A sublinear operator T is said to be (ε, δ)-atomic approximable if there exists
+a sequence {Tj}j of (ε, δ)-atomic operators such that, for every measurable set
+E ⊆ Rn, then |TjχE| ≤ |TχE| and, for every f ∈ L1(Rn) such that ∥f∥∞ ≤ 1,
+|Tf(x)| ≤ lim
+j inf |Tjf(x)|,
+a.e. x ∈ Rn.
+Examples:
+In [6], the author showed that for sublinear operators, the prop-
+erty of being (ε, δ)-atomic is not a strong one. For instance, if
+Tf(x) = K ∗ f(x) =
+�
+Rn K(y − x)f(y) dy,
+x ∈ Rn,
+with K ∈ Lp(Rn) for some 1 ≤ p < ∞, then T is (ε, δ)-atomic. Further, if
+T ∗f(x) = sup
+j∈N
+����
+�
+Rn Kj(x, y)f(y) dy
+����,
+x ∈ Rn,
+with
+lim
+y→x ∥Kj( · , y) − Kj( · , x)∥L1(Rn)+L∞(Rn) = 0,
+then T ∗ is (ε, δ)-atomic approximable (for example, standard maximal Calder´on
+-Zygmund operators are of this type). In general,
+T ∗f(x) = sup
+j
+|Tjf(x)|,
+x ∈ Rn,
+where {Tj}j is a sequence of (ε, δ)-atomic, is (ε, δ)-atomic approximable and
+the same holds for
+Tf(x) =
+� �
+j
+|Tjf(x)|q
+� 1
+q
+,
+x ∈ Rn,
+with q ∈ [1, ∞) and
+Tf(x) =
+�
+j
+Tjf(x),
+x ∈ Rn.
+(See [6, 9] for more examples.)
+Theorem 2.3 ([9]). Let T be a sublinear operator (ε, δ)-atomic approximable.
+Then, given u ∈ A1,
+T : L1
+R(u) −→ L1,∞(u),
+Cu
+=⇒
+T : L1(u) −→ L1,∞(u),
+2nCu∥u∥A1.
+2.3. A Sawyer-type inequality. Here we will study one of the often-called
+Sawyer-type inequalities for weights belonging in the restricted class of weights
+ˆAp. First, to do so, we need the following result.
+Lemma 2.4. Let 1 < p < ∞ and v ∈ �Ap. Take
+1
+p′ < θ ≤ 1 and set u0 =
+(Mh)
+(p−1)(1−θ)
+θ
+. Then,
+(2.3)
+Mu0 : L
+θp′
+θp′−1,1(v) −→ L
+θp′
+θp′−1 ,∞(v),
+
+8
+On weak-type (1, 1) for averaging type operators
+with constant less than or equal to
+θ2p′Cn,p
+1 − p(1 − θ) ∥v∥
+2(θp′−1)
+θ(p′−1)
+ˆ
+Ap
+,
+and where
+Mu0f(x) = sup
+Q∋x
+1
+u0(Q)
+�
+Q
+|f(y)|u0(y) dy,
+x ∈ Rn.
+Proof. Observe that since v ∈ ˆAp, then v is a doubling weight with constant
+∆v ≤ C1 ∥v∥p
+ˆ
+Ap. Therefore, according to [12, Lemma 2.2 (i)], (2.3) is bounded
+with constant less than or equal to
+C1 ∥v∥
+θp′−1
+θ(p′−1)
+ˆ
+Ap
+θp′
+
+ sup
+E⊆Q
+u0(E)
+u0(Q)
+�v(Q)
+v(E)
+� θp′−1
+θp′
+
+ ,
+where the supremum is taken over all cubes Q and all measurable sets E ⊆ Q.
+Now, given a cube Q and a measurable set E ⊆ Q,
+�v(Q)
+v(E)
+� θp′−1
+θp′
+=
+�|Q|
+|E|
+� θp′−1
+θ(p′−1) ��|E|
+|Q|
+�p v(Q)
+v(E)
+� θp′−1
+θp′
+≤ C2 ∥v∥
+θp′−1
+θ(p′−1)
+ˆ
+Ap
+�|Q|
+|E|
+� θp′−1
+θ(p′−1)
+and, as well, due to [12, Lemma 2.5],
+sup
+E⊆Q
+u0(E)
+u0(Q)
+�|Q|
+|E|
+� θp′−1
+θ(p′−1)
+≤
+θC3
+1 − p(1 − θ),
+which yields the desired result.
+□
+The following lemma was proved for the case µ = 1 in [12, Lemma 2.6], and
+the extension to other µ’s has been fundamental for our purposes.
+Lemma 2.5. Let 1 < p < ∞ and let v = (Mh)1−pu ∈ �Ap. Take θ and µ so
+that
+1
+p′ < θ < µ ≤ 1 and set vθ = (Mh)1−puθ. Then,
+���Mµ(χEvθ)
+vθ
+���
+Lp′,∞(v) ≲ Cp,θ,µ(u)v(E)
+1
+p′ ,
+∀E ⊆ Rn,
+where Mµf := M(|f|1/µ)µ and
+(2.4)
+Cp,θ,µ(u) =
+�
+p2
+(p − 1)2(µ − θ)(θ − 1
+p′ )2
+�θ
+∥u∥
+2θ− 2
+p′
+A1
+.
+Proof. Observe that in virtue of the Kolmogorov’s inequality [22] with 1 < r′ =
+1
+θ < p′, it is enough to prove that
+sup
+F ⊆Rn
+1
+v(F)
+1
+r′ − 1
+p′
+� �
+F
+(Mh(x))(p−1)(r′−1)�
+Mµ(χE(Mh)1−puθ)(x)
+�r′
+dx
+� 1
+r′
+≲ Cp,θ,µ(u)v(E)
+1
+p′ .
+
+9
+Then, using the Fefferman-Stein’s inequality [18], since µr′ > 1, we obtain
+that
+�
+F
+(Mh(x))(p−1)(r′−1)�
+Mµ(χE(Mh)1−puθ)(x)
+�r′
+dx
+≲
+µr′
+µr′ − 1
+�
+E
+(Mh(x))(1−p)r′M(χF (Mh)(p−1)(r′−1))(x)u(x)dx.
+Now, since u0 = (Mh)(p−1)(r′−1) ∈ A1, we have that, for every x ∈ E and
+every cube Q ∋ x in Rn,
+1
+|Q|
+�
+Q
+χFu0(y) dy ≤ u0(Q)
+|Q| Mu0(χF )(x) ≤ ||u0||A1u0(x)Mu0(χF )(x)
+≲
+1
+1 − (p − 1)(r′ − 1)u0(x)Mu0(χF )(x),
+(2.5)
+where in the last estimate we have used (2.1). Hence, taking the supremum
+over all cubes Q ∈ Rn such that Q ∋ x in (2.5), with x ∈ E, we deduce that
+�
+E
+(Mh(x))(1−p)r′M(χF (Mh)(p−1)(r′−1))(x)u(x)dx
+≲
+1
+1 − (p − 1)(r′ − 1)
+�
+E
+Mu0(χF )(x)v(x) dx.
+Therefore, since r′ = 1
+θ, the inequality we want to prove will hold if we see
+that
+sup
+E⊆Rn
+1
+v(E)
+1
+p′
+��
+E
+Mu0(χF )(x)v(x) dx
+�θ
+≲
+�(µ − θ)(1 − p(1 − θ))
+µθ
+�θ
+Cp,θ,µ(u)v(F)θ− 1
+p′
+or equivalently,
+sup
+E⊆Rn
+1
+v(E)1−
+�
+1− 1
+θp′
+�
+�
+E
+Mu0(χF )(x)v(x) dx
+≲
+�(µ − θ)(1 − p(1 − θ))
+µθ
+�
+Cp,θ,µ(u)
+1
+θ v(F)1− 1
+θp′ .
+(2.6)
+Finally, using again the Kolmogorov’s inequality in (2.6), it is enough to
+prove that
+Mu0 : L
+θp′
+θp′−1 ,1(v) −→ L
+θp′
+θp′−1,∞(v)
+with constant less than or equal to
+cn,p
+θp′
+�(µ − θ)(1 − p(1 − θ))
+µθ
+�
+Cp,θ,µ(u)
+1
+θ .
+According to Lemma 2.4, this will happen if
+Cp,θ,µ(u) ≳
+�
+p2
+(p − 1)2(µ − θ)(1 − p(1 − θ))2
+�θ
+∥u∥
+2(θp′−1)
+p′
+A1
+,
+from which the desired result follows by taking Cp,θ,µ(u) as in (2.4).
+
+10
+On weak-type (1, 1) for averaging type operators
+□
+3. Proof of the main result
+We are now ready to prove our main result:
+Proof of Theorem 1.6. Let h ∈ L1
+loc(Rn) and u ∈ A1 so that v = (Mh)1−pu ∈
+�Ap. Further, let us take
+1
+p′ < θ < 1,
+µ := 1 − 1 − θ
+t
+and
+vθ := (Mh)1−puθ,
+where t = 1+
+1
+2n+1∥u∥A1 satisfies ut ∈ A1 and ||ut||A1 ≲ ||u||A1 (see (2.2)). Then,
+θ < µ < 1 and, by (2.1), for every measurable set F ⊆ Rn,
+u0 = Mµ(χF vθ)u1−θ = M(χF v1/µ
+θ
+)µ(ut)1−µ ∈ A1,
+||u0||A1 ≤ C||u||A1
+1 − µ .
+Let y > 0 and set F = {x : |g(x)| > y} so that v(F) = λv
+g(y). We can
+assume, without lost of generality, that v(F) < ∞, since on the contrary we
+can take gN = gχB(0,N) and let N go to infinity at the end of our estimate.
+By hypothesis we obtain that
+yλv
+g(y)
+=
+y
+�
+{x : |g(x)|>y}
+v(x) dx ≤ y
+�
+F
+Mµ(χF vθ)(x)u(x)1−θdx
+≤
+ϕ
+�C||u||A1
+1 − µ
+� �
+Rn |f(x)|Mµ(χF vθ)(x)u(x)1−θdx
+=
+ϕ
+�Ct||u||A1
+1 − θ
+� �
+Rn |f(x)|Mµ(χF vθ)(x)
+vθ(x)
+v(x) dx
+≤
+ϕ
+�Ct||u||A1
+1 − θ
+� ����
+Mµ(χF vθ)
+vθ
+����
+Lp′,∞(v)
+||f||Lp,1(v),
+where in the last estimate we have used the H¨older’s inequality for Lorentz
+spaces with respect to the measure v(x) dx.
+Now, by virtue of Lemma 2.5,
+����
+Mµ(χF vθ)
+vθ
+����
+Lp′,∞(v)
+≲
+Cp,θ,µ(u)v(F)
+1
+p′ = Cp,θ,µ(u)λv
+g(y)
+1
+p′ ,
+so taking the supremum over all y > 0, in particular, we obtain that
+∥g∥Lp,∞(v) ≲ Cp,θ,µ(u)ϕ
+�Ct||u||A1
+1 − θ
+�
+||f||Lp,1(v).
+Finally, concerning about the constant Cp,θ,µ(u), we observe that
+Cp,θ,µ(u) =
+�
+p2
+(p − 1)2(µ − θ)(θ − 1
+p′ )2
+�θ
+∥u∥
+2θ− 2
+p′
+A1
+≈
+�
+p2
+(p − 1)2(1 − θ)(θ − 1
+p′)2
+�θ
+∥u∥
+3θ− 2
+p′
+A1
+.
+
+11
+Therefore, letting
+θ = 1
+p′
+�
+1 +
+1
+(p + 1)R
+�
+,
+1 ≤ R < ∞,
+then
+Cp,θ,µ(u) ≲
+�p5(p + 1)3R2
+(p − 1)4
+� 1
+p′
+�
+1+
+1
+(p+1)R
+�
+∥u∥
+1
+p′
+A1 ∥u∥
+3
+Rp′(p+1)
+A1
+≲ R
+2
+p′ ∥u∥
+1
+p′
+A1 ∥u∥
+3
+R
+A1 .
+Furthermore, with the same choice of θ,
+ϕ
+�Ct||u||A1
+1 − θ
+�
+≤ ϕ
+�
+˜C||u||A1
+�
+.
+Thus, the result follows by setting R = 1 + log∥u∥A1 and then taking the
+infimum on ∥u∥A1 over all possible representations of v ∈ �Ap.
+□
+4. Examples and applications to average operators, multipliers
+and integral operators
+4.1. Examples. There are many operators in harmonic analysis for which the
+weak-type (1, 1) boundedness for every weight in A1 has been proved [9, 23,
+26, 27, 28, 35].
+As a consequence of the classical Rubio de Francia extrapolation theory (see
+Theorem 1.1) it is known that they are also bounded on Lp(v) for every v ∈ Ap;
+but, in general, the restricted weak-type
+T : Lp,1(v) −→ Lp,∞(v),
+∀v ∈ �Ap,
+has been unknown up to now for many examples. This is the case, for instance,
+of the Bochner-Riesz operator at the critical index B n−1
+2 , introduced by S.
+Bochner in [4] and defined as follows (see [7] for some partial results in this
+context): let a+ = max{a, 0} denote the positive part of a ∈ R and given λ > 0,
+the Bochner-Riesz operator Bλ on Rn is defined by
+�
+Bλf(ξ) =
+�
+1 − |ξ|2�λ
++ ˆf(ξ),
+ξ ∈ Rn.
+Proposition 4.1 ([28, 35]). For every n > 1,
+B n−1
+2
+: L1(u) −→ L1,∞(u),
+C||u||2
+A1 log(||u||A1 + 1),
+∀u ∈ A1.
+Thereby, in virtue of Theorem 1.6, we completely answer the open question
+formulated in [7] about the restricted weak-type boundedness of B n−1
+2 .
+Corollary 4.2. For every n > 1 and every p > 1,
+B n−1
+2
+: Lp,1(v) −→ Lp,∞(v),
+C||v||3p−1
+ˆ
+Ap
+(1 + log ||v|| ˆ
+Ap)1+ 2
+p′ ,
+∀v ∈ �Ap.
+Same estimates can be obtained for a large list of operators such as those
+appearing in [2, 7, 9]: rough operators, H¨ormander multipliers, radial Fourier
+multipliers, square functions, etc.
+
+12
+On weak-type (1, 1) for averaging type operators
+4.2. Average operators.
+Corollary 4.3. Assume that {Tθ}θ is a family of operators indexed in a prob-
+ability measure space such that the average operator
+TAf(x) =
+�
+Tθf(x)dP(θ),
+x ∈ Rn,
+is well defined and that
+(4.1)
+Tθ : L1(u) −→ L1,∞(u),
+ϕ(||u||A1),
+∀u ∈ A1,
+where ϕ is a positive nondecreasing function on [1, ∞). Then,
+(4.2)
+TA : L1
+R(u) −→ L1,∞(u),
+C1ϕ(C2||u||A1)(1 + log ||u||A1),
+∀u ∈ A1.
+Moreover, if TA is a sublinear (ε, δ)-atomic approximable operator, then
+(4.3)
+TA : L1(u) −→ L1,∞(u),
+˜C1ϕ(C2||u||A1)||u||A1(1 + log ||u||A1).
+Proof. Set 1 < p < ∞. Using Theorem 1.6, we have that (4.1) implies
+Tθ : Lp,1(v) −→ Lp,∞(v),
+Φ(||v|| �
+Ap),
+∀v ∈ �Ap.
+Now, Lp,∞(v) is a Banach function space since there exists a norm || · ||(p,∞,v)
+so that
+||f||Lp,∞(v) ≤ ||f||(p,∞,v) ≤
+p
+p − 1||f||Lp,∞(v).
+Hence, by the Minkowski’s integral inequality, TA satisfies that for every p > 1,
+TA : Lp,1(v) −→ Lp,∞(v),
+p
+p − 1Φ(||v|| �
+Ap),
+∀v ∈ �Ap.
+Therefore, using Theorem 1.3 the desired result (4.2) follows by taking the
+infimum in p > 1. Finally, (4.3) is just a consequence of Theorem 2.3.
+□
+In particular, the next result stated in the introduction follows:
+Proof of Corollary 1.7. This result is just a direct consequence of Corollary 4.3
+since
+�
+cj
+||c||ℓ1 Tj
+�
+j is a family of operators indexed in the counting probability
+measure.
+□
+(I) Fourier multipliers
+Our next application is in the context of restriction multipliers from Rn+k to
+Rn. First, let us recall that a bounded function m defined on Rn is said to be
+normalized if
+(4.4)
+lim
+j
+�
+ψj ∗ m(x) = m(x),
+∀x ∈ Rn,
+where for each j, ψj(x) = ψ(x/j), and ψ ∈ C∞
+c (Rn) (i.e., ψ is an infinitely
+differentiable function with compact support), ˆψ ≥ 0 and || ˆψ||1 = 1.
+It is easy to see that then, for every Lebesgue point x of m, (4.4) holds. In
+particular, every continuous and bounded function is normalized.
+
+13
+Proposition 4.4. Let k ≥ 1 and assume that a normalized bounded function
+m defined in Rn+k satisfies that
+Tm : L1(u) −→ L1,∞(u),
+ϕ(||u||A1),
+∀u ∈ A1(Rn+k),
+where ϕ is a positive nondecreasing function on [1, ∞). Let φ ∈ L1(Rk) and
+define
+mφ(x) =
+�
+Rk m(x, y)φ(y) dy,
+x ∈ Rn.
+Then, for every v ∈ A1(Rn),
+Tmφ : L1
+R(v) −→ L1,∞(v),
+C1ϕ(C2||v||A1)||v||A1(1 + log ||v||A1).
+Proof. Take v ∈ A1(Rn) and define u = v ⊗ χRk, so that
+u :
+Rn × Rk
+−→ R,
+(x, y)
+�−→ u(x, y) = v(x),
+satisfies u ∈ A1(Rn+k) with ||u||A1 ≤ ||v||A1. Then, Tm : L1(u) −→ L1,∞(u)
+and, by [11, Theorem 4.4] (where here is used that m is normalized),
+Tm(·,y) : L1(v) −→ L1,∞(v),
+∀y ∈ Rk,
+with
+sup
+y∈Rk ||Tm(·,y)||L1(v)→L1,∞(v) ≲ ||u||A1||Tm||L1(u)→L1,∞(u) ≤ ||v||A1ϕ(||v||A1).
+Now, take f ∈ C∞
+c (Rn). Then, for every y ∈ Rk we have that m(·, y) ˆf ∈
+L1(Rn) and, as well, mφ ˆf ∈ L1(Rn), so that, by the properties of the Fourier
+transform,
+Tm(·,y)f(x) =
+�
+m(·, y) ˆf
+�∨
+(x)
+and
+Tmφf(x) = (mφ ˆf)∨(x),
+∀x ∈ Rn.
+Hence, by Fubini’s theorem,
+Tmφf(x) =
+�
+Rn mφ(ξ) ˆf(ξ)e2πix·ξ dξ =
+�
+Rn
+��
+Rk m(ξ, y)φ(y) dy
+�
+ˆf(ξ)e2πix·ξ dξ
+=
+�
+Rk
+��
+Rn m(ξ, y) ˆf(ξ)e2πix·ξ dξ
+�
+φ(y) dy =
+�
+Rk Tm(·,y)f(x)φ(y) dy,
+and the result follows as in Corollary 4.3 together with the density of Lp,1(v)
+by functions in C∞
+c (Rn) ∩ Lp,1(v).
+□
+(II) Integral operators
+Let us now consider the operator
+Tf(x) =
+�
+Rm K(x, y)f(y)dy,
+x ∈ Rn,
+where the integral kernel K satisfies some size condition of the form |K(x, y)| ≲
+|x − y|−n.
+
+14
+On weak-type (1, 1) for averaging type operators
+Proposition 4.5. Assume that, for every s > 0,
+Tsf(x) =
+�
+|x−y|≥s
+K(x, y)f(y)dy,
+x ∈ Rn,
+satisfies that
+Ts : L1
+R(u) −→ L1,∞(u),
+ϕ(||u||A1),
+∀u ∈ A1,
+where ϕ is a positive nondecreasing function on [1, ∞). Then, if φ is a bounded
+variation function on (0, ∞) with limx→0+ φ(x) = 0, we have that
+Tφf(x) =
+�
+Rm K(x, y)φ(|x − y|)f(y)dy,
+x ∈ Rn,
+satisfies that
+Tφ : L1
+R(u) −→ L1,∞(u),
+C1ϕ(C2||u||A1)(1 + log ||u||A1),
+∀u ∈ A1.
+Proof. We observe that, by hypothesis,
+φ(|x − y|) =
+� |x−y|
+0
+φ′(s)ds,
+φ′ ∈ L1(Rn),
+and hence, for every x ∈ Rn and every ε > 0, by Fubini’s theorem we have that
+Tφf(x) − φ(ε)Tf(x) =
+� ∞
+0
+��
+|x−y|≥s≥ε
+K(x, y)f(y)dy
+�
+φ′(s) ds
+=
+� ∞
+ε
+Tsf(x)φ′(s) ds,
+is an average operator, and so the result follows by Corollary 4.3 and letting ε
+tend to zero.
+□
+5. Limited extrapolation results
+The motivation of this section comes from the fact that there are also many
+operators in harmonic analysis (such as the Bochner-Riesz) so that
+T : Lp0(v) −→ Lp0(v)
+is not bounded for every v ∈ Ap0 but is bounded for every v in a certain subclass
+of Ap0. Under this weaker hypothesis, only boundedness on Lp(v) of T can be
+deduced whenever p ∈ (p−, p+) for certain values of p− and p+. The purpose of
+this section is to establish some equivalence, similar to Theorem 1.6, between
+the boundedness at the endpoint p− and restricted weak-type boundedness at
+the p level. Indeed, the Rubio de Francia extrapolation results in this case are
+called limited extrapolation (see [1, 8, 10, 15, 17]).
+Definition 5.1. Given 0 ≤ α, β ≤ 1 and 1 ≤ p < ∞, let us define the classes
+of weights
+Ap;(α,β) =
+�
+0 < v ∈ L1
+loc(Rn) : v = vα
+0 vβ(1−p)
+1
+, vj ∈ A1
+�
+with
+||v||Ap;(α,β) = inf
+�
+||v0||α
+A1||v1||β(p−1)
+A1
+: v = vα
+0 vβ(1−p)
+1
+�
+,
+
+15
+and
+ˆAp;(α,β) =
+�
+0 < v ∈ L1
+loc(Rn) : v = vα
+0 (Mh)β(1−p), v0 ∈ A1, h ∈ L1
+loc(Rn)
+�
+with
+||v|| ˆ
+Ap;(α,β) = inf
+�
+||v0||
+α
+1+β(p−1)
+A1
+: v = vα
+0 (Mf)β(1−p)
+�
+.
+Definition 5.2. Given 1 ≤ p0 < ∞ and 0 ≤ α, β ≤ 1, set
+p+ =
+p0
+1 − α,
+p′
+− =
+p′
+0
+1 − β ,
+�
+or p− =
+p0
+1 + β(p0 − 1)
+�
+,
+where p+ = ∞ if α = 1 and p− = 1 if β = 1. Then, 1 ≤ p− ≤ p+ ≤ ∞ and we
+can associate to every p ∈ [p−, p+] the indices
+α(p) = p+ − p
+p+
+and
+β(p) =
+p − p−
+p−(p − 1),
+so that 0 ≤ α(p), β(p) ≤ 1, p+ =
+p
+1−α(p), p′
+− =
+p′
+1−β(p) and α(p0) = α, β(p0) = β.
+Theorem 5.3 ([17]). Let (f, g) be a pair of measurable functions such that for
+some 1 ≤ p0 < ∞ and 0 ≤ α, β ≤ 1 (not both identically zero) we have
+∥g∥Lp0(v) ≤ ϕ
+�
+∥v∥Ap0;(α,β)
+�
+∥f∥Lp0(v) ,
+∀v ∈ Ap0;(α,β),
+where ϕ is a nondecreasing function on [1, ∞). Then, for every p− < p < p+,
+∥g∥Lp(v) ≤ C1ϕ
+�
+C2 ∥v∥
+max
+� p+−p0
+p+−p ,
+p0−p−
+p−p−
+�
+Ap;(α(p),β(p))
+�
+∥f∥Lp(v) ,
+∀v ∈ Ap;(α(p),β(p)),
+with C1 and C2 being two positive constants independent of all parameters in-
+volved.
+Observe that in Theorem 5.3 is not possible to extrapolate till the endpoints
+p− and p+. However, in [10, Theorem 3.7] the authors were able to obtain
+an estimate in the endpoint p−. To do so, they needed to assume that the
+operators satisfy a restricted weak-type boundedness for the class of weights
+ˆAp;(α,β) which is a slightly bigger class than Ap;(α,β).
+Theorem 5.4 ([10]). Let 1 ≤ p0 < ∞, 0 ≤ α, β ≤ 1 (not both identically zero)
+and let T be an operator. Assume that
+T : Lp0,1(v) −→ Lp0,∞(v),
+ϕ(∥v∥ ˆ
+Ap0;(α,β)),
+∀v ∈ ˆAp0;(α,β),
+where ϕ is a positive nondecreasing function on [1, ∞). Then:
+(i) If p− > 1,
+T : Lp−,1 �
+uα(p−)�
+−→ Lp−,∞ �
+uα(p−)�
+,
+Φp−(||u||α(p−)
+A1
+)
+p− − 1
+,
+∀u ∈ A1,
+where Φp− is a positive nondecreasing function on [1, ∞).
+(ii) If p− = 1,
+T : L1
+R
+�
+uα(p−)�
+−→ L1,∞ �
+uα(p−)�
+,
+Φ1(||u||α(p−)
+A1
+),
+∀u ∈ A1.
+
+16
+On weak-type (1, 1) for averaging type operators
+Our following theorem shows that the converse is also true:
+Theorem 5.5. Let (f, g) be a pair of measurable functions such that for some
+1 ≤ p0 < ∞ and 0 < α ≤ 1,
+||g||Lp0,∞(uα) ≤ ϕ
+�
+∥u∥α
+A1
+�
+||f||Lp0,1(uα),
+∀u ∈ A1,
+with ϕ being a nondecreasing function on [1, ∞). Then, for any p0 ≤ p <
+p0
+1−α,
+||g||Lp,∞(v) ≤ Ψ
+�
+||v|| ˆ
+Ap;(α(p),β(p))
+�
+||f||Lp,1(v),
+∀v ∈ ˆAp;(α(p),β(p)),
+where α(p) = 1 − p(1−α)
+p0
+, β(p) =
+p−p0
+p0(p−1) and, for every r ≥ 1,
+Ψ(r) = C1
+�
+1
+p0 − p(1 − α)
+� p−p0
+p
+ϕ
+�
+C2r
+αp
+p0−p(1−α)
+�
+r
+α(p−p0)
+p0−p(1−α) (1 + log r)
+2(p−p0)
+p
+,
+with C1 and C2 being two positive constants independent of all parameters in-
+volved.
+Proof. Let h ∈ L1
+loc(Rn) and u ∈ A1 so that
+v = (Mh)β(p)(1−p)uα(p) ∈ �Ap;(α(p),β(p)).
+Further, take t = 1+
+1
+2n+1∥u∥A1 so that ut ∈ A1 with ||ut||A1 ≲ ||u||A1 (see (2.2))
+and, since t > 1,
+p − p0
+p
+= α − α(p)
+1 − α(p) < tα − α(p)
+t − α(p) < α.
+Hence, we can take
+vθ = (Mh)β(1−p)uα(p)θ
+with
+p − p0
+p
+< θ < tα − α(p)
+t − α(p) .
+Besides, since α(p) ≤ α, letting
+µ = 1 − α(p)(1 − θ)
+αt
+∈ (0, 1),
+then θ < αµ < 1 and, by (2.1), for every measurable set F ⊆ Rn,
+u0 =
+�
+Mαµ(χF vθ)uα(p)(1−θ)� 1
+α = M
+�
+χFv
+1
+αµ
+θ
+�µ
+(ut)1−µ ∈ A1,
+with ||u0||A1 ≤
+C||u||A1
+1−µ .
+Now, let y > 0 and set F = {x : |g(x)| > y} so that v(F) = λv
+g(y). We can
+assume, as it was done in the proof of Theorem 1.6, that v(F) < ∞. Then, by
+
+17
+hypothesis, we obtain that
+yp0λv
+g(y) = yp0
+�
+{x : |g(x)|>y}
+v(x) dx ≤ yp0
+�
+F
+u0(x)αdx
+≤ ϕ
+�
+||u0||α
+A1
+�p0
+
+p0
+� ∞
+0
+��
+{|f(x)|>z}
+u0(x)α dx
+� 1
+p0
+dz
+
+
+p0
+≲ ϕ
+�
+||u0||α
+A1
+�p0
+����
+Mαµ(χFvθ)
+vθ
+����
+L( p
+p0)
+′,∞(v)
+�� ∞
+0
+||χ{|f(x)|>z}||
+1
+p0
+L
+p
+p0 ,1(v)
+dz
+�p0
+≲ ϕ
+�� Cαt||u||A1
+α(p)(1 − θ)
+�α�p0 ����
+Mαµ(χF vθ)
+vθ
+����
+L( p
+p0)
+′,∞(v)
+||f||p0
+Lp,1(v),
+where in the penultimate estimate we have used the H¨older’s inequality for
+Lorentz spaces with respect to the measure v(x) dx.
+Now, since β(p)(1 − p) = 1 − p
+p0, then v ∈ ˆA p
+p0 and, by virtue of Lemma 2.5,
+����
+Mαµ(χF vθ)
+vθ
+����
+L( p
+p0 )
+′,∞(v)
+≲ C p
+p0 ,θ,αµ(u)v(F)
+p−p0
+p
+= C p
+p0 ,θ,αµ(u)λv
+g(y)
+p−p0
+p ,
+so taking the supremum over all y > 0, in particular, we obtain that
+∥g∥Lp,∞(v) ≲ C p
+p0 ,θ,αµ(u)
+1
+p0 ϕ
+� ˜C||u||α
+A1
+(1 − θ)α
+�
+||f||Lp,1(v).
+Finally, concerning about the constant C p
+p0 ,θ,αµ(u), we observe that
+C p
+p0 ,θ,αµ(u) =
+�
+p2
+(p − p0)2(αµ − θ)(θ − p−p0
+p )2
+�θ
+∥u∥
+2θ− 2(p−p0)
+p0
+A1
+≲
+�
+p2
+(p − p0)2(α − θ)(θ − p−p0
+p )2
+�θ
+∥u∥
+3θ− 2(p−p0)
+p0
+A1
+,
+so the behaviour of the constant C p
+p0 ,θ,αµ(u) follows as in the proof of Theo-
+rem 1.6.
+□
+As an application, we present some new weighted estimates for the Bochner-
+Riesz operator below the critical index.
+Proposition 5.6 ([25]). Let n = 2 and 0 < λ < 1
+2. Then,
+Bλ : L
+4
+3+2λ
+�
+u
+2λ
+3+2λ
+�
+−→ L
+4
+3+2λ ,∞ �
+u
+2λ
+3+2λ
+�
+,
+c(n, λ) ∥u∥
+λ(7+4λ)
+6+4λ
+A1
+,
+∀u ∈ A1.
+Proposition 5.7 ([13]). Let n > 2 and
+n−1
+2(n+1) < λ < n−1
+2 . Then
+Bλ : L2 �
+u
+1+2λ
+n
+�
+−→ L2 �
+u
+1+2λ
+n
+�
+,
+ϕ
+�
+||u||
+1+2λ
+n
+A1
+�
+,
+∀u ∈ A1,
+where ϕ is a positive nondecreasing function on [1, ∞).
+
+18
+On weak-type (1, 1) for averaging type operators
+Therefore, as a consequence of Theorem 5.5 and Propositions 5.6 and 5.7, we
+obtain the following result.
+Corollary 5.8. Let n = 2 and 0 < λ < 1
+2. For every
+4
+3+2λ ≤ p < 4
+3,
+Bλ : Lp,1(v) −→ Lp,∞(v),
+∀v ∈ ˆAp;
+�
+4−3p
+4
+, (3+2λ)p−4
+4(p−1)
+�.
+Now, let n > 2 and
+n−1
+2(n+1) < λ < n−1
+2 . For every 2 ≤ p <
+2n
+n−1−2λ,
+Bλ : Lp,1(v) −→ Lp,∞(v),
+∀v ∈ ˆAp;
+�
+2n−p(n−1−2λ)
+2n
+,
+p−2
+2(p−1)
+�.
+References
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+Anal. 128 (1995), no. 1, 163–185.
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+A course on singular integrals and weights. Adv. Courses Math. CRM
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+Departament de Matem`atiques i Inform`atica, Universitat de Barcelona, 08007
+Barcelona, Spain.
+Email address: sergibaena@ub.edu
+Departamento de An´alisis Matem´atico y Matem´atica Aplicada, Universidad
+Complutense de Madrid, Spain.
+Email address: mjcarro@ucm.es
+
diff --git a/ftE4T4oBgHgl3EQfqg2l/content/tmp_files/load_file.txt b/ftE4T4oBgHgl3EQfqg2l/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf,len=688
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='05201v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='FA]  12 Jan 2023  ON WEAK-TYPE (1, 1) FOR AVERAGING TYPE  OPERATORS  SERGI BAENA-MIRET∗∗ AND MAR´IA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' CARRO∗  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' It is known that, due to the fact that L1,∞ is not a Banach  space, if (Tj)j is a sequence of bounded operators so that  Tj : L1 −→ L1,∞,  with norm less than or equal to ||Tj|| and �  j ||Tj|| < ∞, nothing can be  said about the operator T = �  j Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' This is the origin of many difficult and  open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' However, if we assume that  Tj : L1(u) −→ L1,∞(u),  ∀u ∈ A1,  with norm less than or equal to ϕ(||u||A1)||Tj||, where ϕ is a nondecreasing  function and A1 the Muckenhoupt class of weights, then we prove that,  essentially,  T : L1(u) −→ L1,∞(u),  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  We shall see that this is the case of many interesting problems in Harmonic  Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Introduction  Let {Tθ}θ be a family of operators indexed in a probability measure space  such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1)  Tθ : L1(Rn) −→ L1,∞(Rn)  with norm less than or equal to a uniform constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' What can we say about  the boundedness of the average operator  TAf(x) =  �  Tθf(x)dP(θ),  x ∈ Rn,  whenever is well defined?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' The following trivial example shows that, at first  sight, nothing of interest can be concluded: for 0 < θ < 1, set  Tθf(x) =  � 1  0 f(y)dy  |x − θ| ,  x ∈ (0, 1),  so clearly Tθ satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1), but  TAf(x) =  � 1  0  Tθf(x)dθ ≡ ∞,  ∀x ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 42A99, 46E30, 47B34, 42A38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Muckenhoupt weights, restricted weak-type extrapolation, average  operators, Fourier multiplier operators, the Bochner-Riesz operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  The authors were partially supported by grant PID2020-113048GB-I00 funded by  MCIN/AEI/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  E-mail addresses:  ∗mjcarro@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='es, ∗∗sergibaena@ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  1  2  On weak-type (1, 1) for averaging type operators  However, things change completely, and this is one of the main goals of this  paper, if we assume that  Tθ : L1(u) −→ L1,∞(u),  ∀u ∈ A1,  where A1 is the class of Muckenhoupt weights defined as follows: we say that  u ∈ A1 if u is a nonnegative locally integrable function (called weight) so that  there exists a positive constant C such that  Mu(x) ≤ Cu(x),  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' x ∈ Rn,  where M is the Hardy-Littlewood maximal operator defined by  Mf(x) = sup  Q∋x  1  |Q|  �  Q  |f(y)| dy,  f ∈ L1  loc(Rn),  with the supremum being taken over all cubes Q ⊆ Rn containing x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' We  denote by ∥u∥A1 the least constant C satisfying such inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Besides, it is  wellknown ([5, 29]) that  M : L1(u) −→ L1,∞(u)  ⇐⇒  u ∈ A1,  with ||M||L1(u)→L1,∞(u) ≤ C||u||A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Let us start with a very simple and motivating example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let m be a bounded  variation function on R that is right-continuous and normalized by the condition  m(−∞) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then,  m(ξ) =  � ξ  −∞  dm(t) =  �  R  χ(−∞,ξ)(t) dm(t) =  �  R  χ(t,∞)(ξ) dm(t),  ∀ξ ∈ R,  where dm is the Lebesgue-Stieltjes measure associated with m and it is a finite  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Hence, if we consider the Fourier multiplier operator  Tmf(x) =  �  R  m(ξ) ˆf(ξ)e2πixξdξ,  x ∈ R,  for every Schwartz function f, where  ˆf(ξ) =  �  R  f(x)e−2πixξdx,  ξ ∈ R,  is the Fourier transform of the function f, a formal computation shows that  Tmf(x) =  �  R  Htf(x)dm(t),  ∀x ∈ R,  where  Htf(x) = Tχ(t,∞)f(x) =  � ∞  t  ˆf(ξ)e2πixξdξ,  x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, Ht is essentially a Hilbert transform operator (recall that Hf = Tmf with  m(ξ) = −i sgn ξ) because  χ(t,∞)(ξ) = sgn(ξ − t) + 1  2  ,  ∀ξ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Thus, since  Ht : Lp(R) −→ Lp(R),  ∀p > 1,  3  we have, using the Minkowski’s integral inequality and the density of the  Schwartz functions on Lp(R), that every right-continuous bounded variation  function is a Fourier multiplier on Lp(R) for every p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' However, even though  we also have  Ht : L1(R) −→ L1,∞(R),  we cannot deduce (at least not immediately) that the same boundedness holds  for Tm due to the lack of the Minkowski’s integral inequality for the space  L1,∞(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  The main theorem of this paper will show that since  Ht : L1(u) −→ L1,∞(u),  ϕ(||u||A1),  ∀u ∈ A1,  with ϕ being a nondecreasing function on [1, ∞) and independent of t ∈ R, then  for every measurable set E ⊆ Rn,  ||TmχE||L1,∞(u) ≤ C(m)ϕ(C2||u||A1)(1 + log ||u||A1)u(E),  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  The result will be proved using an extended version of the Rubio de Francia’s  extrapolation theorem which deals with the theory of Muckenhoupt weights  (see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Other interesting applications will be given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Let us now recall (see [5, 29]) that for p > 1,  M : Lp(v) −→ Lp(v)  ⇐⇒  v ∈ Ap,  where this class of weights is defined by the condition  ∥v∥Ap = sup  Q⊆Rn  � 1  |Q|  �  Q  v(x) dx  �� 1  |Q|  �  Q  v(x)  1  1−p dx  �p−1  < ∞,  and, given a weight v, Lp(v) is the Lebesgue space defined as the set of mea-  surable functions f such that  ||f||Lp(v) =  ��  Rn |f(x)|pv(x) dx  � 1  p  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Indeed, (see [18]) for every p ≥ 1,  M : Lp(v) −→ Lp,∞(v)  ⇐⇒  v ∈ Ap,  with Lp,∞(v) being the weak Lebesgue space defined as the set of measurable  functions f so that  ∥f∥Lp,∞(v) = sup  y>0  yλv  f(y)  1  p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Here, λv  f is the distribution function of f with respect to v defined by  λv  f(y) = v  ��  x ∈ Rn : |f(x)| > y  ��  ,  y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  (Here we are using the standard notation v(E) =  �  E v(x) dx for every measur-  able set E ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' If v = 1, we shall write λf and |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' See [3] for more details  about this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=')  An important result for our purpose concerning Ap weights is the extrapola-  tion theorem of Rubio de Francia [32, 33] (see also [15, 17, 19, 20, 21]) which,  nowadays, can be formulated as follows:  4  On weak-type (1, 1) for averaging type operators  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1 ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let (f, g) be a pair of measurable functions such that for  some 1 ≤ p0 < ∞,  ||g||Lp0(v) ≤ ϕ(∥v∥Ap0 )||f||Lp0(v),  ∀v ∈ Ap0,  with ϕ being a nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, for every 1 < p < ∞,  ||g||Lp(v) ≤ C1ϕ  �  C2∥v∥  max  �  1, p0−1  p−1  �  Ap  �  ||f||Lp(v),  ∀v ∈ Ap,  with C1 and C2 being two positive constants independent of all parameters in-  volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  We have to emphasize here that although p0 can be 1, it is not possible, in  general, to extrapolate till the endpoint p = 1 (take just T = M ◦ M or see,  for instance, [30] where a counterexample is given in the case of commutators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  However, in the recent papers [9, 12], a Rubio de Francia extrapolation theory  for operators satisfying a weighted restricted weak-type boundedness for the  class of weights �Ap (slightly bigger than the class Ap) has been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' The  main advantage of this new class of weights is that allows to obtain boundedness  estimates at the endpoint p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' We define  �Ap =  �  v ∈ L1  loc(Rn) : ∃ h ∈ L1  loc(Rn) and ∃ u ∈ A1 with v = (Mh)1−pu  �  ,  endowed with the norm  ∥v∥ �  Ap = inf  �  ∥u∥  1  p  A1 : v = (Mh)1−pu  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Clearly, �A1 = A1, while for 1 < p < ∞, Ap ⊊ �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  It holds that (see [9, 14, 24]) for every 1 ≤ p < ∞ and every v ∈ �Ap,  M : Lp,1(v) −→ Lp,∞(v),  ∥M∥Lp,1(v)−→Lp,∞(v) ≤ C∥v∥ �  Ap,  where the Lorentz space Lp,1(v) is defined as the set of measurable functions f  such that  ∥f∥Lp,1(v) = p  � ∞  0  λv  f(y)  1  p dy < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Then, the restricted weak-type Rubio de Francia extrapolation result proved  in [9] can be stated as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let 1 < p0 < ∞ and let T be an operator such that  T : Lp0,1(v) −→ Lp0,∞(v),  ϕ(∥v∥ �  Ap0),  ∀v ∈ �Ap0,  where ϕ is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, T is of weighted  restricted weak-type (1, 1) for every weight in A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' that is, for any measurable  set E ⊆ Rn, there exists a constant C > 0 independent of E such that  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2)  ∥TχE∥L1,∞(u) ≤ C∥u∥  1− 1  p0  A1  ϕ  �  ∥u∥  1  p0  A1  �  u(E),  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  5  For simplicity, whenever an operator T satisfies that for every measurable  set E,  ∥TχE∥L1,∞(u) ≤ Cuu(E),  we shall denote it by  T : L1  R(u) −→ L1,∞(u),  Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' The complete result that T is of weighted weak-type (1, 1) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=',  that the estimate in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2) holds for every f ∈ L1(u)) is, in general, false (see  [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' However, under certain mild condition in the operator T (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2)  the weighted weak-type (1, 1) boundedness can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' We should emphasize here that our operators do not need to be  sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' However, if T is sublinear, it was proved in [34] that  T : L1  R(u) −→ L1,∞(u)  is equivalent to have the boundedness on the space  B∗(u) =  �  f :  � ∞  0  λu  f(t)  �  1 + log ||f||1  λu  f(t)  �  dt < ∞  �  ,  which can be endowed with a quasi-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Our main goal will be consequence of the fact that the converse of Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 is also true, and hence  T : L1  R(u) −→ L1,∞(u), ∀u ∈ A1 ⇐⇒ T : Lp0,1(v) −→ Lp0,∞(v), ∀v ∈ �Ap0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Indeed, if p′ is the conjugate exponent of p > 1 (that is, 1  p + 1  p′ = 1) our main  theorem reads as follows:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let (f, g) be a pair of measurable functions such that  ||g||L1,∞(u) ≤ ϕ(∥u∥A1)||f||L1(u),  ∀u ∈ A1,  with ϕ being a nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, for every 1 < p < ∞,  ||g||Lp,∞(v) ≤ Φ(∥v∥ �  Ap)||f||Lp,1(v),  ∀v ∈ �Ap,  where  Φ(r) = C1ϕ(C2rp)rp−1(1 + log r)  2  p′ ,  r ≥ 1,  with C1 and C2 being two positive constants independent of all parameters in-  volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  As a consequence we obtain the following corollary:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let c = (cj)j ∈ ℓ1 and let {Tj}j be such that  Tj : L1(u) −→ L1,∞(u),  ϕ(||u||A1),  ∀u ∈ A1,  where ϕ is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, for every u ∈ A1,  �  j  cjTj : L1  R(u) −→ L1,∞(u),  C1||c||ℓ1ϕ(C2||u||A1)(1 + log ||u||A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  6  On weak-type (1, 1) for averaging type operators  As usual, we shall use the symbol A ≲ B to indicate that there exists a  universal positive constant C, independent of all important parameters, such  that A ≤ CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' When A ≲ B and B ≲ A, we will write A ≈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' In Section 2, we will see some previous  notions, the necessary definitions and some technical results which shall be used  later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Indeed, there we will prove Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5 which will be essential in the  proof of the main result given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Further, Section 4 contains our  main examples and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Finally, we also include a last section related  with similar results in the context of limited extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Preliminary notions and some technical results  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' A1 weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let us start by recalling some wellknown facts of the class  A1:  i) ([16, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='7]) A weight u belongs to A1 if and only if there exists  h ∈ L1  loc(Rn) and K such that K, K−1 ∈ L∞(Rn) satisfying that, for some  0 < µ < 1,  u(x) = K(x)(Mh(x))µ,  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' x ∈ Rn,  where L∞(Rn) consists of all measurable functions f such that  ||f||∞ := ||f||L∞(Rn) = ess sup f < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  ii) ([12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='12]) For every h ∈ L1  loc(Rn), every u ∈ A1 and 0 < µ < 1,  then (Mh)µu1−µ ∈ A1 with  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1)  ����(Mh)µu1−µ  ����  A1  ≲ ∥u∥A1  1 − µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  iii) ([31, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1]) If t = 1 +  1  2n+1∥u∥A1 , then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2)  ut ∈ A1  and  ||ut||A1 ≲ ||u||A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' (ε, δ)-atomic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' As mentioned above, in general, the following  implication does not hold for every u ∈ A1:  T : L1  R(u) −→ L1,∞(u)  =⇒  T : L1(u) −→ L1,∞(u),  even if T is a sublinear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' However, it was proved in [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5]  that for a quite big class of operators the above implication is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Given δ > 0, a function a ∈ L1(Rn) is called a δ-atom if it  satisfies the following properties:  (i)  �  Rn a(x) dx = 0, and  (ii) there exists a cube Q ⊆ Rn such that |Q| ≤ δ and supp a ⊆ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' (a) A sublinear operator T is called (ε, δ)-atomic if, for every  ε > 0, there exists δ > 0 satisfying that  ∥Ta∥L1(Rn)+L∞(Rn) ≤ ε∥a∥1,  for every δ-atom a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  7  (b) A sublinear operator T is said to be (ε, δ)-atomic approximable if there exists  a sequence {Tj}j of (ε, δ)-atomic operators such that, for every measurable set  E ⊆ Rn, then |TjχE| ≤ |TχE| and, for every f ∈ L1(Rn) such that ∥f∥∞ ≤ 1,  |Tf(x)| ≤ lim  j inf |Tjf(x)|,  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Examples:  In [6], the author showed that for sublinear operators, the prop-  erty of being (ε, δ)-atomic is not a strong one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' For instance, if  Tf(x) = K ∗ f(x) =  �  Rn K(y − x)f(y) dy,  x ∈ Rn,  with K ∈ Lp(Rn) for some 1 ≤ p < ∞, then T is (ε, δ)-atomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Further, if  T ∗f(x) = sup  j∈N  ����  �  Rn Kj(x, y)f(y) dy  ����,  x ∈ Rn,  with  lim  y→x ∥Kj( · , y) − Kj( · , x)∥L1(Rn)+L∞(Rn) = 0,  then T ∗ is (ε, δ)-atomic approximable (for example, standard maximal Calder´on  Zygmund operators are of this type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' In general,  T ∗f(x) = sup  j  |Tjf(x)|,  x ∈ Rn,  where {Tj}j is a sequence of (ε, δ)-atomic, is (ε, δ)-atomic approximable and  the same holds for  Tf(x) =  � �  j  |Tjf(x)|q  � 1  q  ,  x ∈ Rn,  with q ∈ [1, ∞) and  Tf(x) =  �  j  Tjf(x),  x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  (See [6, 9] for more examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=')  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 ([9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let T be a sublinear operator (ε, δ)-atomic approximable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Then, given u ∈ A1,  T : L1  R(u) −→ L1,∞(u),  Cu  =⇒  T : L1(u) −→ L1,∞(u),  2nCu∥u∥A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' A Sawyer-type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Here we will study one of the often-called  Sawyer-type inequalities for weights belonging in the restricted class of weights  ˆAp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' First, to do so, we need the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let 1 < p < ∞ and v ∈ �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Take  1  p′ < θ ≤ 1 and set u0 =  (Mh)  (p−1)(1−θ)  θ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3)  Mu0 : L  θp′  θp′−1,1(v) −→ L  θp′  θp′−1 ,∞(v),  8  On weak-type (1, 1) for averaging type operators  with constant less than or equal to  θ2p′Cn,p  1 − p(1 − θ) ∥v∥  2(θp′−1)  θ(p′−1)  ˆ  Ap  ,  and where  Mu0f(x) = sup  Q∋x  1  u0(Q)  �  Q  |f(y)|u0(y) dy,  x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Observe that since v ∈ ˆAp, then v is a doubling weight with constant  ∆v ≤ C1 ∥v∥p  ˆ  Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Therefore, according to [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2 (i)], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3) is bounded  with constant less than or equal to  C1 ∥v∥  θp′−1  θ(p′−1)  ˆ  Ap  θp′  \uf8ee  \uf8f0 sup  E⊆Q  u0(E)  u0(Q)  �v(Q)  v(E)  � θp′−1  θp′  \uf8f9  \uf8fb ,  where the supremum is taken over all cubes Q and all measurable sets E ⊆ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, given a cube Q and a measurable set E ⊆ Q,  �v(Q)  v(E)  � θp′−1  θp′  =  �|Q|  |E|  � θp′−1  θ(p′−1) ��|E|  |Q|  �p v(Q)  v(E)  � θp′−1  θp′  ≤ C2 ∥v∥  θp′−1  θ(p′−1)  ˆ  Ap  �|Q|  |E|  � θp′−1  θ(p′−1)  and, as well, due to [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5],  sup  E⊆Q  u0(E)  u0(Q)  �|Q|  |E|  � θp′−1  θ(p′−1)  ≤  θC3  1 − p(1 − θ),  which yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  □  The following lemma was proved for the case µ = 1 in [12, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6], and  the extension to other µ’s has been fundamental for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let 1 < p < ∞ and let v = (Mh)1−pu ∈ �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Take θ and µ so  that  1  p′ < θ < µ ≤ 1 and set vθ = (Mh)1−puθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then,  ���Mµ(χEvθ)  vθ  ���  Lp′,∞(v) ≲ Cp,θ,µ(u)v(E)  1  p′ ,  ∀E ⊆ Rn,  where Mµf := M(|f|1/µ)µ and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4)  Cp,θ,µ(u) =  �  p2  (p − 1)2(µ − θ)(θ − 1  p′ )2  �θ  ∥u∥  2θ− 2  p′  A1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Observe that in virtue of the Kolmogorov’s inequality [22] with 1 < r′ =  1  θ < p′, it is enough to prove that  sup  F ⊆Rn  1  v(F)  1  r′ − 1  p′  � �  F  (Mh(x))(p−1)(r′−1)�  Mµ(χE(Mh)1−puθ)(x)  �r′  dx  � 1  r′  ≲ Cp,θ,µ(u)v(E)  1  p′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  9  Then, using the Fefferman-Stein’s inequality [18], since µr′ > 1, we obtain  that  �  F  (Mh(x))(p−1)(r′−1)�  Mµ(χE(Mh)1−puθ)(x)  �r′  dx  ≲  µr′  µr′ − 1  �  E  (Mh(x))(1−p)r′M(χF (Mh)(p−1)(r′−1))(x)u(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, since u0 = (Mh)(p−1)(r′−1) ∈ A1, we have that, for every x ∈ E and  every cube Q ∋ x in Rn,  1  |Q|  �  Q  χFu0(y) dy ≤ u0(Q)  |Q| Mu0(χF )(x) ≤ ||u0||A1u0(x)Mu0(χF )(x)  ≲  1  1 − (p − 1)(r′ − 1)u0(x)Mu0(χF )(x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5)  where in the last estimate we have used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Hence, taking the supremum  over all cubes Q ∈ Rn such that Q ∋ x in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5), with x ∈ E, we deduce that  �  E  (Mh(x))(1−p)r′M(χF (Mh)(p−1)(r′−1))(x)u(x)dx  ≲  1  1 − (p − 1)(r′ − 1)  �  E  Mu0(χF )(x)v(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Therefore, since r′ = 1  θ, the inequality we want to prove will hold if we see  that  sup  E⊆Rn  1  v(E)  1  p′  ��  E  Mu0(χF )(x)v(x) dx  �θ  ≲  �(µ − θ)(1 − p(1 − θ))  µθ  �θ  Cp,θ,µ(u)v(F)θ− 1  p′  or equivalently,  sup  E⊆Rn  1  v(E)1−  �  1− 1  θp′  �  �  E  Mu0(χF )(x)v(x) dx  ≲  �(µ − θ)(1 − p(1 − θ))  µθ  �  Cp,θ,µ(u)  1  θ v(F)1− 1  θp′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6)  Finally, using again the Kolmogorov’s inequality in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6), it is enough to  prove that  Mu0 : L  θp′  θp′−1 ,1(v) −→ L  θp′  θp′−1,∞(v)  with constant less than or equal to  cn,p  θp′  �(µ − θ)(1 − p(1 − θ))  µθ  �  Cp,θ,µ(u)  1  θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4, this will happen if  Cp,θ,µ(u) ≳  �  p2  (p − 1)2(µ − θ)(1 − p(1 − θ))2  �θ  ∥u∥  2(θp′−1)  p′  A1  ,  from which the desired result follows by taking Cp,θ,µ(u) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  10  On weak-type (1, 1) for averaging type operators  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Proof of the main result  We are now ready to prove our main result:  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let h ∈ L1  loc(Rn) and u ∈ A1 so that v = (Mh)1−pu ∈  �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Further, let us take  1  p′ < θ < 1,  µ := 1 − 1 − θ  t  and  vθ := (Mh)1−puθ,  where t = 1+  1  2n+1∥u∥A1 satisfies ut ∈ A1 and ||ut||A1 ≲ ||u||A1 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then,  θ < µ < 1 and, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1), for every measurable set F ⊆ Rn,  u0 = Mµ(χF vθ)u1−θ = M(χF v1/µ  θ  )µ(ut)1−µ ∈ A1,  ||u0||A1 ≤ C||u||A1  1 − µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Let y > 0 and set F = {x : |g(x)| > y} so that v(F) = λv  g(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' We can  assume, without lost of generality, that v(F) < ∞, since on the contrary we  can take gN = gχB(0,N) and let N go to infinity at the end of our estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  By hypothesis we obtain that  yλv  g(y)  =  y  �  {x : |g(x)|>y}  v(x) dx ≤ y  �  F  Mµ(χF vθ)(x)u(x)1−θdx  ≤  ϕ  �C||u||A1  1 − µ  � �  Rn |f(x)|Mµ(χF vθ)(x)u(x)1−θdx  =  ϕ  �Ct||u||A1  1 − θ  � �  Rn |f(x)|Mµ(χF vθ)(x)  vθ(x)  v(x) dx  ≤  ϕ  �Ct||u||A1  1 − θ  � ����  Mµ(χF vθ)  vθ  ����  Lp′,∞(v)  ||f||Lp,1(v),  where in the last estimate we have used the H¨older’s inequality for Lorentz  spaces with respect to the measure v(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, by virtue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5,  ����  Mµ(χF vθ)  vθ  ����  Lp′,∞(v)  ≲  Cp,θ,µ(u)v(F)  1  p′ = Cp,θ,µ(u)λv  g(y)  1  p′ ,  so taking the supremum over all y > 0, in particular, we obtain that  ∥g∥Lp,∞(v) ≲ Cp,θ,µ(u)ϕ  �Ct||u||A1  1 − θ  �  ||f||Lp,1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Finally, concerning about the constant Cp,θ,µ(u), we observe that  Cp,θ,µ(u) =  �  p2  (p − 1)2(µ − θ)(θ − 1  p′ )2  �θ  ∥u∥  2θ− 2  p′  A1  ≈  �  p2  (p − 1)2(1 − θ)(θ − 1  p′)2  �θ  ∥u∥  3θ− 2  p′  A1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  11  Therefore, letting  θ = 1  p′  �  1 +  1  (p + 1)R  �  ,  1 ≤ R < ∞,  then  Cp,θ,µ(u) ≲  �p5(p + 1)3R2  (p − 1)4  � 1  p′  �  1+  1  (p+1)R  �  ∥u∥  1  p′  A1 ∥u∥  3  Rp′(p+1)  A1  ≲ R  2  p′ ∥u∥  1  p′  A1 ∥u∥  3  R  A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Furthermore, with the same choice of θ,  ϕ  �Ct||u||A1  1 − θ  �  ≤ ϕ  �  ˜C||u||A1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Thus, the result follows by setting R = 1 + log∥u∥A1 and then taking the  infimum on ∥u∥A1 over all possible representations of v ∈ �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Examples and applications to average operators, multipliers  and integral operators  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' There are many operators in harmonic analysis for which the  weak-type (1, 1) boundedness for every weight in A1 has been proved [9, 23,  26, 27, 28, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  As a consequence of the classical Rubio de Francia extrapolation theory (see  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1) it is known that they are also bounded on Lp(v) for every v ∈ Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  but, in general, the restricted weak-type  T : Lp,1(v) −→ Lp,∞(v),  ∀v ∈ �Ap,  has been unknown up to now for many examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' This is the case, for instance,  of the Bochner-Riesz operator at the critical index B n−1  2 , introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Bochner in [4] and defined as follows (see [7] for some partial results in this  context): let a+ = max{a, 0} denote the positive part of a ∈ R and given λ > 0,  the Bochner-Riesz operator Bλ on Rn is defined by  �  Bλf(ξ) =  �  1 − |ξ|2�λ  + ˆf(ξ),  ξ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1 ([28, 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' For every n > 1,  B n−1  2  : L1(u) −→ L1,∞(u),  C||u||2  A1 log(||u||A1 + 1),  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Thereby, in virtue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6, we completely answer the open question  formulated in [7] about the restricted weak-type boundedness of B n−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' For every n > 1 and every p > 1,  B n−1  2  : Lp,1(v) −→ Lp,∞(v),  C||v||3p−1  ˆ  Ap  (1 + log ||v|| ˆ  Ap)1+ 2  p′ ,  ∀v ∈ �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Same estimates can be obtained for a large list of operators such as those  appearing in [2, 7, 9]: rough operators, H¨ormander multipliers, radial Fourier  multipliers, square functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  12  On weak-type (1, 1) for averaging type operators  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Average operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Assume that {Tθ}θ is a family of operators indexed in a prob-  ability measure space such that the average operator  TAf(x) =  �  Tθf(x)dP(θ),  x ∈ Rn,  is well defined and that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1)  Tθ : L1(u) −→ L1,∞(u),  ϕ(||u||A1),  ∀u ∈ A1,  where ϕ is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2)  TA : L1  R(u) −→ L1,∞(u),  C1ϕ(C2||u||A1)(1 + log ||u||A1),  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Moreover, if TA is a sublinear (ε, δ)-atomic approximable operator, then  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3)  TA : L1(u) −→ L1,∞(u),  ˜C1ϕ(C2||u||A1)||u||A1(1 + log ||u||A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Set 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6, we have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1) implies  Tθ : Lp,1(v) −→ Lp,∞(v),  Φ(||v|| �  Ap),  ∀v ∈ �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, Lp,∞(v) is a Banach function space since there exists a norm || · ||(p,∞,v)  so that  ||f||Lp,∞(v) ≤ ||f||(p,∞,v) ≤  p  p − 1||f||Lp,∞(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Hence, by the Minkowski’s integral inequality, TA satisfies that for every p > 1,  TA : Lp,1(v) −→ Lp,∞(v),  p  p − 1Φ(||v|| �  Ap),  ∀v ∈ �Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Therefore, using Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 the desired result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2) follows by taking the  infimum in p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Finally, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3) is just a consequence of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  □  In particular, the next result stated in the introduction follows:  Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' This result is just a direct consequence of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3  since  �  cj  ||c||ℓ1 Tj  �  j is a family of operators indexed in the counting probability  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  □  (I) Fourier multipliers  Our next application is in the context of restriction multipliers from Rn+k to  Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' First, let us recall that a bounded function m defined on Rn is said to be  normalized if  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4)  lim  j  �  ψj ∗ m(x) = m(x),  ∀x ∈ Rn,  where for each j, ψj(x) = ψ(x/j), and ψ ∈ C∞  c (Rn) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=', ψ is an infinitely  differentiable function with compact support), ˆψ ≥ 0 and || ˆψ||1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  It is easy to see that then, for every Lebesgue point x of m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' In  particular, every continuous and bounded function is normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  13  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let k ≥ 1 and assume that a normalized bounded function  m defined in Rn+k satisfies that  Tm : L1(u) −→ L1,∞(u),  ϕ(||u||A1),  ∀u ∈ A1(Rn+k),  where ϕ is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let φ ∈ L1(Rk) and  define  mφ(x) =  �  Rk m(x, y)φ(y) dy,  x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Then, for every v ∈ A1(Rn),  Tmφ : L1  R(v) −→ L1,∞(v),  C1ϕ(C2||v||A1)||v||A1(1 + log ||v||A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Take v ∈ A1(Rn) and define u = v ⊗ χRk, so that  u :  Rn × Rk  −→ R,  (x, y)  �−→ u(x, y) = v(x),  satisfies u ∈ A1(Rn+k) with ||u||A1 ≤ ||v||A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, Tm : L1(u) −→ L1,∞(u)  and, by [11, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4] (where here is used that m is normalized),  Tm(·,y) : L1(v) −→ L1,∞(v),  ∀y ∈ Rk,  with  sup  y∈Rk ||Tm(·,y)||L1(v)→L1,∞(v) ≲ ||u||A1||Tm||L1(u)→L1,∞(u) ≤ ||v||A1ϕ(||v||A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, take f ∈ C∞  c (Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, for every y ∈ Rk we have that m(·, y) ˆf ∈  L1(Rn) and, as well, mφ ˆf ∈ L1(Rn), so that, by the properties of the Fourier  transform,  Tm(·,y)f(x) =  �  m(·, y) ˆf  �∨  (x)  and  Tmφf(x) = (mφ ˆf)∨(x),  ∀x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Hence, by Fubini’s theorem,  Tmφf(x) =  �  Rn mφ(ξ) ˆf(ξ)e2πix·ξ dξ =  �  Rn  ��  Rk m(ξ, y)φ(y) dy  �  ˆf(ξ)e2πix·ξ dξ  =  �  Rk  ��  Rn m(ξ, y) ˆf(ξ)e2πix·ξ dξ  �  φ(y) dy =  �  Rk Tm(·,y)f(x)φ(y) dy,  and the result follows as in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 together with the density of Lp,1(v)  by functions in C∞  c (Rn) ∩ Lp,1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  □  (II) Integral operators  Let us now consider the operator  Tf(x) =  �  Rm K(x, y)f(y)dy,  x ∈ Rn,  where the integral kernel K satisfies some size condition of the form |K(x, y)| ≲  |x − y|−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  14  On weak-type (1, 1) for averaging type operators  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Assume that, for every s > 0,  Tsf(x) =  �  |x−y|≥s  K(x, y)f(y)dy,  x ∈ Rn,  satisfies that  Ts : L1  R(u) −→ L1,∞(u),  ϕ(||u||A1),  ∀u ∈ A1,  where ϕ is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, if φ is a bounded  variation function on (0, ∞) with limx→0+ φ(x) = 0, we have that  Tφf(x) =  �  Rm K(x, y)φ(|x − y|)f(y)dy,  x ∈ Rn,  satisfies that  Tφ : L1  R(u) −→ L1,∞(u),  C1ϕ(C2||u||A1)(1 + log ||u||A1),  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' We observe that, by hypothesis,  φ(|x − y|) =  � |x−y|  0  φ′(s)ds,  φ′ ∈ L1(Rn),  and hence, for every x ∈ Rn and every ε > 0, by Fubini’s theorem we have that  Tφf(x) − φ(ε)Tf(x) =  � ∞  0  ��  |x−y|≥s≥ε  K(x, y)f(y)dy  �  φ′(s) ds  =  � ∞  ε  Tsf(x)φ′(s) ds,  is an average operator, and so the result follows by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 and letting ε  tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Limited extrapolation results  The motivation of this section comes from the fact that there are also many  operators in harmonic analysis (such as the Bochner-Riesz) so that  T : Lp0(v) −→ Lp0(v)  is not bounded for every v ∈ Ap0 but is bounded for every v in a certain subclass  of Ap0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Under this weaker hypothesis, only boundedness on Lp(v) of T can be  deduced whenever p ∈ (p−, p+) for certain values of p− and p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' The purpose of  this section is to establish some equivalence, similar to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6, between  the boundedness at the endpoint p− and restricted weak-type boundedness at  the p level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Indeed, the Rubio de Francia extrapolation results in this case are  called limited extrapolation (see [1, 8, 10, 15, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Given 0 ≤ α, β ≤ 1 and 1 ≤ p < ∞, let us define the classes  of weights  Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β) =  �  0 < v ∈ L1  loc(Rn) : v = vα  0 vβ(1−p)  1  , vj ∈ A1  �  with  ||v||Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β) = inf  �  ||v0||α  A1||v1||β(p−1)  A1  : v = vα  0 vβ(1−p)  1  �  ,  15  and  ˆAp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β) =  �  0 < v ∈ L1  loc(Rn) : v = vα  0 (Mh)β(1−p), v0 ∈ A1, h ∈ L1  loc(Rn)  �  with  ||v|| ˆ  Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β) = inf  �  ||v0||  α  1+β(p−1)  A1  : v = vα  0 (Mf)β(1−p)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Given 1 ≤ p0 < ∞ and 0 ≤ α, β ≤ 1, set  p+ =  p0  1 − α,  p′  − =  p′  0  1 − β ,  �  or p− =  p0  1 + β(p0 − 1)  �  ,  where p+ = ∞ if α = 1 and p− = 1 if β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, 1 ≤ p− ≤ p+ ≤ ∞ and we  can associate to every p ∈ [p−, p+] the indices  α(p) = p+ − p  p+  and  β(p) =  p − p−  p−(p − 1),  so that 0 ≤ α(p), β(p) ≤ 1, p+ =  p  1−α(p), p′  − =  p′  1−β(p) and α(p0) = α, β(p0) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 ([17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let (f, g) be a pair of measurable functions such that for  some 1 ≤ p0 < ∞ and 0 ≤ α, β ≤ 1 (not both identically zero) we have  ∥g∥Lp0(v) ≤ ϕ  �  ∥v∥Ap0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β)  �  ∥f∥Lp0(v) ,  ∀v ∈ Ap0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β),  where ϕ is a nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, for every p− < p < p+,  ∥g∥Lp(v) ≤ C1ϕ  �  C2 ∥v∥  max  � p+−p0  p+−p ,  p0−p−  p−p−  �  Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α(p),β(p))  �  ∥f∥Lp(v) ,  ∀v ∈ Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α(p),β(p)),  with C1 and C2 being two positive constants independent of all parameters in-  volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Observe that in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='3 is not possible to extrapolate till the endpoints  p− and p+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' However, in [10, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='7] the authors were able to obtain  an estimate in the endpoint p−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' To do so, they needed to assume that the  operators satisfy a restricted weak-type boundedness for the class of weights  ˆAp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β) which is a slightly bigger class than Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='4 ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let 1 ≤ p0 < ∞, 0 ≤ α, β ≤ 1 (not both identically zero)  and let T be an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Assume that  T : Lp0,1(v) −→ Lp0,∞(v),  ϕ(∥v∥ ˆ  Ap0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β)),  ∀v ∈ ˆAp0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α,β),  where ϕ is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then:  (i) If p− > 1,  T : Lp−,1 �  uα(p−)�  −→ Lp−,∞ �  uα(p−)�  ,  Φp−(||u||α(p−)  A1  )  p− − 1  ,  ∀u ∈ A1,  where Φp− is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  (ii) If p− = 1,  T : L1  R  �  uα(p−)�  −→ L1,∞ �  uα(p−)�  ,  Φ1(||u||α(p−)  A1  ),  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  16  On weak-type (1, 1) for averaging type operators  Our following theorem shows that the converse is also true:  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let (f, g) be a pair of measurable functions such that for some  1 ≤ p0 < ∞ and 0 < α ≤ 1,  ||g||Lp0,∞(uα) ≤ ϕ  �  ∥u∥α  A1  �  ||f||Lp0,1(uα),  ∀u ∈ A1,  with ϕ being a nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then, for any p0 ≤ p <  p0  1−α,  ||g||Lp,∞(v) ≤ Ψ  �  ||v|| ˆ  Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α(p),β(p))  �  ||f||Lp,1(v),  ∀v ∈ ˆAp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α(p),β(p)),  where α(p) = 1 − p(1−α)  p0  , β(p) =  p−p0  p0(p−1) and, for every r ≥ 1,  Ψ(r) = C1  �  1  p0 − p(1 − α)  � p−p0  p  ϕ  �  C2r  αp  p0−p(1−α)  �  r  α(p−p0)  p0−p(1−α) (1 + log r)  2(p−p0)  p  ,  with C1 and C2 being two positive constants independent of all parameters in-  volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let h ∈ L1  loc(Rn) and u ∈ A1 so that  v = (Mh)β(p)(1−p)uα(p) ∈ �Ap;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='(α(p),β(p)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Further, take t = 1+  1  2n+1∥u∥A1 so that ut ∈ A1 with ||ut||A1 ≲ ||u||A1 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='2))  and, since t > 1,  p − p0  p  = α − α(p)  1 − α(p) < tα − α(p)  t − α(p) < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Hence, we can take  vθ = (Mh)β(1−p)uα(p)θ  with  p − p0  p  < θ < tα − α(p)  t − α(p) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Besides, since α(p) ≤ α, letting  µ = 1 − α(p)(1 − θ)  αt  ∈ (0, 1),  then θ < αµ < 1 and, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1), for every measurable set F ⊆ Rn,  u0 =  �  Mαµ(χF vθ)uα(p)(1−θ)� 1  α = M  �  χFv  1  αµ  θ  �µ  (ut)1−µ ∈ A1,  with ||u0||A1 ≤  C||u||A1  1−µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, let y > 0 and set F = {x : |g(x)| > y} so that v(F) = λv  g(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' We can  assume, as it was done in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6, that v(F) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' by  17  hypothesis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' we obtain that  yp0λv  g(y) = yp0  �  {x : |g(x)|>y}  v(x) dx ≤ yp0  �  F  u0(x)αdx  ≤ ϕ  �  ||u0||α  A1  �p0  \uf8ee  \uf8f0p0  � ∞  0  ��  {|f(x)|>z}  u0(x)α dx  � 1  p0  dz  \uf8f9  \uf8fb  p0  ≲ ϕ  �  ||u0||α  A1  �p0  ����  Mαµ(χFvθ)  vθ  ����  L( p  p0)  ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='∞(v)  �� ∞  0  ||χ{|f(x)|>z}||  1  p0  L  p  p0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1(v)  dz  �p0  ≲ ϕ  �� Cαt||u||A1  α(p)(1 − θ)  �α�p0 ����  Mαµ(χF vθ)  vθ  ����  L( p  p0)  ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='∞(v)  ||f||p0  Lp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='1(v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  where in the penultimate estimate we have used the H¨older’s inequality for  Lorentz spaces with respect to the measure v(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, since β(p)(1 − p) = 1 − p  p0, then v ∈ ˆA p  p0 and, by virtue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5,  ����  Mαµ(χF vθ)  vθ  ����  L( p  p0 )  ′,∞(v)  ≲ C p  p0 ,θ,αµ(u)v(F)  p−p0  p  = C p  p0 ,θ,αµ(u)λv  g(y)  p−p0  p ,  so taking the supremum over all y > 0, in particular, we obtain that  ∥g∥Lp,∞(v) ≲ C p  p0 ,θ,αµ(u)  1  p0 ϕ  � ˜C||u||α  A1  (1 − θ)α  �  ||f||Lp,1(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Finally, concerning about the constant C p  p0 ,θ,αµ(u), we observe that  C p  p0 ,θ,αµ(u) =  �  p2  (p − p0)2(αµ − θ)(θ − p−p0  p )2  �θ  ∥u∥  2θ− 2(p−p0)  p0  A1  ≲  �  p2  (p − p0)2(α − θ)(θ − p−p0  p )2  �θ  ∥u∥  3θ− 2(p−p0)  p0  A1  ,  so the behaviour of the constant C p  p0 ,θ,αµ(u) follows as in the proof of Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  □  As an application, we present some new weighted estimates for the Bochner-  Riesz operator below the critical index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6 ([25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let n = 2 and 0 < λ < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then,  Bλ : L  4  3+2λ  �  u  2λ  3+2λ  �  −→ L  4  3+2λ ,∞ �  u  2λ  3+2λ  �  ,  c(n, λ) ∥u∥  λ(7+4λ)  6+4λ  A1  ,  ∀u ∈ A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='7 ([13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let n > 2 and  n−1  2(n+1) < λ < n−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Then  Bλ : L2 �  u  1+2λ  n  �  −→ L2 �  u  1+2λ  n  �  ,  ϕ  �  ||u||  1+2λ  n  A1  �  ,  ∀u ∈ A1,  where ϕ is a positive nondecreasing function on [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  18  On weak-type (1, 1) for averaging type operators  Therefore, as a consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='5 and Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='7, we  obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Let n = 2 and 0 < λ < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' For every  4  3+2λ ≤ p < 4  3,  Bλ : Lp,1(v) −→ Lp,∞(v),  ∀v ∈ ˆAp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  �  4−3p  4  , (3+2λ)p−4  4(p−1)  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Now, let n > 2 and  n−1  2(n+1) < λ < n−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' For every 2 ≤ p <  2n  n−1−2λ,  Bλ : Lp,1(v) −→ Lp,∞(v),  ∀v ∈ ˆAp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  �  2n−p(n−1−2λ)  2n  ,  p−2  2(p−1)  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  References  [1] Auscher, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Martell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' : Weighted norm inequalities, off-diagonal estimates and elliptic  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' General operator theory and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 212 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content='  [2] Baena-Miret, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Carro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' 3, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 43, 22 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [3] Bennett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Sharpley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': Interpolation of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Pure and Applied Mathematics,  129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Academic Press, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=', Boston, MA, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' xiv+469 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' ISBN: 0-12-088730-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [4] Bochner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': Summation of multiple Fourier series by spherical means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 40 (1936), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 2, 175–207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [5] Buckley, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': Estimates for operator norms on weighted spaces and reverse Jensen inequal-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 340 (1993), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 1, 253–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [6] Carro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': From restricted weak type to strong type estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  (2) 70 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 3, 750–762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [7] Carro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Domingo-Salazar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': Weighted weak-type (1, 1) estimates for radial Fourier  multipliers via extrapolation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content='  [9] Carro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' 3, 422–426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content='  North-Holland Mathematics Studies, 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' x+604 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=': Modern Fourier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Third edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Graduate Texts in Mathematics,  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Springer, New York, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' ISBN: 978-1-4939-1229-2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=': Real variable methods in Fourier analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' North-Holland Mathematics  Studies, 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Notas de Matem´atica [Mathematical Notes], 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' North-Holland Publishing  Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=', Amsterdam-New York, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' xiii + 392 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' ISBN: 0-444-86124-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [23] Hyt¨onen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' P´erez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': The L(log L)ε endpoint estimate for maximal singular integral  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content='  [24] Kerman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Torchinsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Lacey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' : Sparse endpoint estimates for Bochner-Riesz multipliers on the  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Wheeden, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' : Results on weighted norm inequalities for multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Ombrosi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  IMRN 2008, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Rivera-R´ıos, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+page_content=' : Factorization and extrapolation of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' ), 7 (2) (1982), 393—395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [33] Rubio de Francia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' : Factorization theory and Ap weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 106 (1984),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 3, 533–547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [34] Soria, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': On an extrapolation theorem of Carleson-Sj¨olin with applications to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' con-  vergence of Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 94 (1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 3, 235–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  [35] Vargas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=': Weighted weak type (1, 1) bounds for rough operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 54 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content=' 2, 297–310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Departament de Matem`atiques i Inform`atica, Universitat de Barcelona, 08007  Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Email address: sergibaena@ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='edu  Departamento de An´alisis Matem´atico y Matem´atica Aplicada, Universidad  Complutense de Madrid, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='  Email address: mjcarro@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
+page_content='es' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ftE4T4oBgHgl3EQfqg2l/content/2301.05201v1.pdf'}
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+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2023
+1
+Tacchi: A Pluggable and Low Computational Cost
+Elastomer Deformation Simulator for Optical
+Tactile Sensors
+Zixi Chen 1,2, Shixin Zhang3, Shan Luo1*, Fuchun Sun2, and Bin Fang2*
+Abstract—Simulation is widely applied in robotics research
+to save time and resources. There have been several works to
+simulate optical tactile sensors that leverage either a smoothing
+method or Finite Element Method (FEM). However, elastomer
+deformation physics is not considered in the former method,
+whereas the latter requires a massive amount of computational
+resources like a computer cluster. In this work, we propose a plug-
+gable and low computational cost simulator using the Taichi pro-
+gramming language for simulating optical tactile sensors, named
+as Tacchi. It reconstructs elastomer deformation using particles,
+which allows deformed elastomer surfaces to be rendered into
+tactile images and reveals contact information without suffering
+from high computational costs. Tacchi facilitates creating realistic
+tactile images in simulation, e.g., ones that capture wear-and-tear
+defects on object surfaces. In addition, the proposed Tacchi can be
+integrated with robotics simulators for a robot system simulation.
+Experiment results showed that Tacchi can produce images with
+better similarity to real images and achieved higher Sim2Real
+accuracy compared to the existing methods. Moreover, it can be
+connected with MuJoCo and Gazebo with only the requirement
+of 1G memory space in GPU compared to a computer cluster
+applied for FEM. With Tacchi, physical robot simulation with
+optical tactile sensors becomes possible. All the materials in this
+paper are available at https://github.com/zixichen007115/Tacchi.
+Index Terms—Force and Tactile Sensing, Robot manipulation,
+Simulation of tactile sensors
+I. INTRODUCTION
+S
+IMULATION plays a significant role in robotic research.
+Experiments in the real world are costly and time-
+consuming while posing potential risks of wear and accidents
+Manuscript received August 16, 2022; Revised October 31, 2022; Accepted
+January 11, 2023.This paper was recommended for publication by Editor A.
+Bera upon evaluation of the Associate Editor and Reviewers’ comments.This
+work was jointly supported by Major Project of the New Generation of Artifi-
+cial Intelligence, China (No. 2018AAA0102900), the National Natural Science
+Foundation of China (Grant No. 62173197) and the EPSRC project ”ViTac:
+Visual-Tactile Synergy for Handling Flexible Materials” (EP/T033517/1). Z.
+Chen and S. Zhang contributed equally to this work. *Corresponding authors:
+Shan Luo and Bin Fang.
+1Z.
+Chen
+and
+S.
+Luo
+are
+with
+the
+Department
+of
+Engineering,
+King’s College London, London WC2R 2LS, United Kingdom. E-mail:
+czx980730@gmail.com; shan.luo@kcl.ac.uk.
+2Z. Chen, B. Fang and F. Sun are with the Institute for Artificial In-
+telligence, Department of Computer Science and Technology, Beijing Na-
+tional Research Center for Information Science and Technology, Tsinghua
+University, Beijing 100084, China. E-mail: czx980730@gmail.com;
+fangbin@tsinghua.edu.cn; fcsun@mail.tsinghua.edu.cn .
+3S. Zhang is with the School of Engineering and Technology, China
+University
+of
+Geosciences
+(Beijing),
+Beijing
+100083,
+China.
+E-mail:
+zhangshixin@email.cugb.edu.cn.
+Digital Object Identifier (DOI): see top of this page.
+(B)
+(C)
+(D)
+(E)
+(A)
+Fig. 1. (A): The GelSight sensor. (B): GelSight diagram. This diagram shows
+that the elastomer layer deforms during interaction with an indenter. (C), (D),
+and (E): Optical tactile images collected from the real sensor [8], pseudo
+tactile image generated from [1] and Tacchi proposed in this work. As shown
+in the figures, Tacchi offers more realistic tactile images in simulation with
+surface defects.
+to the robots, but simulation can provide data as preliminary
+experiments without such risks. Besides, data-driven methods
+like neural networks and deep learning are utilized in robotics
+for sensing [1] and control [2]. An extensive amount of data
+is required, and simulation provides an efficient way to collect
+data with the potential for transferring trained robot agents to
+the real world.
+In the past years, various simulation approaches have been
+proposed for object simulation, such as the Finite Element
+Method (FEM) [3] and key points [4]. There have been some
+commonly used simulators for robotics, such as Gazebo [5],
+Pybullet [6] and MuJoCo [7]. Although they are able to
+simulate robot components like robot arms and grippers, tactile
+sensor simulation is still a challenging task, which hinders
+physical robot simulation with tactile sensing.
+Tactile sensing is indispensable for robot contact control,
+and many tactile sensors have been developed in recent
+years [9], [10]. Thanks to the ability to produce high-resolution
+tactile images, optical tactile sensors [8], [11]–[13] that use
+cameras to capture deformation of soft elastomer have been
+favoured for robotics research. An example is GelSight, as
+shown in Fig. 1-(A). A GelSight sensor is composed of a
+camera, a set of LEDs, and a soft elastomer. During interaction
+with objects, the deformed opaque surface of the elastomer
+reflects light from the LEDs to the camera, and the tactile
+arXiv:2301.08343v1  [cs.RO]  19 Jan 2023
+
+Oindenter
+sliver reflecting layer
+light guiding plate
+gel layer
+LED
+supporting structure
+transparent planeindenter
+reflecting layer
+elastomer
+transparent
+plane
+light guiding
+supporting
+plate
+structure
+(B)
+LED
+camera2
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2023
+images captured by the camera can offer contact information.
+Due to their low cost and high resolution, optical tactile
+sensors have been widely used in real experiments
+[14]–
+[16], but challenges in simulating deformations of their soft
+elastomer layers impede having them in simulation.
+The main challenge of optical tactile stimulation is elas-
+tomer deformation reconstruction. Deformable object sim-
+ulation is a complex problem caused by high degrees of
+freedom. Elastomer deformation is first approximated in [17]
+using pseudo depth maps applied with a smoothing method.
+However, in this method the smoothing parameters are fine-
+tuned only based on the optical comparison, ignoring the
+elastomer elasticity properties. Hence, this method can only
+be applied when a normal force is applied. Meanwhile, in
+some other works [3], simulation software like Abaqus is used
+to provide detailed elastomer deformations. However, they
+suffer from a high computational cost for FEM simulation.
+Considering the high speed of the other robot simulation
+processes, this time-consuming approach is unsuitable for a
+single sensor simulation.
+To address these issues, we propose a pluggable and low
+computational cost optical tactile sensor simulator Tacchi.
+This simulator is named after tactile sensor simulation and
+the applied parallel programming language, Taichi [18]. In
+Tacchi, particles are utilized to represent elastomers of the
+optical tactile sensors, and their motions represent elastomer
+deformation. Surface depth maps are generated from the
+particles on the elastomer surface, which are then rendered
+into tactile images. Experimental results show that Tacchi
+can be connected with robotics simulators like Gazebo and
+MuJoCo. Our method produces higher quality tactile images
+than the existing methods [1], [19] in terms of both tactile
+image quality and Sim2Real transfer learning. In addition,
+Tacchi can add random defects to simulate textures on real
+objects and improve the Sim2Real performance. Examples of
+the real and simulated tactile images from different approaches
+are shown in Fig. 1-(C), (D).
+The contributions of this paper are summarized as follows:
+1) We develop a novel simulator Tacchi that is pluggable
+and requires low computation space. This is the first
+optical tactile sensor simulator that utilizes physical
+deformation simulation. The tactile images from Tacchi
+achieve satisfactory similarity to the real tactile images.
+2) We connect Tacchi and the commonly used simulators,
+Gazebo and MuJoCo, for robot simulation. The exper-
+imental results show that this simulation can achieve
+robot simulation and include tactile sensors.
+3) We investigate the effect of particle number on the
+simulated tactile images with Tacchi, and the results
+show that our approach achieves superior performance
+in Sim2Real learning.
+The rest of the paper is structured as follows: Section II
+includes works related to elastomer simulation and sensor
+simulation; Section III introduces our proposed Tacchi sim-
+ulator and its connection with Gazebo and MuJoCo; Section
+IV describes the experimental setup, including the real and
+simulation setups; Section V shows the experimental results;
+Section VI summarizes the work.
+II. RELATED WORK
+A. Elastomer simulation
+It is challenging to simulate elastomers like gel layers in
+optical tactile sensors due to their deformation properties and
+high degrees of freedom. FEM has been widely applied to
+simulate deformation in [3], [10]. Nodes and facets are applied
+in [20] to represent a rubber sphere and a foam cube. Based
+on such a model, a strategy to grasp an elastomer with a multi-
+finger robot hand is introduced.
+In optical tactile sensors, various methods are applied for
+elastomer simulation. Physics-based methods, which simulate
+the deformation with some physical principles, have been uti-
+lized. For example, FEM is applied in [3] for force distribution
+estimation. Although this work utilizes physics-based simu-
+lation, it requires massive computational resources and the
+simulation is achieved by using a high performance computing
+cluster. In FEM, each element is connected with the other ones,
+and deformation simulation may also lead to mesh distortion
+problems like irregular meshes or even negative element
+volume [21]. Meanwhile, some non-physics-based methods
+try to avoid physical interaction simulation with smoothing
+methods. Pseudo elastomer surfaces are reconstructed in [1] by
+smoothing depth maps with Gaussian kernels. Similarly, [19]
+uses object depths for the contacted area and kernel smoothing
+method for the uncontacted area. PyBullet [6] is utilized in
+[22] as the object and sensor simulator. Smooth depth maps
+are generated by either pre-processing the object meshes or
+applying data augmentation. Although these methods simulate
+tactile images, they do not simulate physical deformation and
+only produce depth maps by smoothing, which do not have
+physical meaning.
+To address these issues, we employ the Material Point
+Method (MPM) for physics-based elastomer simulation. This
+method applies particles to represent objects and an imaginary
+grid for contact simulation. The grid exchanges object infor-
+mation with particles during the simulation. Compared with
+other physics-based methods like FEM, MPM does not suffer
+from mesh distortion thanks to independent particles [23] and
+has a low computational cost thanks to the use of Taichi [18], a
+parallel programming language. Compared with non-physics-
+based methods, MPM simulates physics-based interactions and
+provides realistic results.
+B. Sensor simulation
+Various sensors have been applied in robot systems, and
+sensor simulation plays an essential role before deploying
+them into the real world. Common sensors like cameras and
+force sensors have been integrated into mainstream robot
+simulators such as Gazebo [5] and MuJoCo [7].
+In [1], optical tactile sensors were first simulated by smooth-
+ing depth maps collected from Gazebo. In [22], the meshes of
+both the object and the sensor are first loaded in OpenGL [24]
+and then updated during the simulation in PyBullet. In this
+work, depth maps can either be obtained from PyBullet or
+generated in OpenGL together with tactile images. However,
+these depth maps from widely applied robot simulators do not
+include physics-based deformation.
+
+CHEN et al.: TACCHI: A PLUGGABLE AND LOW COMPUTATIONAL COST ELASTOMER DEFORMATION SIMULATOR FOR OPTICAL TACTILE SENSORS
+3
+initialization
+particle-to-grid
+grid-to-particle
+particle advection
+indenter particle
+elastomer particle
+grid
+elastomer simulation
+(A)
+surface depth map
+cropped depth map
+tactile image
+(D)
+(C)
+(B)
+Fig. 2. Tactile image generation process. (A): Elastomer simulation. Grey particles represent the elastomer, and green particles represent the indenter. The
+particles and grid nodes are first initialized; the particle information is transferred to the grid in particle-to-grid; then the object information (i.e., the indenter’s
+motion in this case) is allocated back to nearby particles by the grid nodes in grid-to-particle; finally, particles are moved according to the object information.
+(B): With the discrete surface particles, a continuous surface depth map can be reconstructed via a 2D interpolation method. (C): The depth map is cropped
+into 480 × 640 to fit the image size from the Gelsight. The cropped image is shown by the red frame. (D): A tactile image is generated based on the cropped
+depth map and the rendering method.
+Engineering simulation software is also utilized for tactile
+sensor simulation. COMSOL Multiphysics is exploited in [25]
+to predict magnetic fields under different conductive target
+situations. In [3], Abaqus is used to simulate elastomer de-
+formation. Although these software programs show excellent
+simulation results, most of them are packaged, and it is
+challenging to connect them with robot simulators due to
+incompatibility and time-consuming FEM algorithms.
+In this work, we develop a novel simulator Tacchi to sim-
+ulate elastomer deformation by utilizing a high-performance
+numerical computation language Taichi [18]. Thanks to its
+parallel programming and concise data structure, Tacchi can
+be run with a low requirement of computational resources, and
+it is simple to connect to simulators like MuJoCo and Gazebo.
+III. METHOD
+In this section, the proposed simulator Tacchi is introduced
+and its connection with the robot simulator MuJoCo and
+Gazebo is also detailed. Specifically, the elastomer simula-
+tion is first introduced in Subsection III-A; a tactile image
+rendering method based on elastomer simulation in Tacchi is
+then proposed in Subsection III-B; the integration of Tacchi
+with a robot simulator is detailed in Subsection III-C; and
+the potential applications of Tacchi in the simulation of other
+sensors and robot tasks are introduced in Subsection III-D.
+A. Elastomer simulation
+In Tacchi, we apply the Material Point Method (MPM)
+for elastomer simulation. MPM applies particles to represent
+objects and an imaginary grid for contact simulation. The grid
+exchanges information with particles during the simulation.
+FEM and MPM use meshes and particles as their elements,
+respectively. For both approaches, each element interacts with
+the nearby elements based on their geometry shapes and
+relative positions. For FEM, there will be mesh distortions
+because deformation may lead to problems like irregular
+meshes or even negative element volume, which results in
+low accuracy of stress [21]. In contrast, for MPM, particles
+exchange information with an imaginary grid, and this method
+does not suffer from mesh distortions.
+We first introduce the elastomer simulation in optical tactile
+sensors. To describe this method, we simulate an elastomer
+pressed by a rigid indenter as an example, as shown in Fig. 2-
+(A). In Tacchi, both the elastomer (grey dots in Fig. 2-(A)) and
+the object that interacts with the elastomer, i.e., an indenter
+(green dots in Fig. 2-(A)) in this example, are represented
+using particles. The object information, e.g., the position and
+the velocity, are stored in these particles.
+In the simulation, it is necessary to decide which particles
+are included in the interaction between the elastomer and the
+object and how they interact. To this end, a fixed imaginary
+grid is introduced, as shown in Fig. 2-(A), so as to have the
+particles exchange object information via the grid nodes.
+In each simulation step (i.e., the motions of the particles
+are updated for one time), as shown in Fig. 2-(A), the object
+information is first collected by the grid nodes from nearby
+particles (we name it as particle-to-grid); the interaction
+between the particles of the elastomer and the indenter, and
+also the object elastic deformation, are then calculated by
+the grid nodes and shared with each particle (we name it as
+grid-to-particle); the states of the particles will be updated
+accordingly afterwards (we name it as particle advection).
+Initialization: Firstly, we initialize particles and grid nodes.
+A set of m particles are used to represent the elastomer and
+the indenter, and a grid of n grid nodes is used to record the
+object information with a fixed coordinate frame. For the p-
+th particle, its position xp ∈ R3 and its velocity vp ∈ R3
+are considered for the simulation of the object motions.
+For deformation simulation, similar to [26], we introduce a
+deformation map φp : R3 → R3 that records the initial
+and final position in a simulation step for each particle. The
+deformation gradient Fp ∈ R3×3 can be derived as
+Fp = ∂φp
+∂xp
+(xp),
+(1)
+where Fp is set as a three-dimensional identity matrix I3×3 in
+the initialization.
+In addition, an affine velocity matrix Cp ∈ R3×3 that
+records the velocity of the neighboring particles [26] is intro-
+duced to reduce information loss during information exchange
+
+0.0000
+0.0000
+0.0002
+-0.0002
+0.0004
+0.0004
+-0.0006
+0.0008
+0.0006
+-0.0010
+0.0008
+600
+500
+-0.0010
+400
+100
+OOE
+200
+300
+200
+400
+100
+500
+6004
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2023
+between the particles and the grid nodes. It is initialized
+as 03×3 since the particles are static and will be updated
+afterward.
+Particle-to-grid: In each simulation step, particle information
+is first transferred to the grid by calculating momentum and
+mass in each grid node. It can be seen as simulating part
+of the object surrounding the grid node by quadratic B-Spline
+weighting interpolation. Specifically, the mass and momentum
+of the nearby particles are collected. The mass Mi of the i-th
+grid node is
+Mi =
+�
+j∈Gi
+�
+p∈Pj
+wjpmp,
+(2)
+where mp denotes the mass of the p-th particle, Gi denotes
+3 × 3 × 3 grid nodes which contain the i-th grid node and
+its neighboring grid nodes, Pj denotes the particles inside
+the j-th grid, and wij is the weight parameter of the i-th
+grid node and the j-th particle for weighting interpolation,
+which is calculated by quadratic B-Spline. By applying such
+weight parameters, close particles contribute more to Mi than
+far ones. We follow [27] to have the same weight parameters
+for mass and the following momentum calculation.
+The grid momentum MGi of the i-th grid node can be
+obtained by calculating the momentum resulted from the
+particle motion MM i and the momentum resulted from the
+elasticity MEi:
+MGi = MM i + MEi.
+(3)
+Here MMi is calculated by collecting the velocity and affine
+velocity of the nearby particles:
+MMi =
+�
+j∈Gi
+�
+p∈Pj
+wjp(mpvp + Cp(Xj − xp)),
+(4)
+where Xj denotes the position of the j-th neighboring node
+of the i-th grid node.
+MEi can be obtained as in [26] by:
+MEi = −△t
+�
+j∈Gi
+�
+p∈Pj
+4
+△X2 wjpV 0
+p Sp(Xj − xp),
+(5)
+where △t is the time interval between two adjacent steps,
+△X is the grid node interval, V 0
+p denotes the initial particle
+volume, Sp is the elasticity force for the p-th particle [26].
+After obtaining Mi and MGi using Eq. 2 and Eq. 3
+respectively, the object velocity close to the i-th grid node
+Vi can be computed as:
+Vi = MGi
+Mi
+.
+(6)
+Grid-to-particles: The states of the particles are then updated
+with the previous states of the grid nodes and the particles.
+It can be seen as simulating part of the object surrounding
+the particle by quadratic B-Spline weighting interpolation.
+Suppose that the states of the k-th step, i.e., the velocity v(k)
+p ,
+the affine velocity C(k)
+p , and the deformation gradient F (k)
+p
+,
+are known, v(k+1)
+p
+, C(k+1)
+p
+and F (k+1)
+p
+in the k + 1 step can
+be obtained as
+v(k+1)
+p
+=
+�
+i∈G′p
+wipV (k)
+i
+,
+(7)
+C(k+1)
+p
+=
+4
+△X2
+�
+i∈G′p
+wipv(k+1)
+p
+(Xi − x(k)
+p ),
+(8)
+F (k+1)
+p
+= (I + △tC(k+1)
+p
+)F (k)
+p
+,
+(9)
+where G′p represents the 3×3×3 grid nodes surrounding the
+p-th particle.
+Particle advection: With Eq. 8 and Eq. 9, the states of
+particles can be updated. However, there are two special cases:
+the velocities of the particles on the bottom layers of the
+elastomer are set as 0 since they are fixed onto the sensor
+(and the sensor is static in this example); as the indenter is
+rigid, the particles representing the indenter share the same
+velocity, i.e., the velocity of the indenter. As a result, v(k+1)
+p
+for these special particles are obtained as:
+v(k+1)
+p
+=
+�
+vi,
+p ∈ I,
+0,
+p ∈ B,
+(10)
+where vi denotes the velocity of the indenter; I denotes
+particles representing the indenter; B denotes the particles on
+the bottom layers of the elastomer.
+Finally, the particles move with the above computed veloc-
+ities and the position of each particle x(k+1)
+p
+in the k + 1 step
+is
+x(k+1)
+p
+= x(k)
+p
++ △tv(k+1)
+p
+.
+(11)
+B. Tactile image rendering
+As shown in Fig. 2-(B), (C), and (D), after particle motion
+surface particles and the Phong’s model [28] are used to
+generate the tactile images. Although there may be various
+methods to produce tactile images from the particles, we fol-
+low [1]’s method by generating depth maps first and rendering
+them. A 2D interpolation method is applied to provide a
+continuous depth map based on discrete particles representing
+the elastomer surface, as shown in Fig. 2-(B). It should be
+highlighted that the obtained depth map here is from the
+physics-based elastomer deformation, instead of the depth map
+of the indenter like [1]. In this way, the resulted tactile image
+will be more realistic as in the real sensor the camera also
+captures the elastomer deformation.
+As shown in Fig. 2-(B), the depth map of the whole square
+elastomer surface is provided in Tacchi. However, due to its
+limited field of view, the camera in the Gelsight [8] sensor
+generates images with a resolution of 480×640, in which only
+a partial area of the sensor is captured instead of the whole
+surface. To this end, only this partial area of the depth map
+will be rendered into tactile images (Fig. 2-(C)). In addition,
+there are misalignments between the desired and real setups,
+due to the calibration errors of the camera and the indentor’s
+movements. In [1], some cropping and alignment approaches
+are proposed to match real and the simulated tactile images so
+as to compensate the misalignments. We follow the Alignment
+(per object) approach [1], with the cropped depth map shown
+in Fig. 2-(C).
+
+CHEN et al.: TACCHI: A PLUGGABLE AND LOW COMPUTATIONAL COST ELASTOMER DEFORMATION SIMULATOR FOR OPTICAL TACTILE SENSORS
+5
+main simulator
+Tacchi
+Initialization
+Step Simulation
+Terminal
+Step
+load object and sensor 
+initialize particles 
+and grid
+move objects
+move particles, render
+main simulator 
+terminal condition
+Tacchi
+terminal condition
+Fig. 3. Connection with a robot simulator. The misalignment denotes the time sequence in each simulation step. In initialization, the robot simulator loads
+models as the main simulator, and Tacchi initializes particles and a grid. During the simulation, the main simulator moves the objects based on the desired
+policy, and Tacchi obtains the velocity of the objects from the main simulator and moves the particles. Tactile images are rendered from the depth maps.
+Terminal conditions are proposed in the main simulator and Tacchi.
+We follow [1] to simulate illumination of the GelSight
+sensor using the Phong’s model [28]. With the obtained depth
+map D and light resources L, the tactile image I can be
+obtained, as shown in Fig. 2-(D):
+I = kaia +
+�
+m∈L
+(kd(�Lm · �
+N)im,d + ks( �Rm · �V )αim,s), (12)
+�Rm = 2(�Lm · �
+N) �
+N − �Lm,
+(13)
+�
+N =< ∂H
+∂x , ∂H
+∂y , −1 >
+=< H
+2r ∗ [−1, 0, 1], H
+2r ∗ [−1, 0, 1]T , −1 >,
+(14)
+where ka, kd, ks, and α are surface reflectance property
+parameters; r is the pixel-to-meter ratio; ia is the background
+light that is not from the light resources; im,d and im,s are
+the intensities of the diffuse and specular reflections of light
+source m respectively; �Lm and �Rm are emission and reflection
+direction of the light resource m; �V is the direction from the
+surface point to camera; �
+N is the elastomer surface normal;
+∗ stands for 2D convolution operation in this equation. These
+parameters for simulating the GelSight are detailed in [1].
+C. Connection with a robot simulator
+This subsection introduces the connection of Tacci with
+a robot simulator. As shown in Fig. 3, Tacchi generates
+tactile images, and the robot simulator is applied as the main
+simulator to simulate the motions of the objects and the robot,
+etc. MuJoCo and Gazebo are used as the main simulator in
+this paper. However, any robot simulator that can provide the
+real-time position or velocity of the sensor and the contacted
+objects can also be used. First, the object and sensor models,
+usually mesh files, are loaded in the main simulator, and they
+are placed in the initial positions. As Tacchi requires particle
+information, point clouds from the object and sensor models
+are used to initialize the particles of the object and the sensor.
+In every simulation step, the object, i.e., the indenter in this
+example, moves according to the control policy in the main
+simulator. The object position or velocity is acquired by Tacchi
+from the main simulator to simulate the object’s motion. Then
+depth maps are generated from the elastomer deformation and
+are rendered into tactile images. During the simulation, if a
+terminal condition is met, the simulation will terminate. For
+example, if the indenter reaches the desired depth in the main
+simulator or Tacchi, the simulation will end.
+D. Application of Tacchi to other sensors and tasks
+This subsection describes how to apply Tacchi to other tasks
+like robot grasping, and other optical tactile sensors [12], [13],
+[29] or soft inductive tactile sensors [25]. As shown in the
+above sections, only the elastomer in the tactile sensor and the
+objects which may contact the sensor are involved in Tacchi.
+To this end, there are only a few points to consider when using
+Tacchi to simulate other sensors or in other tasks:
+• In initialization, as the particles are desired to be inside
+the grid during the simulation, the grid needs to be a little
+larger than the particle range;
+• The grid node density should strike a balance between
+simulation quality and speed;
+• As the particle density will affect object surface rough-
+ness and computation cost, it needs to be determined
+according to the simulation demand;
+• For a new object or elastomer, its point clouds can be
+used, and the particles can be aligned according to the
+point positions in the point clouds. Elasticity parameters
+like Young’s modulus and Poisson’s ratio can be initially
+decided by measuring the applied material or using their
+standard parameters, and then fine-tuned based on the
+sensor outputs.
+IV. EXPERIMENTAL SETUP
+To evaluate Tacchi, we collect tactile images in the real
+world and simulation. Subsection IV-A introduces real exper-
+imental devices and data collection. Subsection IV-B includes
+the simulation setup like Tacchi parameters, indentor loading,
+and main simulator setup.
+
+6
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2023
+(A)
+(B)
+Gelsight
+elastomer
+indenter
+Fig. 4. (A): Real experiment diagram. The GelSight sensor presses against
+the indenter. (B): Twenty one indenters included in this work. The first
+row: cone, cross_lines, curved_surface, cylinder, cylinder_shell,
+cylinder_side, dot_in. The second row: dots, flat_slab, hexagon, line,
+moon, pacman, parallel_lines. The third row: prism, random, sphere,
+sphere2, torus, triangle, wave1.
+(A)
+(B)
+Fig. 5. (A): Tacchi connected with Gazebo. The sensor moves in Gazebo, and
+tactile images are generated by Tacchi. (B): Tacchi connected with MuJoCo.
+The indenter is pressed against the elastomer, and tactile images are generated
+by Tacchi.
+A. Real World setup
+We adopt the dataset from [1] in this work. In this case,
+GelSight and the objects in [1] are included in the real world
+setup. In the experiment, a Gelsight is pressed against an
+indenter, as shown in Fig. 4-(A). The dataset contains 21
+objects with various shapes as indenters, which are shown in
+Fig. 4-(B). 3 × 3 positions are chosen for press experiments,
+and the horizontal step is 1mm. The indention depth improves
+from 0 to 1mm, and the vertical step is 0.1mm, as a result
+11 images are collected at each position. Considering 21
+indenters, this experiment results in 3 × 3 × 11 × 21 = 2, 079
+tactile images. More details can be found in [1].
+B. Virtual World setup
+Fig. 5-(A) and Fig. 5-(B) show the Tacchi simulator con-
+nected with Gazebo and MuJoCo, respectively. In the Tacchi
+interface, an elastomer layer (grey dots) and an indenter (green
+dots) are included in the virtual world. In our work, the Gel-
+Sight elastomer is composed of 101×101×21 particles since
+the shape of the elastomer layer in the real GelSight sensor
+is 20mm × 20mm × 4mm. Young’s modulus and Poisson’s
+ratio are 1.45×105 and 0.45, and these parameters are decided
+according to [8] and some preliminary experiments. The grid
+contains 256 × 256 × 256 grid nodes, and the grid edge is
+33mm. More grid nodes lead to better simulation quality but
+slow down the simulation process. In general, all particles
+should be inside the grid during the simulation, and the grid
+interval should be larger than the particle interval. Based on
+such principles and after some preliminary experiments on
+different grid node densities, we choose such grid parameters.
+Although we only apply GelSight as the optical tactile sensor
+in this work, Tacchi can be utilized to simulate other tactile
+sensors by initializing particles with corresponding shapes.
+To create various indenters in Tacchi, the point positions
+in the 3D point clouds are used to initialize the indenter
+particles in Tacchi. If only 3D mesh files are available, some
+software programs and libraries like Open3D and Point
+Cloud Library can be utilized to extract point clouds from
+mesh files. Though all the points in the point clouds can
+be included, we change the particle density to accommodate
+the computational requirements. Also, the intervals among
+particles can simulate defects on the object surfaces. This
+feature will be exploited in Section V.
+Gazebo and MuJoCo simulation interface are shown in Fig.
+5-(A) and Fig. 5-(B). In the main simulator, we simulate the
+sensor and indenter by loading mesh files. Gazebo provides
+the indenter position, and Tacchi generates the corresponding
+tactile image. There is an interface in MuJoCo that can provide
+the indenter velocity, and the indenter in Tacchi shares the
+same velocity with that in MuJoCo by obtaining velocity
+through this interface. The indenter velocity control in Tacchi
+is achieved by Eq. 11. Although we only include Gazebo and
+MuJoCo in this work, any robot simulator which can provide
+the real-time indenter position or velocity has the potential to
+be the main simulator with Tacchi.
+V. EXPERIMENTAL RESULTS
+This section shows the experimental results of applying Tac-
+chi as the optical tactile sensor simulator. We first introduce the
+computation cost in Subsection V-A. To evaluate our approach,
+Subsection V-B compares the pseudo tactile images from [1],
+[19] and Tacchi. We also show images with different particle
+numbers to explore the effect of this parameter. Sim2Real
+learning is applied for evaluation in Subsection V-C.
+A. Computational Cost
+High computational cost is the main problem of physics-
+based simulation. For example, the FEM simulation is
+achieved with high computational costs using a high perfor-
+mance computing cluster in [3]. However, the parallel pro-
+gramming language Taichi [18] highly decreases the demand
+for computational resources. To simulate the sensor interaction
+with an intended in this work, over 106 particles and 107 grid
+nodes are simulated, but it only requires 1G memory in a
+GTX 1050Ti graphics card. Compared with FEM in [3], Tacchi
+requires much lower computational cost.
+B. Pseudo image quality
+In this subsection, we evaluate the simulation methods in
+[1], [19] and Tacchi with three metrics: Structural Similarity
+(SSIM), Peak Signal-to-Noise Ratio (PSNR), and Mean Abso-
+lute Error (MAE). These metrics are widely applied to evaluate
+image similarity, and the similarity between real and generated
+images can be taken as pseudo image quality. Depth maps
+are generated based on the methods in [1], [19] and Tacchi,
+and the rendering method in [1] is applied for deformation
+
+Gazebo
+tactile_Img
+1.00
+simTime:00.00:03:05134tReaf Time:0000:03:06:748:iteratio0s:105348
+FPS:60.66
+Reset TimeTacchl (1.57Fps)mujoco_py
+Tacchl (4.71:FPS)
+tactile_lmage_MuJoCoCHEN et al.: TACCHI: A PLUGGABLE AND LOW COMPUTATIONAL COST ELASTOMER DEFORMATION SIMULATOR FOR OPTICAL TACTILE SENSORS
+7
+Real
+[1]’s method
+Tacchi _10
+[19]’s method
+(A)
+(B)
+Fig. 6. (A): Tactile images from real dataset in [1] (top column), Tacchi 1 (2nd column), Tacchi 10 (3rd column), Tacchi 100 (4th column), [1]’s method (5th
+column), [19]’s method (6th column). These rows show the tactile images of cross_lines, curved_surfaces, dot_in, pacman, and triangle respectively.
+Tacchi 100, [19] and [1] show similar images but have slight differences caused by surface defects. Tacchi 10 provides random defects on the object surface,
+and Tacchi 1 generates too many defects that harm object shapes. (B): Enlarged areas of dot_in tactile image from real dataset, Tacchi 10, [1], and [19].
+simulation comparison. For Tacchi, we build the indenter
+models with 1 × 104, 10 × 104, and 100 × 104 particles and
+name them Tacchi 1, Tacchi 10, and Tacchi 100, respectively.
+We apply such particle numbers since preliminary experiments
+show that they provide indenters with excessive, appropriate,
+and few defects. Real tactile images, images from Tacchi 1,
+Tacchi 10, Tacchi 100, [1] and [19] are shown in Fig. 6-(A),
+and some images are enlarged for comparison in Fig. 6-(B).
+Objects contain defects and textures on their surfaces, and the
+real images show this feature by rough surfaces in Fig. 6-
+(B). The surfaces of images from [1] and [19] are smooth,
+while Tacchi 10 produces rough surfaces similar to the real
+images. Moreover, the indention edges of the other simulation
+methods, [1] and [19], are straight and clear, but the real edges
+are meandering and blurry, similar to images from Tacchi 10.
+TABLE I
+IMAGE QUALITY COMPARISON
+SSIM ↑
+PSNR ↑
+MAE ↓
+Tacchi 1
+0.819 ± 0.011
+18.43 ± 3.07
+8.73 ± 0.03%
+Tacchi 10
+0.845 ± 0.008
+18.57 ± 2.89
+8.57 ± 0.03%
+Tacchi 100
+0.857 ± 0.006
+18.46 ± 2.95
+8.67 ± 0.03%
+[1]’s method
+0.852 ± 0.006
+18.51 ± 3.26
+8.64 ± 0.04%
+[19]’s method
+0.854 ± 0.005
+18.50 ± 3.36
+8.66 ± 0.04%
+In Table I, Tacchi 100 obtains the highest SSIM, and
+Tacchi 10 has the best performance on the other two metrics.
+This discrepancy can also be seen in [1]. Compared with
+Tacchi 100, Tacchi 10 includes part of points in the point
+clouds and produces rough surfaces similar to real objects. In
+the real world, scratches and defects are common on object
+surfaces, and this feature can improve the performance of
+algorithms. But too many defects can harm the object shapes
+and lead to low image quality like Tacchi 1. Our methods
+obtain better performance than [1] and [19] since their depth
+maps are based on kernel smoothing and ignore physical
+deformation, but real sensors provide depth maps by capturing
+the elastomer deformation, similar to our method.
+C. Sim2Real for object classification
+In this section, we study how Tacchi performs in a Sim2Real
+task. Simulated data is desired to approximate the real data,
+and it can be evaluated by comparing the performance of real
+and generated data on the same task. In this work, we apply
+object recognition as the evaluation task and introduce a neural
+network for the recognition task. To this end, we train the
+ResNet-18 [30] network with an early stopping strategy, and
+each tactile image is cropped to 440 × 440 and resized to
+256 × 256 before being fed into the network. We randomly
+choose 20 images for each indenter as the test dataset and
+10 images as the validation dataset, considering that each
+indenter produces 3 × 3 × 11 = 99 images. Images with
+the indention depth 0mm are discarded since the indenters do
+not press into the elastomer in these images, and the images
+cannot provide object information. The training batch is 32,
+and we train the network with real images, images generated
+by Tacchi 1, Tacchi 10, Tacchi 100, [1] and [19] for at most
+2000 iterations. Then we test them on the test dataset of real
+images. The classification accuracies are shown in Table II.
+In Table II, Tacchi 10 achieves the highest accuracy thanks
+to the realistic rough surfaces. The tactile images of Tacchi 1
+are the most coarse ones and acquire the lowest accuracy.
+Simulation methods based on ideal object models tend to
+provide results with few scratches and defects, such as [1],
+[19] and Tacchi 100. However, most objects in the real world
+contain random defects and scratches. Therefore, ideal simu-
+
+Real
+Tacchi 1
+Tacchi 10
+Tacchi_100
+[1]'s method
+[19]'s method8
+IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2023
+TABLE II
+SIM2REAL ACCURACY
+Test Accuracy
+Real2Real
+99.67 ± 0.23%
+Sim2Real (Tacchi 1)
+51.26 ± 5.91%
+Sim2Real (Tacchi 10)
+61.87 ± 7.18%
+Sim2Real (Tacchi 100)
+49.06 ± 8.18%
+Sim2Real ( [1]’s method)
+44.17 ± 5.21%
+Sim2Real ( [19]’s method)
+48.19 ± 6.89%
+lated models cannot be the same as real objects, and a proper
+number of defects can improve the simulation performance,
+as shown in Tacchi 10, but too many defects affect the object
+information and harm the simulation like Tacchi 1.
+VI. CONCLUSION AND DISCUSSION
+We introduce Tacchi, a pluggable and low computational
+cost elastomer deformation simulator for optical tactile sen-
+sors. It simulates elastomer deformation and produces tactile
+images with the depth maps generated from the deformed
+elastomer. Tacchi can be connected with a robot simulator and
+simulates with a low demand of computation resources. The
+experimental results show that Tacchi provides realistic tactile
+images and achieves high accuracy in Sim2Real tasks, and
+defects on the object surfaces can be simulated by changing
+the number of particles. Overall, Tacchi is an effective optical
+tactile sensor simulator that can be applied in robot simulation.
+Only 3D printed objects are applied in this paper, and
+future works will explore the relationship between real object
+roughness and the particle sampling ratio in the simulation
+with different cases of materials and surfaces. Some optical
+tactile sensors contain markers on the inner elastomer surfaces
+which can be represented by the particles sharing the same po-
+sition. By recording surface particle motion, we can simulate
+marker motion. We will also consider simulation of the surface
+frictions during sliding and rotation motions in future works.
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf,len=708
+page_content='IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' ACCEPTED JANUARY, 2023  1  Tacchi: A Pluggable and Low Computational Cost  Elastomer Deformation Simulator for Optical  Tactile Sensors  Zixi Chen 1,2, Shixin Zhang3, Shan Luo1*, Fuchun Sun2, and Bin Fang2*  Abstract—Simulation is widely applied in robotics research  to save time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' There have been several works to  simulate optical tactile sensors that leverage either a smoothing  method or Finite Element Method (FEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However, elastomer  deformation physics is not considered in the former method,  whereas the latter requires a massive amount of computational  resources like a computer cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In this work, we propose a plug-  gable and low computational cost simulator using the Taichi pro-  gramming language for simulating optical tactile sensors, named  as Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' It reconstructs elastomer deformation using particles,  which allows deformed elastomer surfaces to be rendered into  tactile images and reveals contact information without suffering  from high computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Tacchi facilitates creating realistic  tactile images in simulation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', ones that capture wear-and-tear  defects on object surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In addition, the proposed Tacchi can be  integrated with robotics simulators for a robot system simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Experiment results showed that Tacchi can produce images with  better similarity to real images and achieved higher Sim2Real  accuracy compared to the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Moreover, it can be  connected with MuJoCo and Gazebo with only the requirement  of 1G memory space in GPU compared to a computer cluster  applied for FEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' With Tacchi, physical robot simulation with  optical tactile sensors becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' All the materials in this  paper are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='com/zixichen007115/Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Index Terms—Force and Tactile Sensing, Robot manipulation,  Simulation of tactile sensors  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' INTRODUCTION  S  IMULATION plays a significant role in robotic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Experiments in the real world are costly and time-  consuming while posing potential risks of wear and accidents  Manuscript received August 16, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Revised October 31, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Accepted  January 11, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='This paper was recommended for publication by Editor A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Bera upon evaluation of the Associate Editor and Reviewers’ comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='This  work was jointly supported by Major Project of the New Generation of Artifi-  cial Intelligence, China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2018AAA0102900), the National Natural Science  Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 62173197) and the EPSRC project ”ViTac:  Visual-Tactile Synergy for Handling Flexible Materials” (EP/T033517/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Chen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Zhang contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' *Corresponding authors:  Shan Luo and Bin Fang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  1Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Chen  and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Luo  are  with  the  Department  of  Engineering,  King’s College London, London WC2R 2LS, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' E-mail:  czx980730@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' shan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='luo@kcl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Fang and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Sun are with the Institute for Artificial In-  telligence, Department of Computer Science and Technology, Beijing Na-  tional Research Center for Information Science and Technology, Tsinghua  University, Beijing 100084, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' E-mail: czx980730@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  fangbin@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' fcsun@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='cn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  3S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Zhang is with the School of Engineering and Technology, China  University  of  Geosciences  (Beijing),  Beijing  100083,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  E-mail:  zhangshixin@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='cugb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Digital Object Identifier (DOI): see top of this page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  (B)  (C)  (D)  (E)  (A)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (A): The GelSight sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (B): GelSight diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' This diagram shows  that the elastomer layer deforms during interaction with an indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (C), (D),  and (E): Optical tactile images collected from the real sensor [8], pseudo  tactile image generated from [1] and Tacchi proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' As shown  in the figures, Tacchi offers more realistic tactile images in simulation with  surface defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  to the robots, but simulation can provide data as preliminary  experiments without such risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Besides, data-driven methods  like neural networks and deep learning are utilized in robotics  for sensing [1] and control [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' An extensive amount of data  is required, and simulation provides an efficient way to collect  data with the potential for transferring trained robot agents to  the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In the past years, various simulation approaches have been  proposed for object simulation, such as the Finite Element  Method (FEM) [3] and key points [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' There have been some  commonly used simulators for robotics, such as Gazebo [5],  Pybullet [6] and MuJoCo [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Although they are able to  simulate robot components like robot arms and grippers, tactile  sensor simulation is still a challenging task, which hinders  physical robot simulation with tactile sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Tactile sensing is indispensable for robot contact control,  and many tactile sensors have been developed in recent  years [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Thanks to the ability to produce high-resolution  tactile images, optical tactile sensors [8], [11]–[13] that use  cameras to capture deformation of soft elastomer have been  favoured for robotics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' An example is GelSight, as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 1-(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' A GelSight sensor is composed of a  camera, a set of LEDs, and a soft elastomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' During interaction  with objects, the deformed opaque surface of the elastomer  reflects light from the LEDs to the camera, and the tactile  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='08343v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='RO]  19 Jan 2023  Oindenter  sliver reflecting layer  light guiding plate  gel layer  LED  supporting structure  transparent planeindenter  reflecting layer  elastomer  transparent  plane  light guiding  supporting  plate  structure  (B)  LED  camera2  IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' ACCEPTED JANUARY, 2023  images captured by the camera can offer contact information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Due to their low cost and high resolution, optical tactile  sensors have been widely used in real experiments  [14]–  [16], but challenges in simulating deformations of their soft  elastomer layers impede having them in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  The main challenge of optical tactile stimulation is elas-  tomer deformation reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Deformable object sim-  ulation is a complex problem caused by high degrees of  freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Elastomer deformation is first approximated in [17]  using pseudo depth maps applied with a smoothing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  However, in this method the smoothing parameters are fine-  tuned only based on the optical comparison, ignoring the  elastomer elasticity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Hence, this method can only  be applied when a normal force is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Meanwhile, in  some other works [3], simulation software like Abaqus is used  to provide detailed elastomer deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However, they  suffer from a high computational cost for FEM simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Considering the high speed of the other robot simulation  processes, this time-consuming approach is unsuitable for a  single sensor simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  To address these issues, we propose a pluggable and low  computational cost optical tactile sensor simulator Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  This simulator is named after tactile sensor simulation and  the applied parallel programming language, Taichi [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In  Tacchi, particles are utilized to represent elastomers of the  optical tactile sensors, and their motions represent elastomer  deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Surface depth maps are generated from the  particles on the elastomer surface, which are then rendered  into tactile images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Experimental results show that Tacchi  can be connected with robotics simulators like Gazebo and  MuJoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Our method produces higher quality tactile images  than the existing methods [1], [19] in terms of both tactile  image quality and Sim2Real transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In addition,  Tacchi can add random defects to simulate textures on real  objects and improve the Sim2Real performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Examples of  the real and simulated tactile images from different approaches  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 1-(C), (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  The contributions of this paper are summarized as follows:  1) We develop a novel simulator Tacchi that is pluggable  and requires low computation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' This is the first  optical tactile sensor simulator that utilizes physical  deformation simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The tactile images from Tacchi  achieve satisfactory similarity to the real tactile images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  2) We connect Tacchi and the commonly used simulators,  Gazebo and MuJoCo, for robot simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The exper-  imental results show that this simulation can achieve  robot simulation and include tactile sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  3) We investigate the effect of particle number on the  simulated tactile images with Tacchi, and the results  show that our approach achieves superior performance  in Sim2Real learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  The rest of the paper is structured as follows: Section II  includes works related to elastomer simulation and sensor  simulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Section III introduces our proposed Tacchi sim-  ulator and its connection with Gazebo and MuJoCo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Section  IV describes the experimental setup, including the real and  simulation setups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Section V shows the experimental results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Section VI summarizes the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Elastomer simulation  It is challenging to simulate elastomers like gel layers in  optical tactile sensors due to their deformation properties and  high degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' FEM has been widely applied to  simulate deformation in [3], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Nodes and facets are applied  in [20] to represent a rubber sphere and a foam cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Based  on such a model, a strategy to grasp an elastomer with a multi-  finger robot hand is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In optical tactile sensors, various methods are applied for  elastomer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Physics-based methods, which simulate  the deformation with some physical principles, have been uti-  lized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' For example, FEM is applied in [3] for force distribution  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Although this work utilizes physics-based simu-  lation, it requires massive computational resources and the  simulation is achieved by using a high performance computing  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In FEM, each element is connected with the other ones,  and deformation simulation may also lead to mesh distortion  problems like irregular meshes or even negative element  volume [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Meanwhile, some non-physics-based methods  try to avoid physical interaction simulation with smoothing  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Pseudo elastomer surfaces are reconstructed in [1] by  smoothing depth maps with Gaussian kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Similarly, [19]  uses object depths for the contacted area and kernel smoothing  method for the uncontacted area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' PyBullet [6] is utilized in  [22] as the object and sensor simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Smooth depth maps  are generated by either pre-processing the object meshes or  applying data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Although these methods simulate  tactile images, they do not simulate physical deformation and  only produce depth maps by smoothing, which do not have  physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  To address these issues, we employ the Material Point  Method (MPM) for physics-based elastomer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' This  method applies particles to represent objects and an imaginary  grid for contact simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The grid exchanges object infor-  mation with particles during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Compared with  other physics-based methods like FEM, MPM does not suffer  from mesh distortion thanks to independent particles [23] and  has a low computational cost thanks to the use of Taichi [18], a  parallel programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Compared with non-physics-  based methods, MPM simulates physics-based interactions and  provides realistic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Sensor simulation  Various sensors have been applied in robot systems, and  sensor simulation plays an essential role before deploying  them into the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Common sensors like cameras and  force sensors have been integrated into mainstream robot  simulators such as Gazebo [5] and MuJoCo [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In [1], optical tactile sensors were first simulated by smooth-  ing depth maps collected from Gazebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In [22], the meshes of  both the object and the sensor are first loaded in OpenGL [24]  and then updated during the simulation in PyBullet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In this  work, depth maps can either be obtained from PyBullet or  generated in OpenGL together with tactile images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However,  these depth maps from widely applied robot simulators do not  include physics-based deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  CHEN et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' : TACCHI: A PLUGGABLE AND LOW COMPUTATIONAL COST ELASTOMER DEFORMATION SIMULATOR FOR OPTICAL TACTILE SENSORS  3  initialization  particle-to-grid  grid-to-particle  particle advection  indenter particle  elastomer particle  grid  elastomer simulation  (A)  surface depth map  cropped depth map  tactile image  (D)  (C)  (B)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Tactile image generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (A): Elastomer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Grey particles represent the elastomer, and green particles represent the indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The  particles and grid nodes are first initialized;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' the particle information is transferred to the grid in particle-to-grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' then the object information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', the indenter’s  motion in this case) is allocated back to nearby particles by the grid nodes in grid-to-particle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' finally, particles are moved according to the object information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  (B): With the discrete surface particles, a continuous surface depth map can be reconstructed via a 2D interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (C): The depth map is cropped  into 480 × 640 to fit the image size from the Gelsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The cropped image is shown by the red frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (D): A tactile image is generated based on the cropped  depth map and the rendering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Engineering simulation software is also utilized for tactile  sensor simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' COMSOL Multiphysics is exploited in [25]  to predict magnetic fields under different conductive target  situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In [3], Abaqus is used to simulate elastomer de-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Although these software programs show excellent  simulation results, most of them are packaged, and it is  challenging to connect them with robot simulators due to  incompatibility and time-consuming FEM algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In this work, we develop a novel simulator Tacchi to sim-  ulate elastomer deformation by utilizing a high-performance  numerical computation language Taichi [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Thanks to its  parallel programming and concise data structure, Tacchi can  be run with a low requirement of computational resources, and  it is simple to connect to simulators like MuJoCo and Gazebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' METHOD  In this section, the proposed simulator Tacchi is introduced  and its connection with the robot simulator MuJoCo and  Gazebo is also detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Specifically, the elastomer simula-  tion is first introduced in Subsection III-A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' a tactile image  rendering method based on elastomer simulation in Tacchi is  then proposed in Subsection III-B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' the integration of Tacchi  with a robot simulator is detailed in Subsection III-C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' and  the potential applications of Tacchi in the simulation of other  sensors and robot tasks are introduced in Subsection III-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Elastomer simulation  In Tacchi, we apply the Material Point Method (MPM)  for elastomer simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' MPM applies particles to represent  objects and an imaginary grid for contact simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The grid  exchanges information with particles during the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  FEM and MPM use meshes and particles as their elements,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' For both approaches, each element interacts with  the nearby elements based on their geometry shapes and  relative positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' For FEM, there will be mesh distortions  because deformation may lead to problems like irregular  meshes or even negative element volume, which results in  low accuracy of stress [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In contrast, for MPM, particles  exchange information with an imaginary grid, and this method  does not suffer from mesh distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  We first introduce the elastomer simulation in optical tactile  sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' To describe this method, we simulate an elastomer  pressed by a rigid indenter as an example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-  (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In Tacchi, both the elastomer (grey dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(A)) and  the object that interacts with the elastomer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', an indenter  (green dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(A)) in this example, are represented  using particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The object information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', the position and  the velocity, are stored in these particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In the simulation, it is necessary to decide which particles  are included in the interaction between the elastomer and the  object and how they interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' To this end, a fixed imaginary  grid is introduced, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(A), so as to have the  particles exchange object information via the grid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In each simulation step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', the motions of the particles  are updated for one time), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(A), the object  information is first collected by the grid nodes from nearby  particles (we name it as particle-to-grid);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' the interaction  between the particles of the elastomer and the indenter, and  also the object elastic deformation, are then calculated by  the grid nodes and shared with each particle (we name it as  grid-to-particle);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' the states of the particles will be updated  accordingly afterwards (we name it as particle advection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Initialization: Firstly, we initialize particles and grid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  A set of m particles are used to represent the elastomer and  the indenter, and a grid of n grid nodes is used to record the  object information with a fixed coordinate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' For the p-  th particle, its position xp ∈ R3 and its velocity vp ∈ R3  are considered for the simulation of the object motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  For deformation simulation, similar to [26], we introduce a  deformation map φp : R3 → R3 that records the initial  and final position in a simulation step for each particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The  deformation gradient Fp ∈ R3×3 can be derived as  Fp = ∂φp  ∂xp  (xp),  (1)  where Fp is set as a three-dimensional identity matrix I3×3 in  the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In addition, an affine velocity matrix Cp ∈ R3×3 that  records the velocity of the neighboring particles [26] is intro-  duced to reduce information loss during information exchange  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0008  600  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='0010  400  100  OOE  200  300  200  400  100  500  6004  IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' ACCEPTED JANUARY, 2023  between the particles and the grid nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' It is initialized  as 03×3 since the particles are static and will be updated  afterward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Particle-to-grid: In each simulation step, particle information  is first transferred to the grid by calculating momentum and  mass in each grid node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' It can be seen as simulating part  of the object surrounding the grid node by quadratic B-Spline  weighting interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Specifically, the mass and momentum  of the nearby particles are collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The mass Mi of the i-th  grid node is  Mi =  �  j∈Gi  �  p∈Pj  wjpmp,  (2)  where mp denotes the mass of the p-th particle, Gi denotes  3 × 3 × 3 grid nodes which contain the i-th grid node and  its neighboring grid nodes, Pj denotes the particles inside  the j-th grid, and wij is the weight parameter of the i-th  grid node and the j-th particle for weighting interpolation,  which is calculated by quadratic B-Spline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' By applying such  weight parameters, close particles contribute more to Mi than  far ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' We follow [27] to have the same weight parameters  for mass and the following momentum calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  The grid momentum MGi of the i-th grid node can be  obtained by calculating the momentum resulted from the  particle motion MM i and the momentum resulted from the  elasticity MEi:  MGi = MM i + MEi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  (3)  Here MMi is calculated by collecting the velocity and affine  velocity of the nearby particles:  MMi =  �  j∈Gi  �  p∈Pj  wjp(mpvp + Cp(Xj − xp)),  (4)  where Xj denotes the position of the j-th neighboring node  of the i-th grid node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  MEi can be obtained as in [26] by:  MEi = −△t  �  j∈Gi  �  p∈Pj  4  △X2 wjpV 0  p Sp(Xj − xp),  (5)  where △t is the time interval between two adjacent steps,  △X is the grid node interval, V 0  p denotes the initial particle  volume, Sp is the elasticity force for the p-th particle [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  After obtaining Mi and MGi using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 3  respectively, the object velocity close to the i-th grid node  Vi can be computed as:  Vi = MGi  Mi  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  (6)  Grid-to-particles: The states of the particles are then updated  with the previous states of the grid nodes and the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  It can be seen as simulating part of the object surrounding  the particle by quadratic B-Spline weighting interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Suppose that the states of the k-th step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', the velocity v(k)  p ,  the affine velocity C(k)  p , and the deformation gradient F (k)  p  ,  are known, v(k+1)  p  , C(k+1)  p  and F (k+1)  p  in the k + 1 step can  be obtained as  v(k+1)  p  =  �  i∈G′p  wipV (k)  i  ,  (7)  C(k+1)  p  =  4  △X2  �  i∈G′p  wipv(k+1)  p  (Xi − x(k)  p ),  (8)  F (k+1)  p  = (I + △tC(k+1)  p  )F (k)  p  ,  (9)  where G′p represents the 3×3×3 grid nodes surrounding the  p-th particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Particle advection: With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 8 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 9, the states of  particles can be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However, there are two special cases:  the velocities of the particles on the bottom layers of the  elastomer are set as 0 since they are fixed onto the sensor  (and the sensor is static in this example);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' as the indenter is  rigid, the particles representing the indenter share the same  velocity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', the velocity of the indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' As a result, v(k+1)  p  for these special particles are obtained as:  v(k+1)  p  =  �  vi,  p ∈ I,  0,  p ∈ B,  (10)  where vi denotes the velocity of the indenter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' I denotes  particles representing the indenter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' B denotes the particles on  the bottom layers of the elastomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Finally, the particles move with the above computed veloc-  ities and the position of each particle x(k+1)  p  in the k + 1 step  is  x(k+1)  p  = x(k)  p  + △tv(k+1)  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  (11)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Tactile image rendering  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(B), (C), and (D), after particle motion  surface particles and the Phong’s model [28] are used to  generate the tactile images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Although there may be various  methods to produce tactile images from the particles, we fol-  low [1]’s method by generating depth maps first and rendering  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' A 2D interpolation method is applied to provide a  continuous depth map based on discrete particles representing  the elastomer surface, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' It should be  highlighted that the obtained depth map here is from the  physics-based elastomer deformation, instead of the depth map  of the indenter like [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In this way, the resulted tactile image  will be more realistic as in the real sensor the camera also  captures the elastomer deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(B), the depth map of the whole square  elastomer surface is provided in Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However, due to its  limited field of view, the camera in the Gelsight [8] sensor  generates images with a resolution of 480×640, in which only  a partial area of the sensor is captured instead of the whole  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' To this end, only this partial area of the depth map  will be rendered into tactile images (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In addition,  there are misalignments between the desired and real setups,  due to the calibration errors of the camera and the indentor’s  movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In [1], some cropping and alignment approaches  are proposed to match real and the simulated tactile images so  as to compensate the misalignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' We follow the Alignment  (per object) approach [1], with the cropped depth map shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  CHEN et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' : TACCHI: A PLUGGABLE AND LOW COMPUTATIONAL COST ELASTOMER DEFORMATION SIMULATOR FOR OPTICAL TACTILE SENSORS  5  main simulator  Tacchi  Initialization  Step Simulation  Terminal  Step  load object and sensor  initialize particles  and grid  move objects  move particles, render  main simulator  terminal condition  Tacchi  terminal condition  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Connection with a robot simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The misalignment denotes the time sequence in each simulation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In initialization, the robot simulator loads  models as the main simulator, and Tacchi initializes particles and a grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' During the simulation, the main simulator moves the objects based on the desired  policy, and Tacchi obtains the velocity of the objects from the main simulator and moves the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Tactile images are rendered from the depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Terminal conditions are proposed in the main simulator and Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  We follow [1] to simulate illumination of the GelSight  sensor using the Phong’s model [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' With the obtained depth  map D and light resources L, the tactile image I can be  obtained, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 2-(D):  I = kaia +  �  m∈L  (kd(�Lm · �  N)im,d + ks( �Rm · �V )αim,s), (12)  �Rm = 2(�Lm · �  N) �  N − �Lm,  (13)  �  N =< ∂H  ∂x , ∂H  ∂y , −1 >  =< H  2r ∗ [−1, 0, 1], H  2r ∗ [−1, 0, 1]T , −1 >,  (14)  where ka, kd, ks, and α are surface reflectance property  parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' r is the pixel-to-meter ratio;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' ia is the background  light that is not from the light resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' im,d and im,s are  the intensities of the diffuse and specular reflections of light  source m respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' �Lm and �Rm are emission and reflection  direction of the light resource m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' �V is the direction from the  surface point to camera;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' �  N is the elastomer surface normal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  ∗ stands for 2D convolution operation in this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' These  parameters for simulating the GelSight are detailed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Connection with a robot simulator  This subsection introduces the connection of Tacci with  a robot simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 3, Tacchi generates  tactile images, and the robot simulator is applied as the main  simulator to simulate the motions of the objects and the robot,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' MuJoCo and Gazebo are used as the main simulator in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However, any robot simulator that can provide the  real-time position or velocity of the sensor and the contacted  objects can also be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' First, the object and sensor models,  usually mesh files, are loaded in the main simulator, and they  are placed in the initial positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' As Tacchi requires particle  information, point clouds from the object and sensor models  are used to initialize the particles of the object and the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In every simulation step, the object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=', the indenter in this  example, moves according to the control policy in the main  simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The object position or velocity is acquired by Tacchi  from the main simulator to simulate the object’s motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Then  depth maps are generated from the elastomer deformation and  are rendered into tactile images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' During the simulation, if a  terminal condition is met, the simulation will terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' For  example, if the indenter reaches the desired depth in the main  simulator or Tacchi, the simulation will end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Application of Tacchi to other sensors and tasks  This subsection describes how to apply Tacchi to other tasks  like robot grasping, and other optical tactile sensors [12], [13],  [29] or soft inductive tactile sensors [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' As shown in the  above sections, only the elastomer in the tactile sensor and the  objects which may contact the sensor are involved in Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  To this end, there are only a few points to consider when using  Tacchi to simulate other sensors or in other tasks:  In initialization, as the particles are desired to be inside  the grid during the simulation, the grid needs to be a little  larger than the particle range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  The grid node density should strike a balance between  simulation quality and speed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  As the particle density will affect object surface rough-  ness and computation cost, it needs to be determined  according to the simulation demand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  For a new object or elastomer, its point clouds can be  used, and the particles can be aligned according to the  point positions in the point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Elasticity parameters  like Young’s modulus and Poisson’s ratio can be initially  decided by measuring the applied material or using their  standard parameters, and then fine-tuned based on the  sensor outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' EXPERIMENTAL SETUP  To evaluate Tacchi, we collect tactile images in the real  world and simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Subsection IV-A introduces real exper-  imental devices and data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Subsection IV-B includes  the simulation setup like Tacchi parameters, indentor loading,  and main simulator setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  6  IEEE ROBOTICS AND AUTOMATION LETTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' ACCEPTED JANUARY, 2023  (A)  (B)  Gelsight  elastomer  indenter  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (A): Real experiment diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The GelSight sensor presses against  the indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (B): Twenty one indenters included in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The first  row: cone, cross_lines, curved_surface, cylinder, cylinder_shell,  cylinder_side, dot_in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The second row: dots, flat_slab, hexagon, line,  moon, pacman, parallel_lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The third row: prism, random, sphere,  sphere2, torus, triangle, wave1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  (A)  (B)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (A): Tacchi connected with Gazebo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The sensor moves in Gazebo, and  tactile images are generated by Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (B): Tacchi connected with MuJoCo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  The indenter is pressed against the elastomer, and tactile images are generated  by Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Real World setup  We adopt the dataset from [1] in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In this case,  GelSight and the objects in [1] are included in the real world  setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In the experiment, a Gelsight is pressed against an  indenter, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 4-(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The dataset contains 21  objects with various shapes as indenters, which are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 4-(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 3 × 3 positions are chosen for press experiments,  and the horizontal step is 1mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The indention depth improves  from 0 to 1mm, and the vertical step is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='1mm, as a result  11 images are collected at each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Considering 21  indenters, this experiment results in 3 × 3 × 11 × 21 = 2, 079  tactile images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' More details can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Virtual World setup  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 5-(A) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 5-(B) show the Tacchi simulator con-  nected with Gazebo and MuJoCo, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In the Tacchi  interface, an elastomer layer (grey dots) and an indenter (green  dots) are included in the virtual world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In our work, the Gel-  Sight elastomer is composed of 101×101×21 particles since  the shape of the elastomer layer in the real GelSight sensor  is 20mm × 20mm × 4mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Young’s modulus and Poisson’s  ratio are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='45×105 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='45, and these parameters are decided  according to [8] and some preliminary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The grid  contains 256 × 256 × 256 grid nodes, and the grid edge is  33mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' More grid nodes lead to better simulation quality but  slow down the simulation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In general, all particles  should be inside the grid during the simulation, and the grid  interval should be larger than the particle interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Based on  such principles and after some preliminary experiments on  different grid node densities, we choose such grid parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Although we only apply GelSight as the optical tactile sensor  in this work, Tacchi can be utilized to simulate other tactile  sensors by initializing particles with corresponding shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  To create various indenters in Tacchi, the point positions  in the 3D point clouds are used to initialize the indenter  particles in Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' If only 3D mesh files are available, some  software programs and libraries like Open3D and Point  Cloud Library can be utilized to extract point clouds from  mesh files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Though all the points in the point clouds can  be included, we change the particle density to accommodate  the computational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Also, the intervals among  particles can simulate defects on the object surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' This  feature will be exploited in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Gazebo and MuJoCo simulation interface are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  5-(A) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 5-(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In the main simulator, we simulate the  sensor and indenter by loading mesh files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Gazebo provides  the indenter position, and Tacchi generates the corresponding  tactile image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' There is an interface in MuJoCo that can provide  the indenter velocity, and the indenter in Tacchi shares the  same velocity with that in MuJoCo by obtaining velocity  through this interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The indenter velocity control in Tacchi  is achieved by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Although we only include Gazebo and  MuJoCo in this work, any robot simulator which can provide  the real-time indenter position or velocity has the potential to  be the main simulator with Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' EXPERIMENTAL RESULTS  This section shows the experimental results of applying Tac-  chi as the optical tactile sensor simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' We first introduce the  computation cost in Subsection V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' To evaluate our approach,  Subsection V-B compares the pseudo tactile images from [1],  [19] and Tacchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' We also show images with different particle  numbers to explore the effect of this parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Sim2Real  learning is applied for evaluation in Subsection V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Computational Cost  High computational cost is the main problem of physics-  based simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' For example, the FEM simulation is  achieved with high computational costs using a high perfor-  mance computing cluster in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However, the parallel pro-  gramming language Taichi [18] highly decreases the demand  for computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' To simulate the sensor interaction  with an intended in this work, over 106 particles and 107 grid  nodes are simulated, but it only requires 1G memory in a  GTX 1050Ti graphics card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Compared with FEM in [3], Tacchi  requires much lower computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Pseudo image quality  In this subsection, we evaluate the simulation methods in  [1], [19] and Tacchi with three metrics: Structural Similarity  (SSIM), Peak Signal-to-Noise Ratio (PSNR), and Mean Abso-  lute Error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' These metrics are widely applied to evaluate  image similarity, and the similarity between real and generated  images can be taken as pseudo image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Depth maps  are generated based on the methods in [1], [19] and Tacchi,  and the rendering method in [1] is applied for deformation  Gazebo  tactile_Img  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='00  simTime:00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='00:03:05134tReaf Time:0000:03:06:748:iteratio0s:105348  FPS:60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='66  Reset TimeTacchl (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='57Fps)mujoco_py  Tacchl (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='71:FPS)  tactile_lmage_MuJoCoCHEN et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' : TACCHI: A PLUGGABLE AND LOW COMPUTATIONAL COST ELASTOMER DEFORMATION SIMULATOR FOR OPTICAL TACTILE SENSORS  7  Real  [1]’s method  Tacchi _10  [19]’s method  (A)  (B)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (A): Tactile images from real dataset in [1] (top column), Tacchi 1 (2nd column), Tacchi 10 (3rd column), Tacchi 100 (4th column), [1]’s method (5th  column), [19]’s method (6th column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' These rows show the tactile images of cross_lines, curved_surfaces, dot_in, pacman, and triangle respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Tacchi 100, [19] and [1] show similar images but have slight differences caused by surface defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Tacchi 10 provides random defects on the object surface,  and Tacchi 1 generates too many defects that harm object shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' (B): Enlarged areas of dot_in tactile image from real dataset, Tacchi 10, [1], and [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  simulation comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' For Tacchi, we build the indenter  models with 1 × 104, 10 × 104, and 100 × 104 particles and  name them Tacchi 1, Tacchi 10, and Tacchi 100, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  We apply such particle numbers since preliminary experiments  show that they provide indenters with excessive, appropriate,  and few defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Real tactile images, images from Tacchi 1,  Tacchi 10, Tacchi 100, [1] and [19] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 6-(A),  and some images are enlarged for comparison in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 6-(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Objects contain defects and textures on their surfaces, and the  real images show this feature by rough surfaces in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' 6-  (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The surfaces of images from [1] and [19] are smooth,  while Tacchi 10 produces rough surfaces similar to the real  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Moreover, the indention edges of the other simulation  methods, [1] and [19], are straight and clear, but the real edges  are meandering and blurry, similar to images from Tacchi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  TABLE I  IMAGE QUALITY COMPARISON  SSIM ↑  PSNR ↑  MAE ↓  Tacchi 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='819 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='011  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='43 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='07  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='03%  Tacchi 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='845 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='008  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='57 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='89  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='03%  Tacchi 100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='857 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='006  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='46 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='95  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='03%  [1]’s method  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='852 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='006  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='51 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='26  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='04%  [19]’s method  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='854 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='005  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='50 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='36  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='04%  In Table I, Tacchi 100 obtains the highest SSIM, and  Tacchi 10 has the best performance on the other two metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  This discrepancy can also be seen in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Compared with  Tacchi 100, Tacchi 10 includes part of points in the point  clouds and produces rough surfaces similar to real objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In  the real world, scratches and defects are common on object  surfaces, and this feature can improve the performance of  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' But too many defects can harm the object shapes  and lead to low image quality like Tacchi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Our methods  obtain better performance than [1] and [19] since their depth  maps are based on kernel smoothing and ignore physical  deformation, but real sensors provide depth maps by capturing  the elastomer deformation, similar to our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Sim2Real for object classification  In this section, we study how Tacchi performs in a Sim2Real  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Simulated data is desired to approximate the real data,  and it can be evaluated by comparing the performance of real  and generated data on the same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' In this work, we apply  object recognition as the evaluation task and introduce a neural  network for the recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' To this end, we train the  ResNet-18 [30] network with an early stopping strategy, and  each tactile image is cropped to 440 × 440 and resized to  256 × 256 before being fed into the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' We randomly  choose 20 images for each indenter as the test dataset and  10 images as the validation dataset, considering that each  indenter produces 3 × 3 × 11 = 99 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Images with  the indention depth 0mm are discarded since the indenters do  not press into the elastomer in these images, and the images  cannot provide object information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The training batch is 32,  and we train the network with real images, images generated  by Tacchi 1, Tacchi 10, Tacchi 100, [1] and [19] for at most  2000 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Then we test them on the test dataset of real  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The classification accuracies are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  In Table II, Tacchi 10 achieves the highest accuracy thanks  to the realistic rough surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The tactile images of Tacchi 1  are the most coarse ones and acquire the lowest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Simulation methods based on ideal object models tend to  provide results with few scratches and defects, such as [1],  [19] and Tacchi 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' However, most objects in the real world  contain random defects and scratches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=" Therefore, ideal simu-  Real  Tacchi 1  Tacchi 10  Tacchi_100  [1]'s method  [19]'s method8  IEEE ROBOTICS AND AUTOMATION LETTERS." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' PREPRINT VERSION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' ACCEPTED JANUARY, 2023  TABLE II  SIM2REAL ACCURACY  Test Accuracy  Real2Real  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='23%  Sim2Real (Tacchi 1)  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='26 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='91%  Sim2Real (Tacchi 10)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='87 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='18%  Sim2Real (Tacchi 100)  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='06 ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='18%  Sim2Real ( [1]’s method)  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='17 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='21%  Sim2Real ( [19]’s method)  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='19 ± 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='89%  lated models cannot be the same as real objects, and a proper  number of defects can improve the simulation performance,  as shown in Tacchi 10, but too many defects affect the object  information and harm the simulation like Tacchi 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' CONCLUSION AND DISCUSSION  We introduce Tacchi, a pluggable and low computational  cost elastomer deformation simulator for optical tactile sen-  sors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' It simulates elastomer deformation and produces tactile  images with the depth maps generated from the deformed  elastomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Tacchi can be connected with a robot simulator and  simulates with a low demand of computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' The  experimental results show that Tacchi provides realistic tactile  images and achieves high accuracy in Sim2Real tasks, and  defects on the object surfaces can be simulated by changing  the number of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Overall, Tacchi is an effective optical  tactile sensor simulator that can be applied in robot simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  Only 3D printed objects are applied in this paper, and  future works will explore the relationship between real object  roughness and the particle sampling ratio in the simulation  with different cases of materials and surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Some optical  tactile sensors contain markers on the inner elastomer surfaces  which can be represented by the particles sharing the same po-  sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' By recording surface particle motion, we can simulate  marker motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' We will also consider simulation of the surface  frictions during sliding and rotation motions in future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content='  REFERENCES  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Gomes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Paoletti, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
+page_content=' Luo, “Generation of gelsight tactile  images for sim2real learning,” IEEE Robotics and Automation Letters,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gdE_T4oBgHgl3EQf2xzk/content/2301.08343v1.pdf'}
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+DIVISION AND NEW MULTIPLICATION BETWEEN VECTORS
+A PREPRINT
+J. Enrique H. Ramírez
+Department of Physics
+ABC Federal University
+Sao Paulo, Brazil
+enrique.ramirez@ufabc.edu.br
+E Roca O
+Department of Physics
+University of Oriente
+Santiago de Cuba, Cuba
+eroca@cnt.uo.edu.cu
+January 31, 2023
+ABSTRACT
+The division between two vectors belonging to the same vector space is obtained by elementary
+procedures of vector algebra and is defined by a matrix. This representation is obtained for two and
+three dimensional vector spaces. A new vector multiplication is defined and an the inverse vector.
+Through this multiplication we can obtain the division previously defined.The meaning of vector
+division and multiplication are analyzed.
+Keywords Vector Division · Vector Multiplication · Inverse Vector.
+1
+Introduction
+The division between vectors in the framework of vector
+algebra is a subject not widely treated. Few studies have
+been carried out about this subject and those that have been
+carried out conclude that it is impossible to define this op-
+eration. In 1966, Kenneth O. May wrote an article,[3], in
+which he states that it is impossible to divide two vectors in
+a three-dimensional space. This impossibility is essentially
+due to the ways that have been used to define division be-
+tween vectors, which have not achieved this goal because
+they have tried to obtain it by one of the existing prod-
+ucts between vectors, either by the dot product or by the
+vector product, these products not being proper multipli-
+cations like those defined between numbers. These ideas
+lead to endless solutions. In this work we will show that
+it is possible to define the division between vectors within
+the framework of vector algebra by means of a geometric
+representation.
+As is well known, in the field K, of real numbers, R, or of
+complexes, C, the division operation can be entered as the
+inverse operation of multiplication, i.e., a/b is true c such
+that bc = a; provided that b ̸= 0, c exists and is unique. If
+b = 0, c exists, then and only if a = 0, and in this case c is
+any number in K. Since in the case of vectors there is no
+multiplication operation by which to perform the inverse
+operation, it is common to define the quotient between
+vectors as a particular tensor. Often the quotient between
+two quantities is of a different type and more complicated
+than this, the quotient of two integers is generally not an in-
+teger but a rational number. Similarly, the quotient of two
+vectors is not a vector, as is well known, but a quantity of
+a new type: a tensor [1]; but this means that the division of
+vectors can be exploited only with knowledge of the tensor
+algebra. In [2], division of vectors is treated in the context
+of multivector algebra, which is not treated in the same
+way in basic courses. In this algebra multiplication, and
+the inverse of a multivector, defines the division between
+multivectors which are vectors in a special case. Our main
+goal in this paper is to use elementary procedures to divide
+two vectors, a and b, of a certain vector space V of finite
+dimension with defined inner product. In what follows we
+will assume that b ̸= 0 unless otherwise stated, and we
+denote by |a| = a, etc.
+2
+Vector Division
+Let a and b two vectors in finite n-dimensional space V ,
+over the real numbers , with inner producto defined. We
+can divide vector a into two directions, one parallel and
+the other perpendicular to vector the b as,
+a = a∥ + a⊥,
+(1)
+thus, a∥ = α b and a⊥ = β b⊥(see Fig. 1a ), where α and
+β are two real proportionality coefficients to be determined,
+and b⊥ a perpendicular rotation of the vector b. So we can
+write the above equation as,
+arXiv:2301.12078v1  [math.GM]  28 Jan 2023
+
+A PREPRINT - JANUARY 31, 2023
+a = α b + β b⊥.
+(2)
+Scalar multiplication by b and b⊥ let us to find the α and
+β coefficients,
+α
+=
+a · b
+b2 ,
+β
+=
+a · b⊥
+b2
+.
+We can write b⊥ as b⊥ = R b , were R is a perpendicular
+rotation matrix, so 2 can be writen as,
+a = α I b + β R b,
+(3)
+where I is the unit matrix. Factorization of b in 3 give,
+a = (α I + β R) b,
+(4)
+were we can define,
+Definition 2.1. The division between the vectors a and
+b, which we denote as a/b, where b ̸= 0, is given by the
+matrix E,
+a
+b
+=
+α I + β R.
+(5)
+If V is the vector space R2, then
+E
+=
+�
+α
+−β
+β
+α
+�
+(6)
+=
+a
+b
+�
+cos θ
+− sin θ
+sin θ
+cos θ
+�
+.
+Matrix E rotates vector b and then adjusts its magnitude
+to the magnitude of vector a. In the particular case that
+a = b, then E is the usual rotation.
+In this case the determinant of E (detE) e given by
+detE
+=
+α2 + β2
+(7)
+=
+�a
+b
+�2
+If V is the vector space R3, we have
+E
+=
+�
+α
+Cβ
+−Bβ
+−Cβ
+α
+Aβ
+Bβ
+−Aβ
+α
+�
+(8)
+=
+a
+b
+�
+cos θ
+C sin θ
+−B sin θ
+−C sin θ
+cos θ
+A sin θ
+B sin θ
+−A sin θ
+cos θ
+�
+,
+where A, B and C are the components of the unit vector n
+given by,
+n
+=
+a × b
+∥a × b∥
+(9)
+=
+(A, B, C)
+Here,
+detE
+=
+α
+�
+α2 + β2�
+(10)
+=
+�a
+b
+�3
+cos θ.
+Let’s consider two examples:
+Example 2.1. Let a = (3, -1) and b = (2, 5) be two vectors
+in R2 find a/b.
+α
+=
+a · b
+b2
+=
+1
+29.
+β
+=
+a · b⊥
+b2
+There are two vectors perpendicular to b, b1⊥ = (−5, 2)
+and b2⊥ = (5, −2). The perpendicular rotation matrix R
+will depend on which vector we choose. Let’s choose b1⊥,
+then
+β
+=
+−17
+29
+and
+E
+=
+1
+29I − 17
+29R
+=
+1
+29
+�
+1
+17
+−17
+1
+�
+.
+where R is given by
+R
+=
+�
+0
+−1
+1
+0
+�
+Note that if we choose b2⊥, the corresponding perpendicu-
+lar rotation matrix would be the transpose of the previous
+one, on the other hand we can easily verify that a = E b.
+Example 2.2. Let a = (3, -1, 2) and b = (2, 5, 1) be two
+vectors in R3 find a/b.
+In this case we find that α = 3/30. To find β, let’s take
+the cross product of a and b
+a × b
+=
+−11i + j + 17k
+2
+
+A PREPRINT - JANUARY 31, 2023
+being
+n
+=
+1
+√
+411(−11, 1, 17)
+In this way we find that E is given by
+E
+=
+1
+30
+�
+3
+17
+−1
+−17
+3
+−11
+1
+11
+3
+�
+.
+(11)
+Again we can verify that a = E b.
+2.1
+Division Properties
+D1: a/a = I
+Proof. By Definition 2.1 we have that
+a/a = a · a/a2 I + a · a⊥/a2 R but
+a · a⊥ = 0 and a · a/a2 = 1, then a/a = I.
+D2: 0/a = O
+Proof. By Definition 2.1 we have that
+0/a = 0·a/a2 I+0·a⊥/a2 R then a/a = O.
+D3: a/b = (b/a)−1
+Proof. We are going to prove this property by
+taking the product (a/a)(b/a) and we will show
+that it is equal to the unitary matrix I.
+(a/b)(b/a) = (a · b/b2 I + a · b⊥/b2 R)(a ·
+b/a2 I + a · b⊥/a2 ´R)
+Carrying out the indicated operation and consid-
+ering that R ´R = I and R = − ´R we arrive at the
+expected result.
+D4: (a + b)/c = a/c + b/c
+Proof. By definition,
+(a + b)/c
+=
+((a + b) · c)/c2 I + ((a + b) ·
+c⊥)/c2 R
+Because scalar product follow distributive law we
+have that,
+(a + b)/c = a·c/c2 I + b·c/c2 I +a·c⊥/c2 R+
+b · c⊥/c2 R
+Rearranging the terms we get that
+(a + b)/c = a/c + b/c, as we hope.
+3
+Vector Multiplication
+The deficiency of thedot and cross products is due to the
+fact that through them it is not possible to define the in-
+verse vector. For this reason, we will dedicate this section
+to defining a new multiplication between vectors, through
+which we can obtain the inverse vector, and that said
+multiplication be consistent with the division operation
+between vectors previously defined.
+Definition 3.1 (Vector Multiplication). Let V be a vector
+space with inner product defined and a and b two vectors
+∈ V . The product between the vectors a and b, which we
+denote as a ⊗ b, is given by the matrix E,
+a ⊗ b
+=
+E
+(12)
+=
+α I + β R,
+where I is the identity matrix, R a perpendicular rotation
+matrix and α and β two coefficients given by
+α
+=
+a · b
+β
+=
+a · b⊥ = a⊥ · b.
+The β coefficient is closely related to the perpendicular
+rotation matrix R and to the perpendicular vectors b⊥ or
+a⊥.
+Now we are going to define the inverse vector by consider-
+ing Definition 3.1 .
+Definition 3.2. Let a ̸= 0 a vector in V . The inverse vec-
+tor of a, which we denote as a−1, is , from Definition 3.1 ,
+such that
+a ⊗ a−1
+=
+I.
+(13)
+From Definition 3.2 it follows that a−1 = a/a2, which
+shows that the vectors a and a−1 are collinear. Now let’s
+see the properties of multiplication.
+For two and tree dimensional space we have that
+a ⊗ b
+=
+a b
+�
+cos θ
+− sin θ
+sin θ
+cos θ
+�
+,
+(14)
+and
+a ⊗ b
+=
+�
+α
+Cβ
+−Bβ
+−Cβ
+α
+Aβ
+Bβ
+−Aβ
+α
+�
+(15)
+=
+a b
+�
+cos θ
+C sin θ
+−B sin θ
+−C sin θ
+cos θ
+A sin θ
+B sin θ
+−A sin θ
+cos θ
+�
+3
+
+A PREPRINT - JANUARY 31, 2023
+These matrices rotate the inverse vector a−1 (b−1) towards
+b(a) and then adjust its magnitude converting it into the
+vector b ( a).
+From Eq.(15) we can obtain the well-known vector algebra
+relation,
+(a b)2
+=
+(a · b)2 + (∥a × b∥)2 .
+(16)
+3.1
+Multiplication Properties
+M1: a ⊗ a−1 = I, multiplicative inverse.
+Proof. We will show that there exists a multi-
+plicative inverse of a (which we denote as a−1)
+such that a ⊗ a−1 = I
+Since α = 1 that means that a−1 = 1/a. On
+the other hand because β = 0 the vectors are
+collinear (that is, a−1 = k a), so we have that
+k = 1/a2, in this way we find that a−1 = a/a2.
+M2: a ⊗ b = b ⊗ a, commutative law for multiplica-
+tion.
+Proof. Let a ⊗ b = α I + β R, and b ⊗ a =
+´α I + ´β ´R, with a ̸= 0 and b ̸= 0 and θ the angle
+between the vectors a and b. We must show that
+α = ´α, β = ´β and R = ´R.
+By definition α = a · b and ´α = a · b, so α = ´α.
+On the other hand, β = a · b⊥ and ´β = a · b⊥, so
+β = ´β.
+In the plane of the vectors a and b, there are two
+vectors perpendicular to b (see Fig. 1b ), which
+we will call b⊥1 and b⊥2, where b⊥1 = −b⊥2.
+Since β = ´β, we have that a · b⊥1 = a · b⊥2.
+If we consider that b⊥1 ̸= b⊥2 we got to that
+2 a · b⊥1 = 0, which is a contradiction because
+a ̸= 0 and b⊥1 is not perpendicular to a in the
+general sense. Therefore our assumption that
+b⊥1 ̸= b⊥2 is not correct, then, b⊥1 = b⊥2.
+On the other hand we have that R b−1 = b−1
+⊥1
+and ´R b−1 = b−1
+⊥2 = b−1
+⊥1 then we conclude that
+R = ´R, so we show that a ⊗ b = b ⊗ a.
+M3: a ⊗ u = u ⊗ a = a I, identity element of multi-
+plication.
+Proof. We are going to show that there exists a
+vector u such that a ⊗ u = a I.
+By definition a⊗u = α I +β R, where α = a·u
+and β = a · u⊥ = a⊥ · u, but a · u = a and
+a·u⊥ = 0, so u = 1/cosθ and au tanθ = 0 then
+θ = 0 and u = 1.
+Since u = (a ⊗ u) a−1 we have that
+u = a I a−1 = a a/a2 = a/a.
+That is, we find that u = a/a.
+M4: a ⊗ (b + c) = a ⊗ b + a ⊗ c, distributive with
+respect to vector addition.
+Proof. By definition
+a ⊗ (b + c) = a · (b + c) I + a · (b + c)⊥ R.
+Since the dot product is distributive, we have that
+a⊗(b+c) = a·b I +a·c I +a·b⊥ R+a·c⊥ R
+a⊗(b+c) = (a·b I+a·b⊥ R)+(a·c I+a·c⊥ R).
+Then
+a ⊗ (b + c) = a ⊗ b + a ⊗ c.
+M5: a ⊗ k b = k a ⊗ b, linear with respect to scalar
+multiplication, where k ∈ R.
+Proof. By definition, a⊗k b = α I +β R, where
+α = a · kb = ka · b and β = ka · b⊥ then
+a ⊗ k b = ka · b I + ka · b⊥ R = k (a ⊗ b)
+M6: a ⊗ b = O, then either a = 0 or b = 0.
+Proof. a ⊗ b = O, means that α = 0 and β = 0
+or ab (cosθ − sinθ) = 0 for any θ, then necessar-
+ily a = 0 or b = 0, which means that a = 0 or
+b = 0.
+Let’s see now two examples of multiplication. For this we
+return to the vectors of the examples 2.1 and 2.2 .
+Example 3.1. Let a = (3, -1) and b = (2, 5) be two vectors
+in R2 find a ⊗ b.
+a⊗b = a·b I +a·b⊥ R, where a·b = 1 and a·b⊥ = −17
+(we choose b⊥ = (−5, 2)).
+R =
+�
+0
+−1
+1
+0
+�
+, then
+a ⊗ b =
+�
+1
+17
+−17
+1
+�
+.
+If b = 1/29 (2, 5) (the vector inverse of b) then we can
+see that a ⊗ b match with the division of the example 2.1
+as we hope.
+Example 3.2. Let a = (3, -1, 2) and b = (2, 5, 1) be two
+vectors in R3 find a ⊗ b.
+a ⊗ b = 3 I +
+√
+411 R,
+were α = a · b = 3 and β = a · b⊥ =
+√
+411
+4
+
+A PREPRINT - JANUARY 31, 2023
+R
+=
+1
+√
+411
+�
+0
+17
+−1
+−17
+0
+−11
+1
+11
+0
+�
+,
+then
+a ⊗ b
+=
+�
+3
+17
+−1
+−17
+3
+−11
+1
+11
+3
+�
+.
+If b = 1/30 (2, 5, 1) (the inverse vector of b), then we can
+see that a ⊗ b match with the division of the example 2.2 .
+4
+Conclusions
+It is possible in the framework of vector algebra to de-
+fine the division between vectors in the vector spaces R2
+and R3, which is defined as a matrix. This matrix rep-
+resents rotation together with scaling. It is possible to
+define the corresponding multiplication, which represents,
+in the same way, a rotation. The inverse vector was de-
+fined through this multiplication, through which it was
+possible to obtain the division. Fundamental properties
+of division and multiplication between vectors, analogous
+to the properties of operations with numbers, were given
+along with their proofs. We will continue with the study
+of these operations and their applications in Physics. We
+think that they can be linked to group theories.
+References
+[1] GOLDSTEIN, H. Mecánica clásica. Reverté, 2018.
+[2] HESTENES, D. Vectors, spinors, and complex num-
+bers in classical and quantum physics. American Jour-
+nal of Physics 39, 9 (1971), 1013–1027.
+[3] MAY, K. O. The impossibility of a division algebra
+of vectors in three dimensional space. The American
+Mathematical Monthly 73, 3 (1966), 289–291.
+5
+
+A PREPRINT - JANUARY 31, 2023
+(a) Decomposition of the vector a according to the direction
+parallel and perpendicular to b
+(b) The vector b, its inverse (b−1), and the two possible
+vectors perpendicular to it (b⊥1, b⊥2) are shown. The
+inverse perpendicular vectors are shown.
+Figure 1: The Fig. 1a show the decomposition of the vector a that allows defining the division between the vectors a
+and b and Fig. 1b the inverse and perpendicular vectors of the vector b.
+6
+
+b11
+a
+bli
+b~1
+b
+q
+b 12
\ No newline at end of file
diff --git a/i9FLT4oBgHgl3EQfbS-d/content/tmp_files/load_file.txt b/i9FLT4oBgHgl3EQfbS-d/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/i9FLT4oBgHgl3EQfbS-d/content/tmp_files/load_file.txt
@@ -0,0 +1,169 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf,len=168
+page_content='DIVISION AND NEW MULTIPLICATION BETWEEN VECTORS  A PREPRINT  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Enrique H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Ramírez  Department of Physics  ABC Federal University  Sao Paulo, Brazil  enrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='ramirez@ufabc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='br  E Roca O  Department of Physics  University of Oriente  Santiago de Cuba, Cuba  eroca@cnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='uo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='cu  January 31, 2023  ABSTRACT  The division between two vectors belonging to the same vector space is obtained by elementary  procedures of vector algebra and is defined by a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' This representation is obtained for two and  three dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' A new vector multiplication is defined and an the inverse vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Through this multiplication we can obtain the division previously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='The meaning of vector  division and multiplication are analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Keywords Vector Division · Vector Multiplication · Inverse Vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  1  Introduction  The division between vectors in the framework of vector  algebra is a subject not widely treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Few studies have  been carried out about this subject and those that have been  carried out conclude that it is impossible to define this op-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' In 1966, Kenneth O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' May wrote an article,[3], in  which he states that it is impossible to divide two vectors in  a three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' This impossibility is essentially  due to the ways that have been used to define division be-  tween vectors, which have not achieved this goal because  they have tried to obtain it by one of the existing prod-  ucts between vectors, either by the dot product or by the  vector product, these products not being proper multipli-  cations like those defined between numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' These ideas  lead to endless solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' In this work we will show that  it is possible to define the division between vectors within  the framework of vector algebra by means of a geometric  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  As is well known, in the field K, of real numbers, R, or of  complexes, C, the division operation can be entered as the  inverse operation of multiplication, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=', a/b is true c such  that bc = a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' provided that b ̸= 0, c exists and is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' If  b = 0, c exists, then and only if a = 0, and in this case c is  any number in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Since in the case of vectors there is no  multiplication operation by which to perform the inverse  operation, it is common to define the quotient between  vectors as a particular tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Often the quotient between  two quantities is of a different type and more complicated  than this, the quotient of two integers is generally not an in-  teger but a rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Similarly, the quotient of two  vectors is not a vector, as is well known, but a quantity of  a new type: a tensor [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' but this means that the division of  vectors can be exploited only with knowledge of the tensor  algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' In [2], division of vectors is treated in the context  of multivector algebra, which is not treated in the same  way in basic courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' In this algebra multiplication, and  the inverse of a multivector, defines the division between  multivectors which are vectors in a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Our main  goal in this paper is to use elementary procedures to divide  two vectors, a and b, of a certain vector space V of finite  dimension with defined inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' In what follows we  will assume that b ̸= 0 unless otherwise stated, and we  denote by |a| = a, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  2  Vector Division  Let a and b two vectors in finite n-dimensional space V ,  over the real numbers , with inner producto defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' We  can divide vector a into two directions, one parallel and  the other perpendicular to vector the b as,  a = a∥ + a⊥,  (1)  thus, a∥ = α b and a⊥ = β b⊥(see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' 1a ), where α and  β are two real proportionality coefficients to be determined,  and b⊥ a perpendicular rotation of the vector b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' So we can  write the above equation as,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='12078v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='GM]  28 Jan 2023  A PREPRINT - JANUARY 31, 2023  a = α b + β b⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  (2)  Scalar multiplication by b and b⊥ let us to find the α and  β coefficients,  α  =  a · b  b2 ,  β  =  a · b⊥  b2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  We can write b⊥ as b⊥ = R b , were R is a perpendicular  rotation matrix, so 2 can be writen as,  a = α I b + β R b,  (3)  where I is the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Factorization of b in 3 give,  a = (α I + β R) b,  (4)  were we can define,  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The division between the vectors a and  b, which we denote as a/b, where b ̸= 0, is given by the  matrix E,  a  b  =  α I + β R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  (5)  If V is the vector space R2, then  E  =  �  α  −β  β  α  �  (6)  =  a  b  �  cos θ  − sin θ  sin θ  cos θ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Matrix E rotates vector b and then adjusts its magnitude  to the magnitude of vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' In the particular case that  a = b, then E is the usual rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  In this case the determinant of E (detE) e given by  detE  =  α2 + β2  (7)  =  �a  b  �2  If V is the vector space R3, we have  E  =  �  α  Cβ  −Bβ  −Cβ  α  Aβ  Bβ  −Aβ  α  �  (8)  =  a  b  �  cos θ  C sin θ  −B sin θ  −C sin θ  cos θ  A sin θ  B sin θ  −A sin θ  cos θ  �  ,  where A, B and C are the components of the unit vector n  given by,  n  =  a × b  ∥a × b∥  (9)  =  (A, B, C)  Here,  detE  =  α  �  α2 + β2�  (10)  =  �a  b  �3  cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Let’s consider two examples:  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let a = (3, -1) and b = (2, 5) be two vectors  in R2 find a/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  α  =  a · b  b2  =  1  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  β  =  a · b⊥  b2  There are two vectors perpendicular to b, b1⊥ = (−5, 2)  and b2⊥ = (5, −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The perpendicular rotation matrix R  will depend on which vector we choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let’s choose b1⊥,  then  β  =  −17  29  and  E  =  1  29I − 17  29R  =  1  29  �  1  17  −17  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  where R is given by  R  =  �  0  −1  1  0  �  Note that if we choose b2⊥, the corresponding perpendicu-  lar rotation matrix would be the transpose of the previous  one, on the other hand we can easily verify that a = E b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let a = (3, -1, 2) and b = (2, 5, 1) be two  vectors in R3 find a/b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  In this case we find that α = 3/30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' To find β, let’s take  the cross product of a and b  a × b  =  −11i + j + 17k  2  A PREPRINT - JANUARY 31, 2023  being  n  =  1  √  411(−11, 1, 17)  In this way we find that E is given by  E  =  1  30  �  3  17  −1  −17  3  −11  1  11  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  (11)  Again we can verify that a = E b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1  Division Properties  D1: a/a = I  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1 we have that  a/a = a · a/a2 I + a · a⊥/a2 R but  a · a⊥ = 0 and a · a/a2 = 1, then a/a = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  D2: 0/a = O  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1 we have that  0/a = 0·a/a2 I+0·a⊥/a2 R then a/a = O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  D3: a/b = (b/a)−1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' We are going to prove this property by  taking the product (a/a)(b/a) and we will show  that it is equal to the unitary matrix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  (a/b)(b/a) = (a · b/b2 I + a · b⊥/b2 R)(a ·  b/a2 I + a · b⊥/a2 ´R)  Carrying out the indicated operation and consid-  ering that R ´R = I and R = − ´R we arrive at the  expected result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  D4: (a + b)/c = a/c + b/c  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' By definition,  (a + b)/c  =  ((a + b) · c)/c2 I + ((a + b) ·  c⊥)/c2 R  Because scalar product follow distributive law we  have that,  (a + b)/c = a·c/c2 I + b·c/c2 I +a·c⊥/c2 R+  b · c⊥/c2 R  Rearranging the terms we get that  (a + b)/c = a/c + b/c, as we hope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  3  Vector Multiplication  The deficiency of thedot and cross products is due to the  fact that through them it is not possible to define the in-  verse vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' For this reason, we will dedicate this section  to defining a new multiplication between vectors, through  which we can obtain the inverse vector, and that said  multiplication be consistent with the division operation  between vectors previously defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1 (Vector Multiplication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let V be a vector  space with inner product defined and a and b two vectors  ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The product between the vectors a and b, which we  denote as a ⊗ b, is given by the matrix E,  a ⊗ b  =  E  (12)  =  α I + β R,  where I is the identity matrix, R a perpendicular rotation  matrix and α and β two coefficients given by  α  =  a · b  β  =  a · b⊥ = a⊥ · b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  The β coefficient is closely related to the perpendicular  rotation matrix R and to the perpendicular vectors b⊥ or  a⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Now we are going to define the inverse vector by consider-  ing Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let a ̸= 0 a vector in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The inverse vec-  tor of a, which we denote as a−1, is , from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1 ,  such that  a ⊗ a−1  =  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  (13)  From Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='2 it follows that a−1 = a/a2, which  shows that the vectors a and a−1 are collinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Now let’s  see the properties of multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  For two and tree dimensional space we have that  a ⊗ b  =  a b  �  cos θ  − sin θ  sin θ  cos θ  �  ,  (14)  and  a ⊗ b  =  �  α  Cβ  −Bβ  −Cβ  α  Aβ  Bβ  −Aβ  α  �  (15)  =  a b  �  cos θ  C sin θ  −B sin θ  −C sin θ  cos θ  A sin θ  B sin θ  −A sin θ  cos θ  �  3  A PREPRINT - JANUARY 31, 2023  These matrices rotate the inverse vector a−1 (b−1) towards  b(a) and then adjust its magnitude converting it into the  vector b ( a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' (15) we can obtain the well-known vector algebra  relation,  (a b)2  =  (a · b)2 + (∥a × b∥)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  (16)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1  Multiplication Properties  M1: a ⊗ a−1 = I, multiplicative inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' We will show that there exists a multi-  plicative inverse of a (which we denote as a−1)  such that a ⊗ a−1 = I  Since α = 1 that means that a−1 = 1/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' On  the other hand because β = 0 the vectors are  collinear (that is, a−1 = k a), so we have that  k = 1/a2, in this way we find that a−1 = a/a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  M2: a ⊗ b = b ⊗ a, commutative law for multiplica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let a ⊗ b = α I + β R, and b ⊗ a =  ´α I + ´β ´R, with a ̸= 0 and b ̸= 0 and θ the angle  between the vectors a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' We must show that  α = ´α, β = ´β and R = ´R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  By definition α = a · b and ´α = a · b, so α = ´α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  On the other hand, β = a · b⊥ and ´β = a · b⊥, so  β = ´β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  In the plane of the vectors a and b, there are two  vectors perpendicular to b (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' 1b ), which  we will call b⊥1 and b⊥2, where b⊥1 = −b⊥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Since β = ´β, we have that a · b⊥1 = a · b⊥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  If we consider that b⊥1 ̸= b⊥2 we got to that  2 a · b⊥1 = 0, which is a contradiction because  a ̸= 0 and b⊥1 is not perpendicular to a in the  general sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Therefore our assumption that  b⊥1 ̸= b⊥2 is not correct, then, b⊥1 = b⊥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  On the other hand we have that R b−1 = b−1  ⊥1  and ´R b−1 = b−1  ⊥2 = b−1  ⊥1 then we conclude that  R = ´R, so we show that a ⊗ b = b ⊗ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  M3: a ⊗ u = u ⊗ a = a I, identity element of multi-  plication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' We are going to show that there exists a  vector u such that a ⊗ u = a I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  By definition a⊗u = α I +β R, where α = a·u  and β = a · u⊥ = a⊥ · u, but a · u = a and  a·u⊥ = 0, so u = 1/cosθ and au tanθ = 0 then  θ = 0 and u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Since u = (a ⊗ u) a−1 we have that  u = a I a−1 = a a/a2 = a/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  That is, we find that u = a/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  M4: a ⊗ (b + c) = a ⊗ b + a ⊗ c, distributive with  respect to vector addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' By definition  a ⊗ (b + c) = a · (b + c) I + a · (b + c)⊥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Since the dot product is distributive, we have that  a⊗(b+c) = a·b I +a·c I +a·b⊥ R+a·c⊥ R  a⊗(b+c) = (a·b I+a·b⊥ R)+(a·c I+a·c⊥ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Then  a ⊗ (b + c) = a ⊗ b + a ⊗ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  M5: a ⊗ k b = k a ⊗ b, linear with respect to scalar  multiplication, where k ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' By definition, a⊗k b = α I +β R, where  α = a · kb = ka · b and β = ka · b⊥ then  a ⊗ k b = ka · b I + ka · b⊥ R = k (a ⊗ b)  M6: a ⊗ b = O, then either a = 0 or b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' a ⊗ b = O, means that α = 0 and β = 0  or ab (cosθ − sinθ) = 0 for any θ, then necessar-  ily a = 0 or b = 0, which means that a = 0 or  b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Let’s see now two examples of multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' For this we  return to the vectors of the examples 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let a = (3, -1) and b = (2, 5) be two vectors  in R2 find a ⊗ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  a⊗b = a·b I +a·b⊥ R, where a·b = 1 and a·b⊥ = −17  (we choose b⊥ = (−5, 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  R =  �  0  −1  1  0  �  , then  a ⊗ b =  �  1  17  −17  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  If b = 1/29 (2, 5) (the vector inverse of b) then we can  see that a ⊗ b match with the division of the example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='1  as we hope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Let a = (3, -1, 2) and b = (2, 5, 1) be two  vectors in R3 find a ⊗ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  a ⊗ b = 3 I +  √  411 R,  were α = a · b = 3 and β = a · b⊥ =  √  411  4  A PREPRINT - JANUARY 31, 2023  R  =  1  √  411  �  0  17  −1  −17  0  −11  1  11  0  �  ,  then  a ⊗ b  =  �  3  17  −1  −17  3  −11  1  11  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  If b = 1/30 (2, 5, 1) (the inverse vector of b), then we can  see that a ⊗ b match with the division of the example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  4  Conclusions  It is possible in the framework of vector algebra to de-  fine the division between vectors in the vector spaces R2  and R3, which is defined as a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' This matrix rep-  resents rotation together with scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' It is possible to  define the corresponding multiplication, which represents,  in the same way, a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The inverse vector was de-  fined through this multiplication, through which it was  possible to obtain the division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Fundamental properties  of division and multiplication between vectors, analogous  to the properties of operations with numbers, were given  along with their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' We will continue with the study  of these operations and their applications in Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' We  think that they can be linked to group theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  References  [1] GOLDSTEIN, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Mecánica clásica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Reverté, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  [2] HESTENES, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' Vectors, spinors, and complex num-  bers in classical and quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' American Jour-  nal of Physics 39, 9 (1971), 1013–1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  [3] MAY, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The impossibility of a division algebra  of vectors in three dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The American  Mathematical Monthly 73, 3 (1966), 289–291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  5  A PREPRINT - JANUARY 31, 2023  (a) Decomposition of the vector a according to the direction  parallel and perpendicular to b  (b) The vector b, its inverse (b−1), and the two possible  vectors perpendicular to it (b⊥1, b⊥2) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' The  inverse perpendicular vectors are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  Figure 1: The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' 1a show the decomposition of the vector a that allows defining the division between the vectors a  and b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content=' 1b the inverse and perpendicular vectors of the vector b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
+page_content='  6  b11  a  bli  b~1  b  q  b 12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FLT4oBgHgl3EQfbS-d/content/2301.12078v1.pdf'}
diff --git a/iNE3T4oBgHgl3EQfIwkK/content/tmp_files/2301.04336v1.pdf.txt b/iNE3T4oBgHgl3EQfIwkK/content/tmp_files/2301.04336v1.pdf.txt
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@@ -0,0 +1,1749 @@
+1 
+ 
+ 
+An Overview of Artificial Intelligence-based Soft Upper 
+Limb Exoskeleton for Rehabilitation: A Descriptive 
+Review 
+Sanjukta Halder1, Dr. Amit Kumar2 
+1,2Mechanical department, NIT Patna 
+ 
+Abstract: The upper limb robotic exoskeleton is an electromechanical device which 
+use to recover a patient’s motor dysfunction in the rehabilitation field. It can provide 
+repetitive, comprehensive, focused, positive, and precise training to regain the joints 
+and muscles’ capability. It has been shown that existing robotic exoskeletons are 
+generally used rigid motors and mechanical structures. Soft robotic devices can be a 
+correct substitute for rigid ones. Soft exosuits are flexible, portable, comfortable, 
+user-friendly, low-cost, and travel-friendly. Somehow, they need expertise or 
+therapist to assist those devices. Also, they cannot be adaptable to different patients 
+with non-identical physical parameters and various rehabilitation needs. For that 
+reason, nowadays we need intelligent exoskeletons during rehabilitation which have 
+to learn from patient’s previous data and act according to it with patient’s intention. 
+There also has a big gap between theoretical and practical applications for using 
+those exoskeletons. Most of the intelligent exoskeletons are prototype in manner. To 
+solve this problem, the robotic exoskeleton should be made both criteria as 
+ergonomic and portable. The exoskeletons have to the power of decision-making to 
+avoid the presence of expertise. In this growing field, the present trend is to make 
+the exoskeleton intelligent and make it more reliable to use in clinical practice. 
+ 
+Keywords: Upper Limb Exoskeleton, AI, Human-Machine interaction, AI-based 
+exoskeleton, Rehabilitation therapy,  daily assistance, Soft exoskeleton, Exosuit, 
+intelligent devices. 
+ 
+Introduction: Now-a-days stroke is a very 
+common disorder of blood circulation in 
+human arteries; It obstructs the flow of the 
+vessel. Temporary or permanent disability 
+may occur in adults because of these 
+conditions [1] [3]. Currently, 60% of stroke 
+patients are belongs to India in the world as 
+per India express. Approximately 4000 stroke 
+cases happen daily in India and not more than 
+2-3% are treated, among them 50-80% are in 
+the acute phase and 40-50% are in the chronic 
+phase. After a stroke, rehabilitation must be 
+needed, it can prevent the next stroke which 
+may occur within five years after the first 
+stroke. 
+Traditional 
+treatments 
+such 
+as 
+physiotherapy use simple equipment or 
+monitoring to help patients return to normal 
+life with some uncertainty, even can be 
+limited by wheelchairs [4][6]. Expertise gives 
+an indication that robot-assisted rehabilitation 
+can improve the motor function of a patient's 
+limbs with appropriate treatment [7]. 
+A predetermined way can be traced by a 
+robot, operating different training methods, 
+such as active, passive, assistive, or resistive 
+
+2 
+ 
+ 
+training; for the patient's recovery. The 
+rehabilitation robot can record patient data 
+and estimate exercise capability due to high-
+level 
+control 
+techniques 
+that 
+can 
+be 
+implemented in a range of methods [8]. In 
+Addition, the rehabilitation process can be 
+more fruitful with virtual reality or other 
+techniques such as gaming technology [9]. 
+The rehabilitation robots communicate with 
+patients directly, thus the robots acquire 
+patient data easily and protect it with effective 
+training. Artificial intelligence (AI) presents 
+some algorithms which acquired knowledge 
+from 
+occurrences 
+and 
+forecast 
+exotic 
+conditions, which increases robot-assisted 
+rehabilitation intelligence [1] [10]. Robots 
+can analyze native movements and compute 
+human desires combined with AI. Intelligent 
+systems can acquire knowledge from past 
+happening processes and robotic models can 
+adapt gracefully to dynamic and unknown 
+environments by following these processes 
+[5] [10].  
+Exoskeletons start a new period of the modern 
+neuromuscular rehabilitation process and 
+research in assistive technology [11]. Most of 
+the used actuators of exoskeletons were 
+electrical, and almost all devices were 
+considered fixed and stiff as per previous 
+publications, but the soft exoskeleton has 
+been developed in the last decade and its 
+development is gradually increasing day by 
+day with further research. Although existing 
+review articles discuss robotic structure [11], 
+sensing technology [7], control techniques 
+[8], kinematic evaluation [13], patient’s 
+engagement [9] and human-robot interaction 
+methods [10] etc. By doing these studies, the 
+benefits and drawbacks of every method are 
+also presented.  So, the representative's 
+investigation left some concerns about AI-
+based robot-assisted rehabilitation. 
+The aim of this research is to summarize AI-
+based soft wearable robotic devices and 
+assistance with a focus on upper limb 
+rehabilitation with intelligence. Over the last 
+two decades, robotics technology has been 
+used in rehabilitation and many devices have 
+been proposed depending on this topic. There 
+are many devices that do not have clinical 
+acceptance due to their poor therapeutic 
+utility, many of them were not transportable, 
+airy and convenient. Low-to-middle-income 
+countries cannot adopt exosuits due to high 
+cost whereas stroke happens here more than in 
+high-income countries [12]. In this way, 
+various strategies with AI have to be applied 
+to reduce the cost of production. 
+Literature review:  
+I. 
+Recap of Upper-Limb Exoskeletons: 
+An exoskeleton of the upper limb describes as 
+the covering of the human arm as shown in 
+figure1. Wearable exoskeleton robots have 
+  
+ 
+Figure 1: Mechanical structure of an upper limb exoskeleton ; (a) 
+Schematic design and DOFs, (b) Back View and (c) Front View 
+of the exoskeleton, (d) exoskeleton position when elbow at 90 
+degrees. 
+gradually become popular in clinics because 
+they can support multiple joints and provide 
+flexible and safe rehabilitation training 
+therapy. Still, exoskeletons have further 
+complicated 
+mechanical 
+structures 
+and 
+control 
+techniques 
+and 
+most 
+of 
+the 
+rehabilitation 
+exoskeletons 
+are 
+in 
+the 
+prototype stage [4]. 
+ 
+Upper limb exoskeletons work parallel to the 
+human arm and are attached to the human 
+upper limb at various joints. By this 
+indication, it is clear that robots have to adapt 
+to different arm lengths. Different types of 
+exoskeletons are present as the figure2 based 
+
+b)
+di
+Pineda-Rico etal.3 
+ 
+ 
+on DOFs, Supporting joints, operation modes, 
+control techniques and applications, etc  
+ 
+As per works of literature with figure 3(a), 
+most exoskeletons are rigid in nature. The 
+hard 
+exoskeletons 
+are 
+structured 
+mechanically and offer high resistance to 
+tension or deformation [13]. The full limb will 
+be supported by this type of exoskeleton. It 
+provides some mechanical flexibility. Some 
+of the reviews are introduced about soft 
+exoskeleton systems but in very small 
+amounts. These types of exoskeletons are soft 
+and flexible and made of soft materials. Soft 
+exoskeletons are used to assist the users’ 
+movement without any restrictions [16]. 
+The robotic exoskeletons have many features 
+that highlight their applicability in rehab 
+properly. Most of the exoskeletons are made 
+of rigid materials such as metal or carbon fiber 
+or aluminum alloy. As regards the structural 
+materials, the review of the physical models is 
+made by a combination of different materials 
+[14]. Nowadays cable-driven joints are used to 
+make the exoskeleton lighter, compact and 
+portable. Plastic materials are used to reduce 
+the weight of physical appearances. A very 
+small amount of research used soft or semi-
+hard materials [16]. The distribution of used 
+materials for upper exosuits are laid out in 
+figure 3(d). With the growing technology 
+additive manufacturing or 3D printing 
+technology is also used for manufacturing 
+exoskeletons [2] [15].  
+The characters of the exoskeletons depend 
+upon the type of modes and degrees of 
+freedom (DOFs). Four types of operational 
+modes are observed in motor rehabilitation or 
+neuromotor functioning. They are Active 
+mode, Passive mode, Assistive mode & 
+Resistive mode. The active mode [26][28] 
+gives all possible movements of a limb using 
+a robotic exoskeleton. In passive mode 
+[16][18], limb movements are monitored 
+using the exoskeleton. Assistive mode [34] is 
+partially used to move the limb and resistive 
+mode [29] opposes the movement of the limb. 
+Another operating mode is multi-mode [27] 
+which contributes to mixing modes of 
+operation. Different types of exoskeletons 
+utilize different operational modes for 
+different types of regaining health conditions.  
+ 
+ 
+Figure 3: The general distribution of physical properties of 
+robotics exoskeleton based on (a) exoskeleton types, (b) 
+degrees of freedom(DOFs), (c) operational modes and (d) 
+materials. 
+Figure 2: Types and classifications of the upper-limb exoskeleton 
+ 
+
+Upper-LimbExoskeleton
+Type of joints(upper limb)
+Modes
+DOFs
+Types of
+Control Methods
+Applications
+I.ShoulderExoskeleton
+I.Active
+Materials
+I.Artificial Neural Network
+I.Medical
+I.Single-joint
+Il.ElbowExoskeleton
+Il.Passive
+I. Hard
+Il.AdaptiveAlgorithms
+Il.Military
+exoskeleton
+Ill.WristExoskeleton
+Ill.Resistive
+Il. Soft
+Ill. Fuzzy-logic
+Ill.Commercial
+Il.Multi-joint
+IV.Finger Exoskeleton
+IV.Assistive
+IV.Sliding Modes
+IV.Industry
+exoskeleton
+V.OtherTechniques
+V.AssistiveSoft Exoskeleton
+Multi DOF
+1 DOF
+Hard Exoskeleton
+3DOF
+2DOF
+(a)
+(b)
+3D printing
+Mix /
+Assistive
+Others
+technology
+Others
+Resistive
+Active
+Mix
+Metal
+Passive
+Plastic
+(c)
+(d)4 
+ 
+ 
+The DOF determines the possible movement 
+of a limb over one or different joints. Higher 
+DOFs require more actuators and are more 
+complex in mechanical design and needed 
+much improvement in control techniques to 
+do the specific task. The design is simple to 
+use with one DOF [26] [27] [28] [29] [30], 
+and the driving mechanism allows accurate 
+joint mobility but increases the complexity of 
+the controller. Approximately 45% of 
+research considered single DOF exosuits 
+mostly based on the elbow joint [26] which 
+assists in flexion and extension movements. 
+The internal and external design of two DOFs 
+robots [17] [18] are more complex than 
+single.  Almost 20% of reviewed papers are 
+based on two DOF as shown in figure 3(b). 
+All of those shoulder-elbow models are very 
+common [17]. The robotic structure decides to 
+improve 
+the 
+efficacy 
+of 
+rehabilitation 
+guidance according to the patient’s condition. 
+This study presented 10% of reviews are on 
+the basis of three DOFs models [34] [38] [39]. 
+In specific, the used exoskeleton in paper [42] 
+introduced three DOFs at the elbow joint, arm 
+rotation, and wrist joint. Finally, the most 
+illuminating 
+work 
+was 
+conducted 
+by 
+Varghese et al. [41], where a multi-DoFs 
+shoulder kinematic detection model was 
+published. The work was established based on 
+the physical location of one patient and will 
+be elongated for multiple patients in the 
+future.  The primary aim of exoskeletons is to 
+bear rehabilitation treatment with a compact, 
+airy, and high-performance device.  The 
+tendon-routing framework can be extended to 
+the exosuit's actuator structure and to other 
+multi-DOF joints such as the hip, wrist, and 
+ankle. 
+II. 
+Soft Exoskeleton: 
+ 
+Development of soft exoskeletons has 
+increased over the past decade, particularly in 
+lower limb applications. Soft exoskeletons 
+may replace many or all hard exoskeletons in 
+the future. Light, thin and flexible materials 
+are the substitute for hard, heavy, and rigid 
+materials for soft exoskeletons [2]. The 
+flexible soft exoskeleton is also known as an 
+Exosuits. Some components must be rigid, 
+such as the battery and controller, which are 
+often placed in a bag or separately somewhere 
+to reduce weight [17]. That types of exosuits 
+are great because of their features as they are 
+light, flexible, and easy to transport and 
+install. Patients can use these devices alone 
+and carry them around. Therefore, they are 
+user-friendly and can be used in daily life for 
+improvement easily. Soft exosuits can be used 
+for the treatment of the upper limb, especially 
+for the shoulder [17,18,41], wrist [20,23], 
+elbow [16,21], and finger [19,22] aid. 
+ 
+The shoulder is a vital joint for the upper limb, 
+it is a kind of ball-socket joint. The disabilities 
+of the shoulder can affect the whole upper 
+limb badly. The shoulder joint has three 
+degrees of freedom (DOFs) such as flexion-
+extension in the sagittal plane, abduction-
+adduction in the coronal plane, and medio-
+lateral rotation in the transverse plane. 
+ 
+ 
+Figure 4: Examples of soft exoskeletons for 
+shoulder joints. 
+ 
+Thompson et al [18] represent a cable-driven 
+soft actuation system for shoulder joints. This 
+
+(a)
+(b)
+shoulderanchor
+(a)
+Stage2
+FREEactuator
+Stage1
+(b)
+Stage2
+armanchor
+Stage
+(a)Frontand (b)rear viewof theshoulder exoskeleton prototype
+(a)Nested linearand (b)pennateFREEarchitectures,0,2,and
+O, represent the contraction displacements of the first nested stage, second
+JCeare the GH and elbow joint centers.Cable path is shown in cyan.
+nested stage, and pennate architecture, respectively.
+(A)Thompsonet.al.
+[Above]
+(B)SAMPER-
+snoutder twee cabl
+elcroattachment
+ESCUDEROet.al.
+Elbowoheath
+[Below]
+lbowt5 
+ 
+ 
+model presented a nested linear and pennant 
+architecture 
+by 
+using 
+fiber-reinforced 
+elastomeric 
+perimeter 
+with 
+the 
+same 
+manufacturing factors for increasing shoulder 
+flexion. 
+It 
+estimated 
+forces 
+for 
+low 
+displacements up to 200 kPa pressure, 
+whereas SAMPER-ESCUDERO et al [17] 
+designed a wearable exosuit made of elastic 
+fabric that supports shoulder and elbow 
+stretch. This soft prototype can support the 
+elbow fully and the shoulder partially up to 
+100 degrees. These two types of soft 
+exoskeletons are used for shoulder flexion as 
+shown in figure-4. 
+ 
+The wrist joint is a complicated joint that 
+bridges the gap between the hand and the 
+forearm. It has three degrees of freedom - 
+flexion-extension, radioulnar-deviation, and 
+rotation. In a paper, Benjamin et al [19] spoke 
+for a 3D-printed soft robotic wrist device for 
+rehabilitation with the mobility of two DOF. 
+By using this at least 70% recovery is possible 
+with sufficient torque and bending. They have 
+adopted a folding-based design fabrication 
+process that can be accepted in other 
+applications and Yuejun Xu [23] designed an 
+assistive and portable soft robot and simulated 
+it for wrist dysfunction. This robotic device 
+used soft actuator, elastomeric chambers, and 
+shell reinforcement plates for inducing 
+specific bending under pneumatic pressure.  
+ 
+ 
+ 
+Figure 5: Examples of soft robotic aided devices for wrist (A) 
+and Elbow (B) rehabilitation. 
+ 
+Elbow joint is a kind of hinge joint between 
+the arm and forearm of a human body. It 
+possesses two degrees of freedom as flexion-
+extension and supination- pronation to the 
+forearm. For the aiding of elbow, Chiaradia et 
+al [16] proposed a refined design of the 
+exosuit 
+of 
+which 
+a 
+smooth 
+control 
+architecture 
+represented 
+motion-intention 
+detection with gravity compensation. It has 
+been shown that the device relieves 77% of a 
+user's sustaining moment to move a light 
+weight with a reduction in muscle effort of 
+64.5%. 
+In 
+Oguntosinet 
+et 
+al 
+[21] 
+representation, he revealed a prototype of an 
+inexpensive, airy, and wearable soft robotic-
+aided device that could assist a patient with 
+elbow motion therapies after a stroke occurs. 
+ 
+Finger joint dysfunction is a very crucial 
+problem due to its most complicated 
+anatomical nature. The fingers have 21 
+degrees of freedom (DOFs), including 5DOFs 
+for the thumb and the remaining four fingers 
+with 4 DOFs each for the index, middle, ring, 
+and little fingers. 
+ 
+Figure 6: Examples of soft exosuits for finger rehabilitation. 
+ 
+ 
+Suresh Gobee et al [20] proposed a portable 
+soft robot assistive device for finger 
+rehabilitation. It utilized electromyography 
+(EMG) signals to catch patients’ muscular 
+signals, 
+amplified 
+those 
+signals, 
+and 
+simulated solenoid valves for controlling 
+pneumatic actuators attached to the wearable 
+gloves. Yet Zhang et al [22] demonstrated a 
+conformal mechanism of a multi-material 
+pneumatic soft finger that was optimized to 
+
+@YuejunXu
+@Benjamin W.K.Ang et.al.
+(a)
+(b
+@ Oguntosin et.al.
+@Benjamin et.al.
+Ba
+(c)
+Motorbike Clove
+(d)
+b
+Pneumatic-tubes
+@Gobee et.al.
+20mm
+@Zhanget.al.6 
+ 
+ 
+obtain its maximum bending deflection and 
+also modified for artificial hand gripper, 
+rehabilitation and other practical applications. 
+ 
+Some exosuits are used for only one reason as 
+shoulder flexion or elbow flexion or wrist 
+bend and some exosuits are applied for 
+various reasons as Escudero et al [17] 
+presented a prototype for both shoulder and 
+elbow flexion, Chingyi Nam, et al. [25] 
+elaborated lightweight, ergonomic and low 
+power-demanding soft wearable robot for the 
+blooming system can aid the elbow, wrist, and 
+fingers to fulfil the following arm reaching in 
+a logical order and withdrawing through 
+EMG signals. These and many other reviews 
+show that soft exoskeletons are user-friendly, 
+low-cost, portable, lightweight, and ready to 
+move anywhere. Mainly soft exosuits are 
+made up of fabric or textile or elastomer that 
+bundle up around a user’s limbs and act 
+aligned to the limb [24].  
+ 
+III. 
+Exoskeletons and AI control techniques: 
+AI is a technique to mimic human 
+intelligence. A machine can perform different 
+functions such as planning, learning, and 
+solving problems with artificial intelligence. 
+AI is a wide concept that usually requires 
+humans to carry out tasks performed by a 
+machine. Machine learning algorithm (MLA) 
+is the subset of AI which adopt and learn data 
+iteratively by statistical models and draw 
+useful hypothesis without human feedback.  
+Figure 7: Different techniques of AI used for robotic device 
+control. 
+Finally, Deep Learning is a special category 
+of MLA which uses artificial neural networks 
+to copy like the human brain. The evolution 
+of AI techniques is shown in Figure 7. 
+Advanced data acquisition, processing and 
+control techniques based on AI are much 
+more feasible in modern times that can be 
+more cost-effective than classic methods 
+applied in biomechatronic systems including 
+robotic exoskeletons. [1] [2]. Artificial neural 
+networks (ANN), adaptive algorithms, fuzzy 
+logic, and other mixed techniques are mostly 
+used 
+for 
+exoskeleton 
+purposes. 
+These 
+techniques are commonly used to detect 
+patterns or analyze motion patterns. Figure 8 
+illustrates the use of multiple AI-based data 
+processing and control techniques according 
+to the distribution of literature references. 
+ 
+Figure 8: General distribution of control techniques for 
+soft exoskeletons. 
+ 
+The development and application of 
+artificial intelligence (AI) techniques in this 
+field have become a current trend in late 
+2015 and will change this field with further 
+research in the future. The literature shows 
+that among AI processes that have been used 
+in rehabilitation, 40% focus on artificial 
+neural 
+networks, 
+20% 
+on 
+adaptive 
+algorithms, and 40% on other mixed 
+techniques. Also, it was observed that only 
+16% of the articles concentrated on 
+neuromotor rehabilitation. 
+ 
+
+Mix
+Fuzzy
+Logic/
+Adaptive
+sliding
+Algorithms
+Mode
+Others
+Mix
+Mix
+ArtificialNeuralNetworkMachine Learning
+Artificial Intelligence
+Deep Learning
+Subsetof MLwhichmakethe
+Subsetof Al techniques which use
+Thetechniqueswhichenables
+computationofmulti-layerneuraljstatisticalmethods to enable
+computertomimichuman behaviour
+networksfeasible
+machinestoimprovewith
+forsolvingproblemsandproviding
+*Recurrent NN
+experiences
+data-driven answers.
+*Convolution NN
+*Artificial Neural Network
+*Fuzzy Learning
+*Support Vector Machnine,
+"K-nearest neighbors
+*RandomForest
+*Logic regression
+*Decision Tree
+*Naives Bayes7 
+ 
+ 
+No 
+Ref 
+No 
+Authors’ 
+Name 
+Yea
+r 
+Used 
+AI 
+Technique 
+Scope 
+of 
+movement 
+Results 
+Clinicall
+y Tested 
+1 
+26 
+Tageldee
+n et.al. 
+2016 Fuzzy 
+interfacingc
+ontroller 
+Elbow [1 DOF] 
+Torque 
+Estimation 
+No 
+2 
+27 
+Sangha 
+et.al. 
+2016 Neural 
+Network 
+Wrist [1 dof] 
+Motor Recovery 
+No 
+3 
+28 
+Siddiqi et 
+al 
+2016 SVM 
+with 
+Puk Kernel 
+Thumb [1] 
+Angle 
+Estimation 
+No 
+4 
+29 
+Yu Lei et 
+al. 
+2016 ELM 
+algorithms 
+with SVM 
+Wrist [1] 
+A 
+potential 
+approach to the 
+remote 
+quantitive 
+rehabilitation 
+Yes 
+5 
+30 
+Rethlep 
+et.al. 
+2017 NARX 
+MLPNN 
+Elbow [1] 
+Joint 
+Angle 
+velocity 
+estimation 
+No 
+6 
+34 
+Xin 
+Li 
+et.al. 
+2017 TDRNN 
+Wrist[2] 
+& 
+Hand[1] 
+Angle 
+Estimation 
+Yes 
+7 
+39 
+ Zhang 
+2018  BP Neural 
+Network 
+elbow & Wrist 
+[3] 
+ Angle 
+Estimation 
+No 
+8 
+38 
+Peng Xia 
+et.al 
+2018 CNN neural 
+Network 
+3 DOF  
+Improvement of 
+limb movement 
+No 
+9 
+36 
+Yudha P 
+Pane 
+et.al. 
+2019 Reinforcem
+ent learning 
+based 
+compensati
+on 
+Shoulder[1],Elb
+ow[1], Base[1], 
+Wrist[3] 
+Enhance 
+the 
+control 
+performance 
+No 
+10 
+33 
+Yan 
+Chen 
+et.al. 
+2019 LSTM 
+Neural 
+Network 
+Shoulder 
+Joint 
+Angle 
+estimation 
+No 
+11 
+32 
+Siqi Cai 
+et.al. 
+2019 SVM 
+Machine 
+ML Method 
+5 
+Standard 
+Upper 
+limb 
+Motion 
+Self-
+Rehabilitation 
+(ReRobot) 
+No 
+12 
+41 
+ Varghese 
+et al 
+2019 ANN 
+Shoulder (Multi-
+DOF)  
+Bio-Inspired 
+Kinematic 
+Sensing Suit 
+No 
+13 
+43 
+Zhu et al. 2021 CNN-
+LSTM 
+Multi movement 
+(Upper 
+and 
+LowerLimb both 
+movement)  
+Predicts 
+the 
+human 
+knee/ankle joint 
+trajectory 
+No 
+14 
+31 
+Hussain 
+et al. 
+2022 NARX-NN 
+Elbow (2) 
+Wrist (2) 
+Predicts 
+the 
+torques 
+during 
+different eating 
+activities. 
+No 
+
+8 
+ 
+ 
+Some of published AI-based robotic devices 
+for exoskeletons are closely observed from 
+this growing field and based on those some 
+used techniques such as fuzzy logic, and 
+neural networks, etc are discussed in the table 
+as Tageldeen et.al. [26] in 2016 made a fuzzy 
+logic technique to estimate joint (elbow) 
+torque from relevant sEMG signals for 
+accurate control of exoskeleton. A compact 
+wearable robotic Orthosis is developed by 
+Sangha et.al.[22] for Wrist Assistance with 
+the help of neural network; where Siddiqi et 
+al [ 28] used support vector machine (SVM) 
+technique with puk kernel for thumb angle 
+estimation with an average accuracy of 
+86.5%. Yu Lei et al. [24] proposed a fugl-
+meyer evaluation model using machine 
+learning algorithm and RRelief algorithm 
+implemented to discover the optimal feature 
+subset. 
+ 
+The research is growing in the rehabilitation 
+field with artificial intelligence every day 
+because upper limb disorders harm our daily 
+routine. No doubt eating activity is one of 
+them. To eliminate that type of problem 
+Hussain et al. [31] build a 4 DOFs dynamic 
+model for predicting torque during eating 
+activity process by using Nonlinear Auto-
+Regressive network with exogenous input 
+Neural Network (NARX-NN). 
+We can see many types of AI techniques are 
+used to control the exoskeletons. Among 
+those traditional AI techniques make easy 
+paths for these robot-aided devices. From this 
+table, it can be understood that AI is suitable 
+for quantitative patient feedback by pre-
+processing, monitoring progress, and tracking 
+outcomes in the retrieval process. Some of 
+them are clinically tested, and most of them 
+are in the prototype phase. In context, AI 
+cannot bind patients to a predetermined 
+motion and motivate patients with great 
+accuracy and reliability with their own 
+intention. Most of the AI-based exoskeletons 
+are not clinically tested, they need more 
+observations over it and will make them 
+usable. 
+ 
+IV. Intelligent soft exosuits: 
+An intelligent soft rehabilitation 
+exoskeleton is a combination of bionic 
+structure design and artificial intelligence 
+control algorithm. We can see from this 
+research that intelligent soft-exoskeleton or 
+exosuits is good for rehabilitation because of 
+their lightweight, portable, low-cost, user-
+friendly, and travel-friendly nature with the 
+user’s intention. Artificial intelligence makes 
+the 
+exoskeleton 
+more 
+intelligent 
+and 
+ergonomic. In 2019, Varghese et al. [41] 
+presented a kinematic perception outfit for the 
+complex joint of shoulder joint. This 
+prototype proposed bio-inspired cable-driven 
+architecture which is trained by a mapping 
+based on ANN transfer sensor data to joint 
+angles but it does not friendly with 
+nonlinearity as friction, hysteresis, creep etc.  
+Hence the existing exoskeletons 
+cannot provide rehabilitation training without 
+any therapist due to low intelligence. Since 
+limb movements are uncertain, it is important 
+to recognize and analyze motor intentions and 
+provide proper training accordingly. Artificial 
+intelligence can control training by detecting 
+motion intention by physiological cues and 
+predict patient’s movements including the 
+ability of users. An intelligent controller is 
+able to build their reference trajectory from 
+the patient's state or learn to adjust the control 
+criteria. Qingcong et. al. [44] represent a 
+neuro-fuzzy adaptive controller which is drew 
+on radial basis function neural network 
+(RBFNN) for trajectory tracking. Therefore, 
+intelligent or adaptive controllers are more 
+suitable for rehabilitation training. Some 
+basic technologies are discussed below: 
+ 
+ 
+
+9 
+ 
+ 
+i. 
+Actuation 
+Process 
+and 
+Power 
+Transmission: 
+ 
+According to research status, the most used 
+the following actuation systems are cable-
+driven modules, pneumatic actuators, shape 
+memory alloy-based actuators, and passive 
+actuators. The cable-driven actuator uses 
+cables and sheaths to transfer the power [46], 
+where sheaths are usually installed on the 
+surface of the body and follow the 
+correspondence path from the prime mover to 
+the anchor joint. In a pneumatic actuator, the 
+volume or shape is changed according to the 
+adjusting internal pressure and executes such 
+actions as elongation, bending, or shortening 
+[47]. But most of the pneumatic actuator-
+driven exosuits need external connections for 
+heavy air sources, thatswhy they are not 
+unportable. 
+Shape 
+memory 
+alloy-based 
+actuator is very popular as they are 
+lightweight, produce noiseless actuation [48]. 
+Copaci et.al. [49] represent an exoskeleton 
+using a nonconventional actuation system 
+based on Bowden cable, Teflon sheath, and 
+SMA wire for elbow flexion. Passive 
+actuators mainly consist of spring and elastic 
+fabric, where quasi-passive actuators are 
+using spring, elastic fabric and quasi-passive 
+elements such as variable dampers or 
+electromagnetic clutch [50]. 
+ 
+ii. Control techniques: 
+A powered exosuit must detect the user's 
+motion and provide information for human-
+robot interactions that are controlled by an 
+appropriate control algorithm. The adaptive 
+controller can adjust some control parameters 
+to accommodate dynamic changes in human-
+machine interaction. An adaptive [51] 
+proposed auxiliary controller with inverse 
+dynamic 
+model 
+attempted 
+to 
+provide 
+feedforward support for online learning. 
+Some extra torque can be added to the human 
+joint to reduce the gravity load in a limb by 
+using the gravity compensation control 
+algorithm 
+[52]. 
+Backlash 
+compensation 
+control methods are mostly applied to cable-
+driven exosuits to improve motion control 
+performance. Transmission is delayed by 
+friction between wire and sheath. To solve 
+this problem some researchers established a 
+backlash observing controller and non-linear 
+adaptive controller with hysteresis model 
+[53]. Another most popular control system is 
+bio-signal-based control. This control method 
+is processed with motion intention by the help 
+of physiological signals such as EMG, ECG, 
+EOG etc. A neural network controller is 
+developed to enhance torque estimation by 
+the combination of feedback signals, SEMG, 
+IMU, force sensor and motor encoder, etc. 
+[54]. A Kalman filter is used to identify joint 
+torque, a NN module to perceive the user’s 
+intent and PID controller for hybrid position 
+or torque feedback to achieve motion 
+guidance. 
+Finally, a set of soft wearable robots has been 
+designed for specific functional requirements 
+to support various human joints. In order to 
+improve human-machine compatibility and 
+assistive performance, related researchers 
+have been trying to create different methods 
+by 
+accumulating 
+rich 
+technological 
+achievements in various processes. The 
+resulting 
+prototypes 
+can 
+contribute 
+exceptionally well to many aspects of people's 
+daily lives. 
+Discussion: 
+The role of human upper limbs is very 
+important for performing personal activities 
+in daily life. A patient's daily activities can be 
+greatly affected due to improper functioning 
+of this organ due to neurological disorders or 
+surgery. The design of the exoskeleton still 
+requires many adaptations, such as moving 
+towards flexible, adaptable, wearable, and 
+lightweight structures, as some devices still 
+do not fully meet these criteria. Wire-actuated 
+and soft-structured exoskeletons are also 
+preferred in rehabilitation processes because 
+they are more friendly to everyday life due to 
+their lightness and performance as Thompson 
+
+10 
+ 
+ 
+et al [18] represent a cable-driven soft 
+actuation system for shoulder and elbow 
+joints for increasing muscle strength to 
+working labour. This technology can be 
+brought into the rehabilitation process with 
+some modifications. Benjamin’s 3D-printed 
+soft robotics wrist sleeve is capable of 70% 
+recovery and supports the fabrication process 
+of fold-based designs which may be able to 
+adapt in other applications. In Nam et al. [25] 
+paper said a lightweight, ergonomic and low 
+power-demanding soft wearable robot for 
+assisting elbow, wrist, and fingers by using 
+surface EMG signals. 
+The incorporation of different modes of 
+operation of the exoskeleton has been shown 
+to improve patient’s motor recovery.  In the 
+paper of sangha et. al. [27] represents a multi-
+mode orthosis for wrist rehabilitation. They 
+used passive, active, resistive and active 
+assistive modes for motor recovery with the 
+help of neural networks. Resistance mode 
+helped users reach full motion with current 
+muscle tone or unable to reach full range and 
+active assistive mode was used to assist the 
+user in reducing muscle abnormalities and 
+increasing interaction. The results displayed 
+that the robotic orthosis was capable to done 
+jobs successfully under any functional 
+parameters. Now the continued development 
+of new devices is suggested multi-mode 
+approaches. 
+Future research is trending toward creating 
+exoskeletons that actively learn from input 
+experiences and automatically adapt to the 
+environment or user as artificial intelligence 
+provides the basis for developing more 
+reliable systems. A variety of AI techniques 
+are incorporated into the study to find possible 
+solutions among which artificial neural 
+networks are popular. Since ANN has strong 
+non-linear properties, it shows a large effect 
+on continuous motion estimation results. 
+ANN makes robots movement smoother and 
+safer. As chiaradia [16] represents a smooth 
+control exosuits for elbow which detect 
+motion 
+intention 
+by 
+using 
+gravity 
+compensation.Some 
+studies 
+have 
+also 
+become home base for self-rehabilitation after 
+the Covid pandemic [36]. 
+Almost all enlisted robotics exoskeletons are 
+capable of muscle power strength but are not 
+intelligent enough to function alone. Yet 
+existing intelligent soft exoskeletons are not 
+suitable for clinical use due to some of their 
+shortcomings, with some exceptions. The 
+literature appears that there is a need to 
+develop integration methods for intelligent 
+systems and specialized hardware that cover 
+user needs in a global, comprehensive way.  
+ 
+Conclusion: 
+The soft exoskeleton is currently a boon for 
+everyday life. It protects our organs from 
+damage. A variety of exosuits have been 
+presented, each of them a new concept and 
+attitude to the device, which has been in the 
+form of prototypes, early clinical studies, and 
+sometimes extensive clinical and commercial 
+research to solve movement disorders and 
+rehabilitation problems. These concepts 
+require more extensive clinical trials to 
+become a complete and efficient plan. For 
+this, related researchers are trying to develop 
+different types of algorithms to achieve 
+different soft upper limb exosuits. 
+The 
+present 
+scenario 
+of 
+upper 
+limb 
+rehabilitation 
+systems 
+shows 
+that 
+transportable robotic exoskeletons composed 
+of 
+intelligent 
+control 
+and 
+information 
+processing systems can play an important role 
+in bridging the gap between prototype and 
+clinical 
+acceptance. 
+Additionally, 
+exoskeleton capabilities can be greatly 
+enhanced by improving materials and 
+incorporating a better mechanical design, and 
+improving motion sensing techniques.
+ 
+
+11 
+ 
+ 
+References: 
+1. Manuel Andrés Vélez-Guerrero, Mauro 
+Callejas-Cuervo and Stefano Mazzoleni 
+; 
+“Artificial 
+Intelligence-Based 
+Wearable Robotic Exoskeletons for 
+Upper Limb Rehabilitation: A Review”; 
+MDPI; : 18 March 2021. 
+2. Elena Bardi, Marta Gandolla, Francesco 
+Braghin, Ferruccio Resta, Alessandra L. 
+G. Pedrocchi and Emilia Ambrosini; 
+“Upper limb soft robotic wearable 
+devices: a systematic review”; Journal 
+of 
+NeuroEngineering 
+and 
+Rehabilitation,2022. 
+3. Chen, T.; Keravnou-Papailiou, E.; 
+Antoniou, G. “Medical analytics for 
+healthcare 
+intelligence—Recent 
+advances and future directions”. Artif. 
+Intell. Med. 2021. 
+4. Md 
+Rasedul 
+Islam*, 
+Christopher 
+Spiewak, Mohammad Habibur Rahman 
+and Raouf Fareh; “A Brief Review on 
+Robotic 
+Exoskeletons 
+for 
+Upper 
+Extremity Rehabilitation to Find the 
+Gap between Research Porotype and 
+Commercial Type”; Adv Robot Autom, 
+an open access journal Volume 6 Issue 3 
+1000177 ISSN: 2168-9695;2017. 
+5. Alan Francisco Pérez Vidal, Jesse Yoe 
+Rumbo Morales, Gerardo Ortiz Torres, 
+Felipe de Jesús Sorcia Vázquez, Alan 
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf,len=673
+page_content='1  An Overview of Artificial Intelligence-based Soft Upper  Limb Exoskeleton for Rehabilitation: A Descriptive  Review  Sanjukta Halder1, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Amit Kumar2  1,2Mechanical department, NIT Patna  Abstract: The upper limb robotic exoskeleton is an electromechanical device which  use to recover a patient’s motor dysfunction in the rehabilitation field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It can provide  repetitive, comprehensive, focused, positive, and precise training to regain the joints  and muscles’ capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It has been shown that existing robotic exoskeletons are  generally used rigid motors and mechanical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Soft robotic devices can be a  correct substitute for rigid ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Soft exosuits are flexible, portable, comfortable,  user-friendly, low-cost, and travel-friendly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Somehow, they need expertise or  therapist to assist those devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Also, they cannot be adaptable to different patients  with non-identical physical parameters and various rehabilitation needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' For that  reason, nowadays we need intelligent exoskeletons during rehabilitation which have  to learn from patient’s previous data and act according to it with patient’s intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  There also has a big gap between theoretical and practical applications for using  those exoskeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Most of the intelligent exoskeletons are prototype in manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' To  solve this problem, the robotic exoskeleton should be made both criteria as  ergonomic and portable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The exoskeletons have to the power of decision-making to  avoid the presence of expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In this growing field, the present trend is to make  the exoskeleton intelligent and make it more reliable to use in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Keywords: Upper Limb Exoskeleton, AI, Human-Machine interaction, AI-based  exoskeleton, Rehabilitation therapy,  daily assistance, Soft exoskeleton, Exosuit,  intelligent devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Introduction: Now-a-days stroke is a very  common disorder of blood circulation in  human arteries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It obstructs the flow of the  vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Temporary or permanent disability  may occur in adults because of these  conditions [1] [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Currently, 60% of stroke  patients are belongs to India in the world as  per India express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Approximately 4000 stroke  cases happen daily in India and not more than  2-3% are treated, among them 50-80% are in  the acute phase and 40-50% are in the chronic  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' After a stroke, rehabilitation must be  needed, it can prevent the next stroke which  may occur within five years after the first  stroke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Traditional  treatments  such  as  physiotherapy use simple equipment or  monitoring to help patients return to normal  life with some uncertainty, even can be  limited by wheelchairs [4][6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" Expertise gives  an indication that robot-assisted rehabilitation  can improve the motor function of a patient's  limbs with appropriate treatment [7]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  A predetermined way can be traced by a  robot, operating different training methods,  such as active, passive, assistive, or resistive  2  training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" for the patient's recovery." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The  rehabilitation robot can record patient data  and estimate exercise capability due to high-  level  control  techniques  that  can  be  implemented in a range of methods [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In  Addition, the rehabilitation process can be  more fruitful with virtual reality or other  techniques such as gaming technology [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The rehabilitation robots communicate with  patients directly, thus the robots acquire  patient data easily and protect it with effective  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Artificial intelligence (AI) presents  some algorithms which acquired knowledge  from  occurrences  and  forecast  exotic  conditions, which increases robot-assisted  rehabilitation intelligence [1] [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Robots  can analyze native movements and compute  human desires combined with AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Intelligent  systems can acquire knowledge from past  happening processes and robotic models can  adapt gracefully to dynamic and unknown  environments by following these processes  [5] [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Exoskeletons start a new period of the modern  neuromuscular rehabilitation process and  research in assistive technology [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Most of  the used actuators of exoskeletons were  electrical, and almost all devices were  considered fixed and stiff as per previous  publications, but the soft exoskeleton has  been developed in the last decade and its  development is gradually increasing day by  day with further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Although existing  review articles discuss robotic structure [11],  sensing technology [7], control techniques  [8], kinematic evaluation [13], patient’s  engagement [9] and human-robot interaction  methods [10] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' By doing these studies, the  benefits and drawbacks of every method are  also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" So, the representative's  investigation left some concerns about AI-  based robot-assisted rehabilitation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The aim of this research is to summarize AI-  based soft wearable robotic devices and  assistance with a focus on upper limb  rehabilitation with intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Over the last  two decades, robotics technology has been  used in rehabilitation and many devices have  been proposed depending on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' There  are many devices that do not have clinical  acceptance due to their poor therapeutic  utility, many of them were not transportable,  airy and convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Low-to-middle-income  countries cannot adopt exosuits due to high  cost whereas stroke happens here more than in  high-income countries [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In this way,  various strategies with AI have to be applied  to reduce the cost of production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Literature review:  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Recap of Upper-Limb Exoskeletons:  An exoskeleton of the upper limb describes as  the covering of the human arm as shown in  figure1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Wearable exoskeleton robots have  Figure 1: Mechanical structure of an upper limb exoskeleton ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' (a)  Schematic design and DOFs, (b) Back View and (c) Front View  of the exoskeleton, (d) exoskeleton position when elbow at 90  degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  gradually become popular in clinics because  they can support multiple joints and provide  flexible and safe rehabilitation training  therapy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Still, exoskeletons have further  complicated  mechanical  structures  and  control  techniques  and  most  of  the  rehabilitation  exoskeletons  are  in  the  prototype stage [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Upper limb exoskeletons work parallel to the  human arm and are attached to the human  upper limb at various joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' By this  indication, it is clear that robots have to adapt  to different arm lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Different types of  exoskeletons are present as the figure2 based  b)  di  Pineda-Rico etal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='3  on DOFs, Supporting joints, operation modes,  control techniques and applications, etc  As per works of literature with figure 3(a),  most exoskeletons are rigid in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The  hard  exoskeletons  are  structured  mechanically and offer high resistance to  tension or deformation [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The full limb will  be supported by this type of exoskeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It  provides some mechanical flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Some  of the reviews are introduced about soft  exoskeleton systems but in very small  amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' These types of exoskeletons are soft  and flexible and made of soft materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Soft  exoskeletons are used to assist the users’  movement without any restrictions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The robotic exoskeletons have many features  that highlight their applicability in rehab  properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Most of the exoskeletons are made  of rigid materials such as metal or carbon fiber  or aluminum alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' As regards the structural  materials, the review of the physical models is  made by a combination of different materials  [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Nowadays cable-driven joints are used to  make the exoskeleton lighter, compact and  portable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Plastic materials are used to reduce  the weight of physical appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' A very  small amount of research used soft or semi-  hard materials [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The distribution of used  materials for upper exosuits are laid out in  figure 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' With the growing technology  additive manufacturing or 3D printing  technology is also used for manufacturing  exoskeletons [2] [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The characters of the exoskeletons depend  upon the type of modes and degrees of  freedom (DOFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Four types of operational  modes are observed in motor rehabilitation or  neuromotor functioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' They are Active  mode, Passive mode, Assistive mode &  Resistive mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The active mode [26][28]  gives all possible movements of a limb using  a robotic exoskeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In passive mode  [16][18], limb movements are monitored  using the exoskeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Assistive mode [34] is  partially used to move the limb and resistive  mode [29] opposes the movement of the limb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Another operating mode is multi-mode [27]  which contributes to mixing modes of  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Different types of exoskeletons  utilize different operational modes for  different types of regaining health conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Figure 3: The general distribution of physical properties of  robotics exoskeleton based on (a) exoskeleton types, (b)  degrees of freedom(DOFs), (c) operational modes and (d)  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Figure 2: Types and classifications of the upper-limb exoskeleton  Upper-LimbExoskeleton  Type of joints(upper limb)  Modes  DOFs  Types of  Control Methods  Applications  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='ShoulderExoskeleton  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Active  Materials  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Artificial Neural Network  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Medical  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Single-joint  Il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='ElbowExoskeleton  Il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Passive  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Hard  Il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='AdaptiveAlgorithms  Il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Military  exoskeleton  Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='WristExoskeleton  Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Resistive  Il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Soft  Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Fuzzy-logic  Ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Commercial  Il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Multi-joint  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Finger Exoskeleton  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Assistive  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Sliding Modes  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Industry  exoskeleton  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='OtherTechniques  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='AssistiveSoft Exoskeleton  Multi DOF  1 DOF  Hard Exoskeleton  3DOF  2DOF  (a)  (b)  3D printing  Mix /  Assistive  Others  technology  Others  Resistive  Active  Mix  Metal  Passive  Plastic  (c)  (d)4  The DOF determines the possible movement  of a limb over one or different joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Higher  DOFs require more actuators and are more  complex in mechanical design and needed  much improvement in control techniques to  do the specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The design is simple to  use with one DOF [26] [27] [28] [29] [30],  and the driving mechanism allows accurate  joint mobility but increases the complexity of  the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Approximately 45% of  research considered single DOF exosuits  mostly based on the elbow joint [26] which  assists in flexion and extension movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The internal and external design of two DOFs  robots [17] [18] are more complex than  single.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Almost 20% of reviewed papers are  based on two DOF as shown in figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  All of those shoulder-elbow models are very  common [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The robotic structure decides to  improve  the  efficacy  of  rehabilitation  guidance according to the patient’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  This study presented 10% of reviews are on  the basis of three DOFs models [34] [38] [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  In specific, the used exoskeleton in paper [42]  introduced three DOFs at the elbow joint, arm  rotation, and wrist joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Finally, the most  illuminating  work  was  conducted  by  Varghese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [41], where a multi-DoFs  shoulder kinematic detection model was  published.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The work was established based on  the physical location of one patient and will  be elongated for multiple patients in the  future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The primary aim of exoskeletons is to  bear rehabilitation treatment with a compact,  airy, and high-performance device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" The  tendon-routing framework can be extended to  the exosuit's actuator structure and to other  multi-DOF joints such as the hip, wrist, and  ankle." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Soft Exoskeleton:  Development of soft exoskeletons has  increased over the past decade, particularly in  lower limb applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Soft exoskeletons  may replace many or all hard exoskeletons in  the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Light, thin and flexible materials  are the substitute for hard, heavy, and rigid  materials for soft exoskeletons [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The  flexible soft exoskeleton is also known as an  Exosuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Some components must be rigid,  such as the battery and controller, which are  often placed in a bag or separately somewhere  to reduce weight [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' That types of exosuits  are great because of their features as they are  light, flexible, and easy to transport and  install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Patients can use these devices alone  and carry them around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Therefore, they are  user-friendly and can be used in daily life for  improvement easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Soft exosuits can be used  for the treatment of the upper limb, especially  for the shoulder [17,18,41], wrist [20,23],  elbow [16,21], and finger [19,22] aid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The shoulder is a vital joint for the upper limb,  it is a kind of ball-socket joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The disabilities  of the shoulder can affect the whole upper  limb badly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The shoulder joint has three  degrees of freedom (DOFs) such as flexion-  extension in the sagittal plane, abduction-  adduction in the coronal plane, and medio-  lateral rotation in the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Figure 4: Examples of soft exoskeletons for  shoulder joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Thompson et al [18] represent a cable-driven  soft actuation system for shoulder joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' This  (a)  (b)  shoulderanchor  (a)  Stage2  FREEactuator  Stage1  (b)  Stage2  armanchor  Stage  (a)Frontand (b)rear viewof theshoulder exoskeleton prototype  (a)Nested linearand (b)pennateFREEarchitectures,0,2,and  O, represent the contraction displacements of the first nested stage, second  JCeare the GH and elbow joint centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Cable path is shown in cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  nested stage, and pennate architecture, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  (A)Thompsonet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  [Above]  (B)SAMPER-  snoutder twee cabl  elcroattachment  ESCUDEROet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Elbowoheath  [Below]  lbowt5  model presented a nested linear and pennant  architecture  by  using  fiber-reinforced  elastomeric  perimeter  with  the  same  manufacturing factors for increasing shoulder  flexion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  It  estimated  forces  for  low  displacements up to 200 kPa pressure,  whereas SAMPER-ESCUDERO et al [17]  designed a wearable exosuit made of elastic  fabric that supports shoulder and elbow  stretch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' This soft prototype can support the  elbow fully and the shoulder partially up to  100 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' These two types of soft  exoskeletons are used for shoulder flexion as  shown in figure-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The wrist joint is a complicated joint that  bridges the gap between the hand and the  forearm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It has three degrees of freedom -  flexion-extension, radioulnar-deviation, and  rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In a paper, Benjamin et al [19] spoke  for a 3D-printed soft robotic wrist device for  rehabilitation with the mobility of two DOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  By using this at least 70% recovery is possible  with sufficient torque and bending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' They have  adopted a folding-based design fabrication  process that can be accepted in other  applications and Yuejun Xu [23] designed an  assistive and portable soft robot and simulated  it for wrist dysfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' This robotic device  used soft actuator, elastomeric chambers, and  shell reinforcement plates for inducing  specific bending under pneumatic pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Figure 5: Examples of soft robotic aided devices for wrist (A)  and Elbow (B) rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Elbow joint is a kind of hinge joint between  the arm and forearm of a human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It  possesses two degrees of freedom as flexion-  extension and supination- pronation to the  forearm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' For the aiding of elbow, Chiaradia et  al [16] proposed a refined design of the  exosuit  of  which  a  smooth  control  architecture  represented  motion-intention  detection with gravity compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" It has  been shown that the device relieves 77% of a  user's sustaining moment to move a light  weight with a reduction in muscle effort of  64." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  In  Oguntosinet  et  al  [21]  representation, he revealed a prototype of an  inexpensive, airy, and wearable soft robotic-  aided device that could assist a patient with  elbow motion therapies after a stroke occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Finger joint dysfunction is a very crucial  problem due to its most complicated  anatomical nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The fingers have 21  degrees of freedom (DOFs), including 5DOFs  for the thumb and the remaining four fingers  with 4 DOFs each for the index, middle, ring,  and little fingers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Figure 6: Examples of soft exosuits for finger rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Suresh Gobee et al [20] proposed a portable  soft robot assistive device for finger  rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It utilized electromyography  (EMG) signals to catch patients’ muscular  signals,  amplified  those  signals,  and  simulated solenoid valves for controlling  pneumatic actuators attached to the wearable  gloves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Yet Zhang et al [22] demonstrated a  conformal mechanism of a multi-material  pneumatic soft finger that was optimized to  @YuejunXu  @Benjamin W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Ang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  (a)  (b  @ Oguntosin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  @Benjamin et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Ba  (c)  Motorbike Clove  (d)  b  Pneumatic-tubes  @Gobee et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  20mm  @Zhanget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='6  obtain its maximum bending deflection and  also modified for artificial hand gripper,  rehabilitation and other practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Some exosuits are used for only one reason as  shoulder flexion or elbow flexion or wrist  bend and some exosuits are applied for  various reasons as Escudero et al [17]  presented a prototype for both shoulder and  elbow flexion, Chingyi Nam, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [25]  elaborated lightweight, ergonomic and low  power-demanding soft wearable robot for the  blooming system can aid the elbow, wrist, and  fingers to fulfil the following arm reaching in  a logical order and withdrawing through  EMG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' These and many other reviews  show that soft exoskeletons are user-friendly,  low-cost, portable, lightweight, and ready to  move anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Mainly soft exosuits are  made up of fabric or textile or elastomer that  bundle up around a user’s limbs and act  aligned to the limb [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Exoskeletons and AI control techniques:  AI is a technique to mimic human  intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' A machine can perform different  functions such as planning, learning, and  solving problems with artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  AI is a wide concept that usually requires  humans to carry out tasks performed by a  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Machine learning algorithm (MLA)  is the subset of AI which adopt and learn data  iteratively by statistical models and draw  useful hypothesis without human feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Figure 7: Different techniques of AI used for robotic device  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Finally, Deep Learning is a special category  of MLA which uses artificial neural networks  to copy like the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The evolution  of AI techniques is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Advanced data acquisition, processing and  control techniques based on AI are much  more feasible in modern times that can be  more cost-effective than classic methods  applied in biomechatronic systems including  robotic exoskeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [1] [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Artificial neural  networks (ANN), adaptive algorithms, fuzzy  logic, and other mixed techniques are mostly  used  for  exoskeleton  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  These  techniques are commonly used to detect  patterns or analyze motion patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Figure 8  illustrates the use of multiple AI-based data  processing and control techniques according  to the distribution of literature references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Figure 8: General distribution of control techniques for  soft exoskeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The development and application of  artificial intelligence (AI) techniques in this  field have become a current trend in late  2015 and will change this field with further  research in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The literature shows  that among AI processes that have been used  in rehabilitation, 40% focus on artificial  neural  networks,  20%  on  adaptive  algorithms, and 40% on other mixed  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Also, it was observed that only  16% of the articles concentrated on  neuromotor rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Mix  Fuzzy  Logic/  Adaptive  sliding  Algorithms  Mode  Others  Mix  Mix  ArtificialNeuralNetworkMachine Learning  Artificial Intelligence  Deep Learning  Subsetof MLwhichmakethe  Subsetof Al techniques which use  Thetechniqueswhichenables  computationofmulti-layerneuraljstatisticalmethods to enable  computertomimichuman behaviour  networksfeasible  machinestoimprovewith  forsolvingproblemsandproviding  Recurrent NN  experiences  data-driven answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Convolution NN  Artificial Neural Network  Fuzzy Learning  Support Vector Machnine,  "K-nearest neighbors  RandomForest  Logic regression  Decision Tree  Naives Bayes7  No  Ref  No  Authors’  Name  Yea  r  Used  AI  Technique  Scope  of  movement  Results  Clinicall  y Tested  1  26  Tageldee  n et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2016 Fuzzy  interfacingc  ontroller  Elbow [1 DOF]  Torque  Estimation  No  2  27  Sangha  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2016 Neural  Network  Wrist [1 dof]  Motor Recovery  No  3  28  Siddiqi et  al  2016 SVM  with  Puk Kernel  Thumb [1]  Angle  Estimation  No  4  29  Yu Lei et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2016 ELM  algorithms  with SVM  Wrist [1]  A  potential  approach to the  remote  quantitive  rehabilitation  Yes  5  30  Rethlep  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2017 NARX  MLPNN  Elbow [1]  Joint  Angle  velocity  estimation  No  6  34  Xin  Li  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2017 TDRNN  Wrist[2]  &  Hand[1]  Angle  Estimation  Yes  7  39  Zhang  2018  BP Neural  Network  elbow & Wrist  [3]  Angle  Estimation  No  8  38  Peng Xia  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al  2018 CNN neural  Network  3 DOF  Improvement of  limb movement  No  9  36  Yudha P  Pane  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2019 Reinforcem  ent learning  based  compensati  on  Shoulder[1],Elb  ow[1], Base[1],  Wrist[3]  Enhance  the  control  performance  No  10  33  Yan  Chen  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2019 LSTM  Neural  Network  Shoulder  Joint  Angle  estimation  No  11  32  Siqi Cai  et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2019 SVM  Machine  ML Method  5  Standard  Upper  limb  Motion  Self-  Rehabilitation  (ReRobot)  No  12  41  Varghese  et al  2019 ANN  Shoulder (Multi-  DOF)  Bio-Inspired  Kinematic  Sensing Suit  No  13  43  Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 2021 CNN-  LSTM  Multi movement  (Upper  and  LowerLimb both  movement)  Predicts  the  human  knee/ankle joint  trajectory  No  14  31  Hussain  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2022 NARX-NN  Elbow (2)  Wrist (2)  Predicts  the  torques  during  different eating  activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  No  8  Some of published AI-based robotic devices  for exoskeletons are closely observed from  this growing field and based on those some  used techniques such as fuzzy logic, and  neural networks, etc are discussed in the table  as Tageldeen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [26] in 2016 made a fuzzy  logic technique to estimate joint (elbow)  torque from relevant sEMG signals for  accurate control of exoskeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' A compact  wearable robotic Orthosis is developed by  Sangha et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [22] for Wrist Assistance with  the help of neural network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' where Siddiqi et  al [ 28] used support vector machine (SVM)  technique with puk kernel for thumb angle  estimation with an average accuracy of  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Yu Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [24] proposed a fugl-  meyer evaluation model using machine  learning algorithm and RRelief algorithm  implemented to discover the optimal feature  subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The research is growing in the rehabilitation  field with artificial intelligence every day  because upper limb disorders harm our daily  routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' No doubt eating activity is one of  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' To eliminate that type of problem  Hussain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [31] build a 4 DOFs dynamic  model for predicting torque during eating  activity process by using Nonlinear Auto-  Regressive network with exogenous input  Neural Network (NARX-NN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  We can see many types of AI techniques are  used to control the exoskeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Among  those traditional AI techniques make easy  paths for these robot-aided devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' From this  table, it can be understood that AI is suitable  for quantitative patient feedback by pre-  processing, monitoring progress, and tracking  outcomes in the retrieval process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Some of  them are clinically tested, and most of them  are in the prototype phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In context, AI  cannot bind patients to a predetermined  motion and motivate patients with great  accuracy and reliability with their own  intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Most of the AI-based exoskeletons  are not clinically tested, they need more  observations over it and will make them  usable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Intelligent soft exosuits:  An intelligent soft rehabilitation  exoskeleton is a combination of bionic  structure design and artificial intelligence  control algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' We can see from this  research that intelligent soft-exoskeleton or  exosuits is good for rehabilitation because of  their lightweight, portable, low-cost, user-  friendly, and travel-friendly nature with the  user’s intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Artificial intelligence makes  the  exoskeleton  more  intelligent  and  ergonomic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In 2019, Varghese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [41]  presented a kinematic perception outfit for the  complex joint of shoulder joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' This  prototype proposed bio-inspired cable-driven  architecture which is trained by a mapping  based on ANN transfer sensor data to joint  angles but it does not friendly with  nonlinearity as friction, hysteresis, creep etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Hence the existing exoskeletons  cannot provide rehabilitation training without  any therapist due to low intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Since  limb movements are uncertain, it is important  to recognize and analyze motor intentions and  provide proper training accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Artificial  intelligence can control training by detecting  motion intention by physiological cues and  predict patient’s movements including the  ability of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" An intelligent controller is  able to build their reference trajectory from  the patient's state or learn to adjust the control  criteria." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Qingcong et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [44] represent a  neuro-fuzzy adaptive controller which is drew  on radial basis function neural network  (RBFNN) for trajectory tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Therefore,  intelligent or adaptive controllers are more  suitable for rehabilitation training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Some  basic technologies are discussed below:  9  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Actuation  Process  and  Power  Transmission:  According to research status, the most used  the following actuation systems are cable-  driven modules, pneumatic actuators, shape  memory alloy-based actuators, and passive  actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The cable-driven actuator uses  cables and sheaths to transfer the power [46],  where sheaths are usually installed on the  surface of the body and follow the  correspondence path from the prime mover to  the anchor joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In a pneumatic actuator, the  volume or shape is changed according to the  adjusting internal pressure and executes such  actions as elongation, bending, or shortening  [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' But most of the pneumatic actuator-  driven exosuits need external connections for  heavy air sources, thatswhy they are not  unportable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Shape  memory  alloy-based  actuator is very popular as they are  lightweight, produce noiseless actuation [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Copaci et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [49] represent an exoskeleton  using a nonconventional actuation system  based on Bowden cable, Teflon sheath, and  SMA wire for elbow flexion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Passive  actuators mainly consist of spring and elastic  fabric, where quasi-passive actuators are  using spring, elastic fabric and quasi-passive  elements such as variable dampers or  electromagnetic clutch [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" Control techniques:  A powered exosuit must detect the user's  motion and provide information for human-  robot interactions that are controlled by an  appropriate control algorithm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The adaptive  controller can adjust some control parameters  to accommodate dynamic changes in human-  machine interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' An adaptive [51]  proposed auxiliary controller with inverse  dynamic  model  attempted  to  provide  feedforward support for online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Some extra torque can be added to the human  joint to reduce the gravity load in a limb by  using the gravity compensation control  algorithm  [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Backlash  compensation  control methods are mostly applied to cable-  driven exosuits to improve motion control  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Transmission is delayed by  friction between wire and sheath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' To solve  this problem some researchers established a  backlash observing controller and non-linear  adaptive controller with hysteresis model  [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Another most popular control system is  bio-signal-based control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' This control method  is processed with motion intention by the help  of physiological signals such as EMG, ECG,  EOG etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' A neural network controller is  developed to enhance torque estimation by  the combination of feedback signals, SEMG,  IMU, force sensor and motor encoder, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' A Kalman filter is used to identify joint  torque, a NN module to perceive the user’s  intent and PID controller for hybrid position  or torque feedback to achieve motion  guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Finally, a set of soft wearable robots has been  designed for specific functional requirements  to support various human joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In order to  improve human-machine compatibility and  assistive performance, related researchers  have been trying to create different methods  by  accumulating  rich  technological  achievements in various processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" The  resulting  prototypes  can  contribute  exceptionally well to many aspects of people's  daily lives." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Discussion:  The role of human upper limbs is very  important for performing personal activities  in daily life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=" A patient's daily activities can be  greatly affected due to improper functioning  of this organ due to neurological disorders or  surgery." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The design of the exoskeleton still  requires many adaptations, such as moving  towards flexible, adaptable, wearable, and  lightweight structures, as some devices still  do not fully meet these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Wire-actuated  and soft-structured exoskeletons are also  preferred in rehabilitation processes because  they are more friendly to everyday life due to  their lightness and performance as Thompson  10  et al [18] represent a cable-driven soft  actuation system for shoulder and elbow  joints for increasing muscle strength to  working labour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' This technology can be  brought into the rehabilitation process with  some modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Benjamin’s 3D-printed  soft robotics wrist sleeve is capable of 70%  recovery and supports the fabrication process  of fold-based designs which may be able to  adapt in other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In Nam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [25]  paper said a lightweight, ergonomic and low  power-demanding soft wearable robot for  assisting elbow, wrist, and fingers by using  surface EMG signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The incorporation of different modes of  operation of the exoskeleton has been shown  to improve patient’s motor recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In the  paper of sangha et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' [27] represents a multi-  mode orthosis for wrist rehabilitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' They  used passive, active, resistive and active  assistive modes for motor recovery with the  help of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Resistance mode  helped users reach full motion with current  muscle tone or unable to reach full range and  active assistive mode was used to assist the  user in reducing muscle abnormalities and  increasing interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The results displayed  that the robotic orthosis was capable to done  jobs successfully under any functional  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Now the continued development  of new devices is suggested multi-mode  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Future research is trending toward creating  exoskeletons that actively learn from input  experiences and automatically adapt to the  environment or user as artificial intelligence  provides the basis for developing more  reliable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' A variety of AI techniques  are incorporated into the study to find possible  solutions among which artificial neural  networks are popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Since ANN has strong  non-linear properties, it shows a large effect  on continuous motion estimation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  ANN makes robots movement smoother and  safer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' As chiaradia [16] represents a smooth  control exosuits for elbow which detect  motion  intention  by  using  gravity  compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Some  studies  have  also  become home base for self-rehabilitation after  the Covid pandemic [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Almost all enlisted robotics exoskeletons are  capable of muscle power strength but are not  intelligent enough to function alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Yet  existing intelligent soft exoskeletons are not  suitable for clinical use due to some of their  shortcomings, with some exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' The  literature appears that there is a need to  develop integration methods for intelligent  systems and specialized hardware that cover  user needs in a global, comprehensive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Conclusion:  The soft exoskeleton is currently a boon for  everyday life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' It protects our organs from  damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' A variety of exosuits have been  presented, each of them a new concept and  attitude to the device, which has been in the  form of prototypes, early clinical studies, and  sometimes extensive clinical and commercial  research to solve movement disorders and  rehabilitation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' These concepts  require more extensive clinical trials to  become a complete and efficient plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' For  this, related researchers are trying to develop  different types of algorithms to achieve  different soft upper limb exosuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  The  present  scenario  of  upper  limb  rehabilitation  systems  shows  that  transportable robotic exoskeletons composed  of  intelligent  control  and  information  processing systems can play an important role  in bridging the gap between prototype and  clinical  acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Additionally,  exoskeleton capabilities can be greatly  enhanced by improving materials and  incorporating a better mechanical design, and  improving motion sensing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  11  References:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Manuel Andrés Vélez-Guerrero, Mauro  Callejas-Cuervo and Stefano Mazzoleni  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Artificial  Intelligence-Based  Wearable Robotic Exoskeletons for  Upper Limb Rehabilitation: A Review”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  MDPI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' : 18 March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Elena Bardi, Marta Gandolla, Francesco  Braghin, Ferruccio Resta, Alessandra L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Pedrocchi and Emilia Ambrosini;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Upper limb soft robotic wearable  devices: a systematic review”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Journal  of  NeuroEngineering  and  Rehabilitation,2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Keravnou-Papailiou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Antoniou, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Medical analytics for  healthcare  intelligence—Recent  advances and future directions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Artif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Md  Rasedul  Islam*,  Christopher  Spiewak, Mohammad Habibur Rahman  and Raouf Fareh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “A Brief Review on  Robotic  Exoskeletons  for  Upper  Extremity Rehabilitation to Find the  Gap between Research Porotype and  Commercial Type”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Adv Robot Autom,  an open access journal Volume 6 Issue 3  1000177 ISSN: 2168-9695;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Alan Francisco Pérez Vidal, Jesse Yoe  Rumbo Morales, Gerardo Ortiz Torres,  Felipe de Jesús Sorcia Vázquez, Alan  Cruz Rojas, Jorge Aurelio Brizuela  Mendoza and Julio César Rodríguez  Cerda;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Soft  Exoskeletons:  Development,  Requirements,  and  Challenges of the Last Decade”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' MDPI,  journal,actuators, July 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Mekki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Delgado, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Fry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Putrino,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Huang,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Robotic  Rehabilitation and Spinal Cord Injury: A  Narrative Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Neurotherapeutics  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' De la Tejera, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Bustamante-Bello,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Ramirez-Mendoza, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Izquierdo-  Reyes,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Systematic  review  of  exoskeletons  towards  a  general  categorization  model  proposal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Applied Science Journal, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Miao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Reviewing  high-level  control  techniques on robot-assisted upper-limb  rehabilitation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Advanced  Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Volume 32, Issue 24,2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' David D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Luxton and Laurel D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Riek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Artificial Intelligence and Robotics in  Rehabilitation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Chapter 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Handbook  of Rehabilitation Psychology, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Qingsong Ai, Zemin Liu, Wei Meng,  Quan Liu, and Sheng Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Xie;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Machine  Learning in Robot-Assisted Upper Limb  Rehabilitation:  A  Focused  Review”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  IEEE  Transactions  on  Cognitive  and  Developmental  Systems (Early Access);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 19th July,2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Muhammad Ahsan Gull, Shaoping Bai  and Thomas Bak,” A Review on Design  of  Upper  Limb  Exoskeletons”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  MDPI,2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Andrea Demofonti, Giorgio Carpino,  Loredana Zollo, Michelle J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Johnson,  “Affordable Robotics for Upper Limb  Stroke Rehabilitation in Developing  Countries: A Systematic Review”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' IEEE  Transactions on Medical Robotics and  Bionics , Volume: 3, Issue: 1, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Zaira  Pineda-Rico,  Jose  Alfonso  Sanchez de Lucio, Francisco Javier  Martinez Lopez, Pedro Cruz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Design of  an exoskeleton for upper limb robot-  assisted rehabilitation based on co-  simulation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Journal  of  Vibroengineering, Vol 18, Issue 5, Aug  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Manuel Andres Velez-Guerrero, Mauro  Callejaw-Cuervo  and  Stefano  Mazzoleni ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='“Design, Development,  and  Testing  of  an  Intelligent  Wearable  Robotic  Exoskeleton  Prototype  for  Upper  Limb  Rehabilitation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Journal  of  Sensors,MDPI,10 Aug,2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Emanuele Palazzi, Luca Luzi, Eldison  Dimo,  Matteo  Meneghetti,  Rudy  Vicario, Rafael Ferro Luzia, Rocco  Vertechy and Andrea Calanca ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “An  Affordable Upper-Limb Exoskeleton  Concept  for  Rehabilitation  Applications”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Journal  Of  Technologies,MDPI, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  12  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Domenico  Chiaradia,  Michele  Xiloyannis, Chris W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Antuvan, Antonio  Frisoli, Lorenzo Masia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Design and  Embedded Control of a Soft Elbow  Exosuit”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  8  IEEE  International  Conference  on  Soft  Robotics  (RoboSoft);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Italy, April 24-28, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Jose Luis Samper-Escudero, Antonio  Gimenez-Fernandez,  Miguel  Ángel  Sanchez-Uran, and Manuel Ferre;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “A  Cable-Driven Exosuit for Upper Limb  Flexion Based on Fibres Compliance”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  IEEEAccess, August 21, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Nicholas Thompson, Ayush Sinha and  Girish  Krishnan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Characterizing  Architectures  of  Soft  Pneumatic  Actuators for a Cable-Driven Shoulder  Exoskeleton”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' International Conference  on Robotics and Automation (ICRA),  Canada, May 20-24, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Suresh Gobee, Vickneswari Durairajah,  Naqibullah Mohammadullah;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Portable  Soft-exoskeleton  for  Finger  Rehabilitation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Association  for  Computing Machinery(ACM Digital  Library);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  ,  Bangkok,  Thailand  ,  September 14–16, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Benjamin W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Ang, Chen-Hua Yeow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Design and Characterization of a 3D  Printed Soft Robotic Wrist Sleeve with  2 DoF for Stroke Rehabilitation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' IEEE  International  Conference  on  Soft  Robotics (RoboSoft), , Korea, April 14-  18, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Victoria Oguntosin, William S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Harwin,  Sadao Kawamura, Slawomir J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Nasuto  and Yoshikatsu Hayashi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' ”Development  of a Wearable Assistive Soft Robotic  Device for Elbow Rehabilitation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' IEEE  International  Conference  on  Rehabilitation Robotics (ICORR),2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Hongying Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Senthil Kumar,  Feifei Chen, Jerry Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Fuh, and  Michael  Yu  Wang;”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Topology  Optimized Multimaterial Soft Fingers  for  Applications  on  Grippers,  Rehabilitation and Artificial Hands”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  IEEE/ASME TRANSACTIONS ON  MECHATRONICS,2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Yuejun Xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Design and Simulation of  a Soft Robotic Device for Wrist  Rehabilitation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  International  Conference on Advanced Electronic  Materials, Computers and Software  Engineering (AEMCSE);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 24-26 April  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Michele  Xiloyannis  ,  Domenico  Chiaradia, Antonio Frisoli and Lorenzo  Masia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Physiological and kinematic  effects of a soft exosuit on arm  movements”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Journal  of  NeuroEngineering  and  Rehabilitation,2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Chingyi Nam, Wei Rong, Waiming Li,  Chingyee  Cheung,  Wingkit  Ngai,  Tszching Cheung, Mankit Pang, Li Li,  Junyan Hu, Honwah Wai, and Xiaoling  Hu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “An Exoneuromusculoskeleton for  Self-Help Upper Limb Rehabilitation  After  Stroke”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  SOFT  ROBOTICS  Volume 00, Number 00, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Momen  Kamal  Tageldeen,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Perumal2*, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Elamvazuthi3 and 4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Ganesan ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='“Design and Control of an  Upper Arm Exoskeleton using Fuzzy  Logic Techniques”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' IEEE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Sohail Sangha, Ahmed M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Elnady,  Student Member, IEEE, and Carlo  Menon, Member, IEEE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “A Compact  Robotic Orthosis for Wrist Assistance”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  6th IEEE RAS/EMBS International  Conference on Biomedical Robotics and  Biomechatronics (BioRob) June 26-29,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Abdul Rahman Siddiqi, Shahrul Naim  Sidek, Aida Khorshidtalab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Signal  Processing  of  EMG  Signal  for  Continuous Thumb-Angle Estimation”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  978-1-4799-1762-4/15/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='00  ©2015  IEEE, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Lei Yua,b,∗, Daxi Xiong a, Liquan Guo  a, Jiping Wang a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' ” A remote  quantitative  Fugl-Meyer  assessment  framework for stroke patients based on  wearable sensor networks”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' computer  methods and programs in biomedicine  128 ( 2 0 1 6 ) 100–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Retheep Raj, K S Sivanandan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Elbow  joint angle and elbow movement  velocity  estimation  using  NARX-  multiple  layer  perceptron  neural  network model with surface EMG time  13  domain parameters”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' NIH, Pubmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='gov;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Zakia Hussain & Norsinnira Zainul  Azlan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Estimation of the Torques  Produced by Human Upper Limb during  Eating Activities Using NARXNN”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Applied Artificial Intelligence, An  International Journal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 16 Feb 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Siqi Cai, Yan Chen, Shuangyuan  Huang, Yan Wu, Haiqing Zheng, Xin Li  3 and Longhan Xie1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' SVM-Based  Classification of sEMG Signals for  Upper-Limb  Self-Rehabilitation  Training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 4th June,2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Yan  Chen,  Song  Yu,  Ke  Ma,  Shuangyuan Huang, Guofeng Li, Siqi  Cai, and Longhan Xie;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “A Continuous  Estimation Model of Upper Limb Joint  Angles  by  Using  Surface  Electromyography and Deep Learning  Method”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' IEEE Access,2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Xin Li, Qiang Huang, Jinying Zhu,  Wentao Sun and Haotian She;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “A Novel  Proportional and Simultaneous Control  Method for Prosthetic Hand” Journal of  Mechanics in Medicine and Biology  Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 17, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 8 ,2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Manuel Andrés Vélez-Guerrero, Mauro  Callejas-Cuervo and Stefano Mazzoleni  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Artificial  Intelligence-Based  Wearable Robotic Exoskeletons for  Upper Limb Rehabilitation: A Review”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  MDPI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='18 March 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Yudha  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Pane,  Subramanya  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Nageshrao,  Jens  Kober,  Robert  Babuška;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “Reinforcement  learning  based compensation methods for robot  manipulators”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Engineering  Applications of Artificial Intelligence  ,2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Kai Zhang, Xiaofeng Chen, Fei Liu,  Haili Tang, Jing Wang, and Weina Wen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  “System Framework of Robotics in  Upper  Limb  Rehabilitation  on  Poststroke Motor Recovery”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Hindawi  Behavioural Neurology Volume 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Peng Xia, Jie Hu Ying-hong Peng  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='“EMG‐Based Estimation of Limb  Movement Using Deep Learning With  Recurrent  Convolutional  Neural  Network”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  International  Center  for  Artificial Organs and Transplantation and  Wiley Periodicals , Volume 42, Issue 5 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  May 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Lei Zhang;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='“An upper limb movement  estimation from electromyography by  using BP neural network”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Biomedical  Signal Processing and Control, Volume  49, March 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Chloë  Nicholson-Smith,  Vahid  Mehrabi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Farokh Atashzar, and Rajni  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Patel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “A Multi-Functional Lower-  and Upper-Limb Stroke Rehabilitation  Robot”, IEEE Transactions on Medical  Robotics and Bionics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 2, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Dorin-Sabin Copai, Dolores Blanco,  Alejandro  Martin-Clemente,  Luis  Moreno;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “Flexible shape memory alloy  actuators for soft robotics: Modelling  and  control”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Sage  Journals,  International Journal of Advanced  Robotic System, January,2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Dorin Copaci, Enrique Cano, Luis  Moreno and Dolores Blanco;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' “New  Design of a Soft Robotics Wearable  Elbow Exoskeleton based on Shape  Memory alloy wire actuarors”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Hindawi  Applied  Bionics  and  Biomechanics,2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Poliero, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Di Natali, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Sposito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content='  Ortiz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Graf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Pauli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Bottenberg,  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Caldwell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Soft  Wearable Device for Lower Limb  Assistance: Assessment of an Optimized  Energy Efficient Actuation Prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In  Proceedings of the 1st IEEE-RAS  International  Conference  on  Soft  Robotics (RoboSoft), Livorno, Italy,  24–28 April 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Stoppler, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Ivlev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Adaptive Assistive Control of a Soft  Elbow Trainer with Self-Alignment  using Pneumatic Bending Joint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' In  Proceedings of the 14th IEEE/RAS-  EMBS International Conference on  Rehabilitation  Robotics  (ICORR),  Nanyang  Technology  University,  Singapore, 11–14 August 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Xiloyannis,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Solazzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Masia, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+page_content=' Frisoli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  An Assistive Soft Wrist Exosuit for  Flexion Movements With an Ergonomic  Reinforced Glove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' AI  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Dinh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Xiloyannis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Cappello,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Antuvan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Yen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Masia,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Adaptive backlash compensation in  upper limb soft wearable exoskeletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Auton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Wu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content=' Neural-  network-enhanced torque estimation  control of a soft wearable exoskeleton  for  elbow  assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
+page_content='  Journal  of  Mechatronics, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNE3T4oBgHgl3EQfIwkK/content/2301.04336v1.pdf'}
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+Broad-Range Directional Detection of Light Dark Matter in Cryogenic Ice
+Nora Taufertsh¨ofer,1, 2, 3 Maurice Garcia-Sciveres,4 and Sin´ead M. Griffin1, 2, ∗
+1Molecular Foundry Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
+2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
+3Institute for Theoretical Physics, University of W¨urzburg, Am Hubland, D-97074 W¨urzburg, Germany
+4Physics Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
+(Dated: January 13, 2023)
+We propose hexagonal ice (H2O) as a new target for light dark matter (DM) direct detection. Ice,
+a polar material, is suitable for single phonon detection through DM scattering for which we consider
+light dark photon and light scalar mediator models. We report a rate sensitivity down to a DM
+mass of ∼ keV, constituting a broader mass range than other promising candidates. We find bet-
+ter sensitivity for near-term experimental thresholds from the presence of high-frequency phonons.
+These advantages, and ice’s availability, make it highly promising for single-phonon detection.
+New proposals for the direct detection of dark mat-
+ter (DM) have steadily pushed into the sub-GeV ‘light’
+DM range[1]. In particular, for DM masses in the kev-
+GeV range, freeze-in DM scattering is well motivated
+through both a light scalar mediator and a dark photon
+mediator[2, 3]. The smaller kinetic energy for this lower
+mass DM motivates the exploration of low-energy excita-
+tions in condensed matter systems as possible direct de-
+tection channels[4]. Recent proposals for DM detection
+in crystal targets include quasiparticle excitations rang-
+ing from phonon and electron scattering in conventional
+semiconductors[5, 6] to more exotic quantum materials
+such as Dirac semimetals[7], and axions in topological
+insulators and multiferroics[8, 9].
+Of these proposals for DM direct detection via quasi-
+particle scattering, single phonon production presents a
+promising detection scheme for several reasons[6, 10].
+Typical optical phonons in crystals range from 10-100
+meV, making them kinematically matched with sub-
+MeV DM. Dark photon mediators can mix with standard
+model photons resulting in a coupling to optical phonon
+modes in polar semiconductors. Furthermore, common
+polar semiconductors (e.g. GaAs) have sizeable bandgaps
+resulting in small screening. The intrinsic anisotropy of
+polar semiconductors allows for a directional response
+giving signal discrimination based on daily/annual mod-
+ulation of the DM wind[11]. Importantly, high-quality
+detector-grade crystals of a range of polar semiconduc-
+tors are readily available owing to their use in microelec-
+tronics and quantum computing (e.g. GaAs[6], SiC[12]),
+and phonon sensing technologies are steadily improving
+towards the low sensitivities required for detection[13].
+With the next-generation of low-mass DM experiments
+in planning, it remains to identify the best possible tar-
+get material for each detection scheme. Previous work
+addressed this at the DM-target interaction level by
+performing a comparison of DM-phonon reach, isolat-
+ing materials-specific factors that optimize for low-mass
+sensitivity and greatest cross section[14]. Beyond this,
+other materials factors for target selection should also
+be considered that take into account current synthe-
+sis/fabrication capabilities and sensing constraints for
+near-term experiments. While experimental thresholds
+for phonon sensing continue to improve towards the sin-
+gle phonon excitation regime (∼10-100 meV), near-term
+experiments with larger thresholds could access higher
+energy responses. For instance, beyond the single phonon
+excitation regime, multiphonon responses can be ac-
+cessed for DM scattering above the single phonon spectra
+though at a suppressed rate[15]. However, an alternative
+approach to this is to search for single phonon and other
+quasiparticle responses that lie within the current sensing
+thresholds (>100 meV).
+In this Letter, we propose cryogenic water ice (H2O)
+as a particularly appealing target material for direct de-
+tection of low-mass DM from single-phonon excitations.
+We find that the mixed nature of bonding in H2O re-
+sults in a broad range of phonon frequencies suitable for
+both lower-mass detection, and for higher frequency sin-
+gle phonon excitations that are within the range of cur-
+rent sensing thresholds. Moreover, with the ubiquity and
+long history of fabrication and characterization of solid
+H2O, it can be readily fabricated as large single crystals
+comprising earth-abundant (inexpensive) elements.
+This paper is laid out as follows; we summarize the
+calculation of spin-independent DM scattering for single-
+phonons and the features of ice H2O (specifically its XIh
+polymorph) of relevance to DM-phonon interactions. We
+then present our calculations of the reach for a light dark
+photon mediator and a light scalar mediator with com-
+parisons to best-performing targets for these DM mod-
+els, and their calculated directional dependence. Finally
+we discuss other material limiting considerations for tar-
+get materials and prospects for its use for near-term and
+next-generation DM detection.
+First, we briefly summarize the formalism for calcu-
+lating spin-independent DM scattering processes includ-
+ing a light dark photon mediator and a light scalar
+mediator[16]. For a detector with target density ρT the
+total rate per target mass is given by
+arXiv:2301.04778v1  [hep-ph]  12 Jan 2023
+
+2
+R = 1
+ρT
+ρχ
+mχ
+�
+d3vfχ(v)Γ(v)
+(1)
+where ρχ is the local DM energy density, mχ the DM
+mass, and fχ(v) is the DM velocity distribution in the
+lab frame which is modelled with a boosted Maxwell-
+Boltzmann distribution 1.
+Γ(v) is the scattering rate
+per non-relativistic DM particle which is defined by an
+integral over the momentum transfer q = p − p′ with
+initial DM momentum p = mχv:
+Γ(v) = π¯σ
+µ2
+�
+d3q
+(2π)3 (q0/q)4S(q, ωq)
+(2)
+Here, ¯σ is a model-dependent reference cross section
+and we define ¯σ :=
+µ2
+π |M(q0)|2 with M the target-
+independent 2 → 2 scattering matrix element (see Sup-
+plemental Material). µ is the reduced mass of an elec-
+tron or nucleon and DM particle for light-dark-photon-
+or light-scalar-mediated scattering, respectively. q0 is a
+reference momentum and S(q, ωq) the target-dependent
+dynamic structure factor.
+For single phonon excitations we recall the dynamical
+structure factor:
+S(q, ωq) =
+π
+Ω
+�
+ν
+1
+ων,k
+���
+�
+j
+e−Wj(q)
+√mj
+eiG·x0
+j (Yj · ϵ∗
+ν,j,k)
+���
+2
+δ(ωq − ων,k)
+(3)
+where ωq = q · v −
+q2
+2mχ explicitly depends on the energy
+deposition.
+The sums run over all phonon branches ν
+and over all ions j in the primitive cell. The ionic masses
+are mj with equilibrium positions x0
+j and Ω is the prim-
+itive cell volume. ϵν,j,k defines the phonon polarization
+vectors. G is the reciprocal lattice vector that satisfies
+G = q−k with k within the first Brillouin zone. DM cou-
+plings come in with the model-specific Yj terms defined
+below. In the continuum limit for k, the Debye-Waller
+factor is given by:
+Wj(q) =
+Ω
+4mj
+�
+ν
+�
+1BZ
+d3k
+(2π)3
+|q · ϵν,j,k|2
+ων,k
+.
+(4)
+The materials-dependent quantities, namely the phonon
+frequencies ων,k for branch ν and momentum k, and the
+1 Due to Earth’s rotation the orientation of the incoming DM par-
+ticles with regard to the target crystal changes over a day, which
+leads to a directional modulation of the rate.
+We choose the
+z-axis in the crystal frame to be parallel to Earth’s velocity at
+daytime t = 0[17].
+phonon polarization vectors ϵν,j,k, can be calculated us-
+ing first-principles methods such as Density Functional
+Theory[18].
+Finally, we consider two well-motivated models of light
+DM, namely a kinematically mixed dark photon and a
+light scalar mediator[6]. In the former case, the standard
+model photon kinematically mixes with the dark photon
+owing to their same quantum numbers, resulting in ‘mil-
+licharged’ DM under U(1) of the standard model. In the
+q → 0 limit valid for light DM scattering[16], Yj is given
+by
+Yj,dark photon = −
+q2
+q · ε∞ · q(q · Z∗
+j)
+(5)
+where ε∞ is the high-frequency dielectric tensor and Z∗
+j
+is the Born effective charge tensor of the jth atom in
+the unit cell, both of which can be calculated using first-
+principles methods.
+We also consider DM that only couples to nucleons via
+a light scalar mediator with identical coupling for proton
+and neutron. The now scalar-valued Yj is given by
+Yj,hadrophilic scalar = AjFNj(q)
+(6)
+with Aj the atomic mass number and FNj an isotropic
+nuclear form factor of the jth atom, respectively. The
+derivation of (6) is valid for the q → 0 limit, which is
+fulfilled for the single phonon excitation regime. Further,
+for this type of scattering FNj can be set to one.
+The pervasiveness of liquid and solid H2O has gener-
+ated extensive studies of the phase diagram of ice. So far,
+twenty polymorphs of crystalline H2O have been identi-
+fied under various conditions of temperature and pres-
+sure, with the most recent, ice XIX reported in 2021[19].
+Common to all of these is the local atomic-scale arrange-
+ment of hydrogen atoms in bonded units that fulfil the
+‘Bernal-Fowler ice rules’: each oxygen forms a covalent
+bond with two hydrogens and a weaker van-der-Waals
+bond with two other hydrogens[20]. Since each oxygen is
+also shared between two hydrogen atoms, this results in
+an average oxygen bonding environment comprising ‘two-
+in’ (strong) and ‘two-out’ (weak) bonds with hydrogen.
+Such an atomic arrangement is geometrically frustrated
+on a tetrahedral lattice[21], resulting in disordered rela-
+tive arrangements of the strong and weak bonding net-
+works known as proton disorder. Common ice (Ih) forms
+such a hydrogen-disordered network of H-O-H bonds at
+ambient pressure, giving rise to a residual entropy as first
+predicted by Pauling[22].
+However, spontaneous ordering of the hydrogen net-
+works can occur on cooling, or by the introduction of a
+dopant to overcome the kinetic barrier to ordering. For
+example KOH doping of Ih results in the formation of a
+fully ordered phase of hexagonal ice, XIh below ∼ 72
+K[23, 24].
+In fact, the hydrogen ordering in the XIh
+structure gives it a net dipole moment, making it po-
+tentially ferroelectric[25]. Since XIh is the stable form
+
+3
+°
+X S R A
+Z °
+Y X1A1 T Y
+0
+100
+200
+300
+400
+! [meV]
+Z
+T
+°0.04
+°0.02
+0.00
+0.02
+0.04
+°
+X S R A
+Z °
+Y X1A1 T Y
+0
+100
+200
+300
+400
+! [meV]
+Z
+T
+°0.04
+°0.02
+0.00
+0.02
+0.04
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+Fig. 1: (a) Primitive unit cell of hexagonal ice XIh and its first Brillouin zone with high-symmetry points labelled.
+(b) Calculated phonon band structure for hexagonal ice XIh. The cartoons on the right hand side show the
+movement of ions for three selected phonon modes representative of the types of optical phonon modes in ice:
+translational modes (bottom), libration/bending (middle), and stretch (top).
+of ice at ambient pressures, and can be synthesized in its
+ordered form, we focus on this polymorph for the remain-
+ing discussion, noting, however, that other structures of
+ice should have comparable DM reach for single phonon-
+based detection owing to their similar bonding networks.
+We also considered heavy ice D2O XIh but we did not
+find a significant difference to H2O (results reported in
+the Supplemental Material).
+We now present our DFT-calculated structural and
+phonon properties of ice XIh. The full calculation details
+can be found in the Supplemental Material.
+With its
+combination of covalent bonding, hydrogen bonding, and
+van-der-Waals bonding, the challenge of treating ice H2O
+with DFT functionals has been explored extensively[26].
+We benchmark our choice of exchange correlation func-
+tional for accurate structural and vibrational properties.
+Consistent with previous work[27], we find that the van-
+der-Waals corrected nonlocal functional (optPBE-vdw)
+of Klimeˇs et al. performs best[28, 29], resulting in the
+lattice constants a = 4.470 ˚A and c = 7.212 ˚A. This cor-
+responds to 0.6% and 1.5% deviation from experimental
+measurements at T = 2 K[30].
+In Fig. 1(b) we plot the calculated phonon dispersion
+for ice XIh. All polymorphs of ice exhibit a large range
+of phonon frequencies as a result of the varied nature
+of bonding in H2O crystals[31–33]. The low-energy op-
+tical phonon range is made up of ‘translational’ modes
+where individual H2O molecules behave as atom-like clus-
+ters, and vibrate with respect to one another.
+Such
+low-frequency phonon modes are common in molecular
+crystals such as H2O where molecular units are weakly
+bonded to each other to form the crystal network[34].
+The next highest energy set of phonon modes corre-
+sponds to libration and bending of H-O bonds, with the
+highest-frequency range caused by the stretching of the
+O-H bond. As the lightest element in the periodic table,
+hydrogen sets an upper limit on the range of these high-
+frequency stretch modes that can be found in a material.
+We present our calculated reach for light-dark-photon-
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+<latexit sha1_base64="zDPGs4KaAfqPlXMptLPHIbKS3B4=">AB8nicdVDLSgNBEJyNrxhfUY9eBoPgxWU3DxNvQUH0FsE8
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+Fig. 2: Projected reach for light-dark-photon-mediated
+DM scattering via single phonon excitations with
+1kg-year exposure and no background. Solid,
+long-dashed and dashed lines refer to a 1 meV, 20 meV
+and 100 meV detector threshold. For comparison,
+additional reach data [14] is shown. Stellar constraints
+from red giants (RG) and the horizontal branch (HB)
+are taken from [35], the freeze-in as referenced in [14].
+mediated scattering of single phonons in ice XIh in Fig. 2,
+including comparisons to top-performing targets previ-
+ously studied[14]. All reach calculations were done us-
+ing the code PhonoDark[36, 37]. The reach of ice XIh
+extends further into both the low-mass DM range (well
+into the constrained regions) and has better reach in the
+high-mass DM range than any other previously suggested
+single target material. To understand the exceptionally
+broadband sensitivity of ice XIh, we evaluate its perfor-
+mance with respect to our previously suggested quality
+factors for dark photon mediators in the high- and low-
+frequency phonon ranges[14]. The lowest DM mass acces-
+sible is determined by the energy of the lowest-frequency
+optical mode where mχ ∼ 1
+3ωmin
+0
+× 106, suggesting that
+materials with low-lying optical phonons such as CsI
+
+4
+(ωmin
+0
+∼ 7 meV) have the best low-mass reach. In our
+case, the van-der-Waals bonded molecular units also have
+very low lying optical modes (ωmin
+0
+∼ 6 meV), making ice
+XIh optimal for the low-mass DM range. We expect all
+similar molecular crystals with such low-frequency trans-
+lational phonon modes to have competitive reach for low-
+mass DM.
+For the high-mass range, a quality factor, Q, is defined
+as[14]:
+Q ≡
+Z∗2
+A1A2ε2∞
+�meV
+ωLO
+�
+(7)
+where Z∗ are the Born effective charges, A1, A2 are the
+atomic masses, ε∞ is the high-frequency dielectric con-
+stant, and ωLO is the longitudinal optical phonon fre-
+quency.
+A high Q corresponds to better reach in the
+high-mass DM regime. We calculate each of these values
+for ice XIh using DFT, with the full set of results given in
+the Supplemental Material. The best performing materi-
+als previously proposed include LiF with Q = 270×10−7
+and SiO2 with Q = 200 × 10−7. For ice XIh, we find
+Q = 533 × 10−7 for its highest-frequency stretch modes
+(>375 meV), steadily increasing for lower-frequency ωLO
+clusters. The substantially enhanced Q of ice XIh in the
+high-mass range is due to the low masses of hydrogen and
+oxygen. This optimized reach is dominated by hydrogen’s
+extremely low mass, despite these small masses also re-
+sulting in higher frequency phonon modes. However, this
+quality factor does not take the detection threshold into
+account which is especially important for near-term ex-
+periments.
+We conclude that for optimal high-mass DM single-
+phonon-based detection with dark photon mediators,
+chemistries that include light elements such as H and
+He, and especially H-H bonds, will give the best reach in
+the high-mass range owing to their high-frequency sin-
+gle phonon modes. In the low-mass range, we find that
+both compounds containing heavy elements and molec-
+ular crystals that possess low-frequency optical phonon
+modes will give the best sensitivity. With ice combin-
+ing both of these properties (molecular units and H-H
+bonds), it displays better sensitivity for dark photon me-
+diators than previously suggested targets across a broad
+range of DM masses with CsI being the only exception
+for a very small low-mass range (Fig. 2).
+The reach for light-hadrophilic-scalar-mediated scat-
+tering of single phonons in ice XIh is plotted in Fig. 3
+for several detector thresholds, and compared to previ-
+ous top candidates from Ref. [14]. For the light scalar
+mediator, we also find that ice XIh has broad-band sen-
+sitivity for both low- and high-mass DM. Looking at the
+reach curves, we find a competition between which of di-
+amond or ice is favorable for different mass ranges and
+thresholds. For all thresholds considered, we find that
+ice has sensitivity to lower masses than diamond, which
+become comparable by a threshold of 100 meV. In the
+low mass range, the lowest DM mass sensitivity is gen-
+erally governed by ωmin/cs where ωmin is the detector
+threshold and cs is the material’s speed of sound. There-
+fore, materials with higher cs such as diamond will have
+the best sensitivity to lower-mass DM given low detec-
+tor thresholds. We calculated the directional averaged
+speed of sound of ice XIh to be 4376 ms−1 (details given
+in the Supplemental Material), which is a factor of three
+smaller than in diamond. However, for materials with
+smaller cs the acoustic phonons can be kinematically in-
+accessible, especially as the threshold increases to higher
+energies. When this happens, the lowest DM sensitivity
+is then set by the lowest optical phonon available. For
+ice XIh we find that the minimum reachable DM mass
+from kinematic considerations is ∼ 29 keV for a 1 meV
+threshold, while the lowest optical phonon is 6 meV. This
+manifests as a change in the slope as seen in Fig. 3 where
+optical phonons, rather than acoustic phonons, dominate
+the low mass response.
+In the high-mass,
+high-threshold range (detector
+threshold of 400 meV), we only include ice as it is the
+only material with single phonons above ∼160 meV. We
+note that the definition of ¯σn ∝ m−4
+χ
+leads to the gener-
+ally better reach at high masses[38].
+We finally calculate the daily modulation rate assum-
+ing that the z-axis of the target crystal is aligned with the
+DM wind at t = 0. Each rate is normalised to the average
+rate ¯R over one day. For a dark photon mediator we find
+a strong modulation for a mass range of mχ = 104 − 105
+eV (Fig. 4 (a)). We also find a significant daily modula-
+tion for the scalar mediator case (Fig. 4 (b)), especially
+in the low 10 keV range. Finally, we extract the phonon
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+Fig. 3: Projected reach for light-scalar-mediated DM
+scattering via single phonon excitations with 1kg-year
+exposure and no background. Solid, long-dashed,
+dashed and dotted lines refer to a 1 meV, 20 meV,
+100 meV and 400 meV detector threshold. Additional
+reach data is taken from [14], light-colored lines show
+the expected curve progression for scale ranges not
+shown in the reference data.
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+Fig. 4: Daily modulation for H2O XIh ice with (a) a light dark photon and (b) a light scalar mediator for various mχ.
+lifetimes τ from measured phonon linewidths Γph of ice
+XI[31, 39] from τ = (πΓph)−1 giving a lifetime range of
+0.5 - 2.1 ps (see Supplemental Material), consistent with
+other systems with high-frequency optical phonons[40].
+Our results suggest that ice H2O is a competitive can-
+didate target for single-phonon based light DM detection
+both in the low-mass and high-mass range (here we con-
+sidered ice XIh, but our conclusions are generally appli-
+cable to other ice polymorphs). Molecular units provide
+ultra-low-mass optical phonons (down to 6 meV) giving
+excellent reach in the low-mass DM range for dark pho-
+ton mediators. We expect similar reach to low masses
+for polar semiconductors containing heavy ions (as e.g.
+CsI) or molecular crystals where the relative effective
+masses of the ions/molecules result in low-frequency op-
+tical phonons. For near-term experiments, we find that
+the broad range of single phonons in ice up to 400 meV
+gives it enhanced reach for higher experimental thresh-
+olds compared to alternative proposals such as multi-
+phonon excitations.
+However, the lower density of ice
+due to its van-der-Waals bonding results in a reduced
+reach compared to other solid-state targets; this can be
+addressed by the growth of large single crystals of ice
+which is already well explored[41]. Finally, while ice H2O
+is cheap, earth-abundant, and has been extensively char-
+acterized, it remains an engineering challenge as to the
+best way to incorporate a material that is a liquid at room
+temperature into a (cryogenic) detector architecture.
+We thank Dan Carney, Tanner Trickle, Kathryn Zurek,
+Katherine Inzani and Thomas Harrelson for helpful dis-
+cussions.
+This work was supported by the US De-
+partment of Energy under contract DEAC02-05CH11231
+and Quantum Information Science Enabled Discovery
+(QuantISED) for High Energy Physics grant KA2401032.
+Computational resources were provided by the National
+Energy Research Scientific Computing Center and the
+Molecular Foundry, DOE Office of Science User Facilities
+supported by the Office of Science, U.S. Department of
+Energy under Contract No. DEAC02-05CH11231. The
+work performed at the Molecular Foundry was supported
+by the Office of Science, Office of Basic Energy Sciences,
+of the U.S. Department of Energy under the same con-
+tract.
+∗ sgriffin@lbl.gov
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diff --git a/jdE3T4oBgHgl3EQf5Au-/content/tmp_files/load_file.txt b/jdE3T4oBgHgl3EQf5Au-/content/tmp_files/load_file.txt
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+page_content='Broad-Range Directional Detection of Light Dark Matter in Cryogenic Ice  Nora Taufertsh¨ofer,1, 2, 3 Maurice Garcia-Sciveres,4 and Sin´ead M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content=' ∗  1Molecular Foundry Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' California 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' USA  2Materials Sciences Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' California 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' USA  3Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' University of W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Am Hubland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' D-97074 W¨urzburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Germany  4Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' California 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' USA  (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 2023)  We propose hexagonal ice (H2O) as a new target for light dark matter (DM) direct detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Ice,  a polar material, is suitable for single phonon detection through DM scattering for which we consider  light dark photon and light scalar mediator models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We report a rate sensitivity down to a DM  mass of ∼ keV, constituting a broader mass range than other promising candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We find bet-  ter sensitivity for near-term experimental thresholds from the presence of high-frequency phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  These advantages, and ice’s availability, make it highly promising for single-phonon detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  New proposals for the direct detection of dark mat-  ter (DM) have steadily pushed into the sub-GeV ‘light’  DM range[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' In particular, for DM masses in the kev-  GeV range, freeze-in DM scattering is well motivated  through both a light scalar mediator and a dark photon  mediator[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The smaller kinetic energy for this lower  mass DM motivates the exploration of low-energy excita-  tions in condensed matter systems as possible direct de-  tection channels[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Recent proposals for DM detection  in crystal targets include quasiparticle excitations rang-  ing from phonon and electron scattering in conventional  semiconductors[5, 6] to more exotic quantum materials  such as Dirac semimetals[7], and axions in topological  insulators and multiferroics[8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Of these proposals for DM direct detection via quasi-  particle scattering, single phonon production presents a  promising detection scheme for several reasons[6, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Typical optical phonons in crystals range from 10-100  meV, making them kinematically matched with sub-  MeV DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Dark photon mediators can mix with standard  model photons resulting in a coupling to optical phonon  modes in polar semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Furthermore, common  polar semiconductors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' GaAs) have sizeable bandgaps  resulting in small screening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The intrinsic anisotropy of  polar semiconductors allows for a directional response  giving signal discrimination based on daily/annual mod-  ulation of the DM wind[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Importantly, high-quality  detector-grade crystals of a range of polar semiconduc-  tors are readily available owing to their use in microelec-  tronics and quantum computing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' GaAs[6], SiC[12]),  and phonon sensing technologies are steadily improving  towards the low sensitivities required for detection[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  With the next-generation of low-mass DM experiments  in planning, it remains to identify the best possible tar-  get material for each detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Previous work  addressed this at the DM-target interaction level by  performing a comparison of DM-phonon reach, isolat-  ing materials-specific factors that optimize for low-mass  sensitivity and greatest cross section[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Beyond this,  other materials factors for target selection should also  be considered that take into account current synthe-  sis/fabrication capabilities and sensing constraints for  near-term experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' While experimental thresholds  for phonon sensing continue to improve towards the sin-  gle phonon excitation regime (∼10-100 meV), near-term  experiments with larger thresholds could access higher  energy responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For instance, beyond the single phonon  excitation regime, multiphonon responses can be ac-  cessed for DM scattering above the single phonon spectra  though at a suppressed rate[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' However, an alternative  approach to this is to search for single phonon and other  quasiparticle responses that lie within the current sensing  thresholds (>100 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  In this Letter, we propose cryogenic water ice (H2O)  as a particularly appealing target material for direct de-  tection of low-mass DM from single-phonon excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We find that the mixed nature of bonding in H2O re-  sults in a broad range of phonon frequencies suitable for  both lower-mass detection, and for higher frequency sin-  gle phonon excitations that are within the range of cur-  rent sensing thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Moreover, with the ubiquity and  long history of fabrication and characterization of solid  H2O, it can be readily fabricated as large single crystals  comprising earth-abundant (inexpensive) elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  This paper is laid out as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' we summarize the  calculation of spin-independent DM scattering for single-  phonons and the features of ice H2O (specifically its XIh  polymorph) of relevance to DM-phonon interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We  then present our calculations of the reach for a light dark  photon mediator and a light scalar mediator with com-  parisons to best-performing targets for these DM mod-  els, and their calculated directional dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Finally  we discuss other material limiting considerations for tar-  get materials and prospects for its use for near-term and  next-generation DM detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  First, we briefly summarize the formalism for calcu-  lating spin-independent DM scattering processes includ-  ing a light dark photon mediator and a light scalar  mediator[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For a detector with target density ρT the  total rate per target mass is given by  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='04778v1  [hep-ph]  12 Jan 2023  2  R = 1  ρT  ρχ  mχ  �  d3vfχ(v)Γ(v)  (1)  where ρχ is the local DM energy density, mχ the DM  mass, and fχ(v) is the DM velocity distribution in the  lab frame which is modelled with a boosted Maxwell-  Boltzmann distribution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Γ(v) is the scattering rate  per non-relativistic DM particle which is defined by an  integral over the momentum transfer q = p − p′ with  initial DM momentum p = mχv:  Γ(v) = π¯σ  µ2  �  d3q  (2π)3 (q0/q)4S(q, ωq)  (2)  Here, ¯σ is a model-dependent reference cross section  and we define ¯σ :=  µ2  π |M(q0)|2 with M the target-  independent 2 → 2 scattering matrix element (see Sup-  plemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' µ is the reduced mass of an elec-  tron or nucleon and DM particle for light-dark-photon-  or light-scalar-mediated scattering, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' q0 is a  reference momentum and S(q, ωq) the target-dependent  dynamic structure factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  For single phonon excitations we recall the dynamical  structure factor:  S(q, ωq) =  π  Ω  �  ν  1  ων,k  ���  �  j  e−Wj(q)  √mj  eiG·x0  j (Yj · ϵ∗  ν,j,k)  ���  2  δ(ωq − ων,k)  (3)  where ωq = q · v −  q2  2mχ explicitly depends on the energy  deposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  The sums run over all phonon branches ν  and over all ions j in the primitive cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The ionic masses  are mj with equilibrium positions x0  j and Ω is the prim-  itive cell volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' ϵν,j,k defines the phonon polarization  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' G is the reciprocal lattice vector that satisfies  G = q−k with k within the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' DM cou-  plings come in with the model-specific Yj terms defined  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' In the continuum limit for k, the Debye-Waller  factor is given by:  Wj(q) =  Ω  4mj  �  ν  �  1BZ  d3k  (2π)3  |q · ϵν,j,k|2  ων,k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  (4)  The materials-dependent quantities, namely the phonon  frequencies ων,k for branch ν and momentum k, and the  1 Due to Earth’s rotation the orientation of the incoming DM par-  ticles with regard to the target crystal changes over a day, which  leads to a directional modulation of the rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We choose the  z-axis in the crystal frame to be parallel to Earth’s velocity at  daytime t = 0[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  phonon polarization vectors ϵν,j,k, can be calculated us-  ing first-principles methods such as Density Functional  Theory[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Finally, we consider two well-motivated models of light  DM, namely a kinematically mixed dark photon and a  light scalar mediator[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' In the former case, the standard  model photon kinematically mixes with the dark photon  owing to their same quantum numbers, resulting in ‘mil-  licharged’ DM under U(1) of the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' In the  q → 0 limit valid for light DM scattering[16], Yj is given  by  Yj,dark photon = −  q2  q · ε∞ · q(q · Z∗  j)  (5)  where ε∞ is the high-frequency dielectric tensor and Z∗  j  is the Born effective charge tensor of the jth atom in  the unit cell, both of which can be calculated using first-  principles methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We also consider DM that only couples to nucleons via  a light scalar mediator with identical coupling for proton  and neutron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The now scalar-valued Yj is given by  Yj,hadrophilic scalar = AjFNj(q)  (6)  with Aj the atomic mass number and FNj an isotropic  nuclear form factor of the jth atom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The  derivation of (6) is valid for the q → 0 limit, which is  fulfilled for the single phonon excitation regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Further,  for this type of scattering FNj can be set to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  The pervasiveness of liquid and solid H2O has gener-  ated extensive studies of the phase diagram of ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' So far,  twenty polymorphs of crystalline H2O have been identi-  fied under various conditions of temperature and pres-  sure, with the most recent, ice XIX reported in 2021[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Common to all of these is the local atomic-scale arrange-  ment of hydrogen atoms in bonded units that fulfil the  ‘Bernal-Fowler ice rules’: each oxygen forms a covalent  bond with two hydrogens and a weaker van-der-Waals  bond with two other hydrogens[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Since each oxygen is  also shared between two hydrogen atoms, this results in  an average oxygen bonding environment comprising ‘two-  in’ (strong) and ‘two-out’ (weak) bonds with hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Such an atomic arrangement is geometrically frustrated  on a tetrahedral lattice[21], resulting in disordered rela-  tive arrangements of the strong and weak bonding net-  works known as proton disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Common ice (Ih) forms  such a hydrogen-disordered network of H-O-H bonds at  ambient pressure, giving rise to a residual entropy as first  predicted by Pauling[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  However, spontaneous ordering of the hydrogen net-  works can occur on cooling, or by the introduction of a  dopant to overcome the kinetic barrier to ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For  example KOH doping of Ih results in the formation of a  fully ordered phase of hexagonal ice, XIh below ∼ 72  K[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  In fact, the hydrogen ordering in the XIh  structure gives it a net dipole moment, making it po-  tentially ferroelectric[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Since XIh is the stable form  3  °  X S R A  Z °  Y X1A1 T Y  0  100  200  300  400  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' [meV]  Z  T  °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='04  °0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content='04  °  X S R A  Z °  Y X1A1 T Y  0  100  200  300  400  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content='aO5Wg</latexit>!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' [meV]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 1: (a) Primitive unit cell of hexagonal ice XIh and its first Brillouin zone with high-symmetry points labelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  (b) Calculated phonon band structure for hexagonal ice XIh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The cartoons on the right hand side show the  movement of ions for three selected phonon modes representative of the types of optical phonon modes in ice:  translational modes (bottom), libration/bending (middle), and stretch (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  of ice at ambient pressures, and can be synthesized in its  ordered form, we focus on this polymorph for the remain-  ing discussion, noting, however, that other structures of  ice should have comparable DM reach for single phonon-  based detection owing to their similar bonding networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We also considered heavy ice D2O XIh but we did not  find a significant difference to H2O (results reported in  the Supplemental Material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We now present our DFT-calculated structural and  phonon properties of ice XIh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The full calculation details  can be found in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  With its  combination of covalent bonding, hydrogen bonding, and  van-der-Waals bonding, the challenge of treating ice H2O  with DFT functionals has been explored extensively[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We benchmark our choice of exchange correlation func-  tional for accurate structural and vibrational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Consistent with previous work[27], we find that the van-  der-Waals corrected nonlocal functional (optPBE-vdw)  of Klimeˇs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' performs best[28, 29], resulting in the  lattice constants a = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='470 ˚A and c = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='212 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' This cor-  responds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='6% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='5% deviation from experimental  measurements at T = 2 K[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 1(b) we plot the calculated phonon dispersion  for ice XIh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' All polymorphs of ice exhibit a large range  of phonon frequencies as a result of the varied nature  of bonding in H2O crystals[31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The low-energy op-  tical phonon range is made up of ‘translational’ modes  where individual H2O molecules behave as atom-like clus-  ters, and vibrate with respect to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Such  low-frequency phonon modes are common in molecular  crystals such as H2O where molecular units are weakly  bonded to each other to form the crystal network[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  The next highest energy set of phonon modes corre-  sponds to libration and bending of H-O bonds, with the  highest-frequency range caused by the stretching of the  O-H bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' As the lightest element in the periodic table,  hydrogen sets an upper limit on the range of these high-  frequency stretch modes that can be found in a material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content='aL9TpvTVmLmX30A9bJ3ytkMI=</latexit>Freeze - In  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 2: Projected reach for light-dark-photon-mediated  DM scattering via single phonon excitations with  1kg-year exposure and no background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Solid,  long-dashed and dashed lines refer to a 1 meV, 20 meV  and 100 meV detector threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For comparison,  additional reach data [14] is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Stellar constraints  from red giants (RG) and the horizontal branch (HB)  are taken from [35], the freeze-in as referenced in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  mediated scattering of single phonons in ice XIh in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 2,  including comparisons to top-performing targets previ-  ously studied[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' All reach calculations were done us-  ing the code PhonoDark[36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The reach of ice XIh  extends further into both the low-mass DM range (well  into the constrained regions) and has better reach in the  high-mass DM range than any other previously suggested  single target material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' To understand the exceptionally  broadband sensitivity of ice XIh, we evaluate its perfor-  mance with respect to our previously suggested quality  factors for dark photon mediators in the high- and low-  frequency phonon ranges[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The lowest DM mass acces-  sible is determined by the energy of the lowest-frequency  optical mode where mχ ∼ 1  3ωmin  0  × 106, suggesting that  materials with low-lying optical phonons such as CsI  4  (ωmin  0  ∼ 7 meV) have the best low-mass reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' In our  case, the van-der-Waals bonded molecular units also have  very low lying optical modes (ωmin  0  ∼ 6 meV), making ice  XIh optimal for the low-mass DM range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We expect all  similar molecular crystals with such low-frequency trans-  lational phonon modes to have competitive reach for low-  mass DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  For the high-mass range, a quality factor, Q, is defined  as[14]:  Q ≡  Z∗2  A1A2ε2∞  �meV  ωLO  �  (7)  where Z∗ are the Born effective charges, A1, A2 are the  atomic masses, ε∞ is the high-frequency dielectric con-  stant, and ωLO is the longitudinal optical phonon fre-  quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  A high Q corresponds to better reach in the  high-mass DM regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We calculate each of these values  for ice XIh using DFT, with the full set of results given in  the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The best performing materi-  als previously proposed include LiF with Q = 270×10−7  and SiO2 with Q = 200 × 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For ice XIh, we find  Q = 533 × 10−7 for its highest-frequency stretch modes  (>375 meV), steadily increasing for lower-frequency ωLO  clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The substantially enhanced Q of ice XIh in the  high-mass range is due to the low masses of hydrogen and  oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' This optimized reach is dominated by hydrogen’s  extremely low mass, despite these small masses also re-  sulting in higher frequency phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' However, this  quality factor does not take the detection threshold into  account which is especially important for near-term ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We conclude that for optimal high-mass DM single-  phonon-based detection with dark photon mediators,  chemistries that include light elements such as H and  He, and especially H-H bonds, will give the best reach in  the high-mass range owing to their high-frequency sin-  gle phonon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' In the low-mass range, we find that  both compounds containing heavy elements and molec-  ular crystals that possess low-frequency optical phonon  modes will give the best sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' With ice combin-  ing both of these properties (molecular units and H-H  bonds), it displays better sensitivity for dark photon me-  diators than previously suggested targets across a broad  range of DM masses with CsI being the only exception  for a very small low-mass range (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  The reach for light-hadrophilic-scalar-mediated scat-  tering of single phonons in ice XIh is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 3  for several detector thresholds, and compared to previ-  ous top candidates from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For the light scalar  mediator, we also find that ice XIh has broad-band sen-  sitivity for both low- and high-mass DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Looking at the  reach curves, we find a competition between which of di-  amond or ice is favorable for different mass ranges and  thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For all thresholds considered, we find that  ice has sensitivity to lower masses than diamond, which  become comparable by a threshold of 100 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' In the  low mass range, the lowest DM mass sensitivity is gen-  erally governed by ωmin/cs where ωmin is the detector  threshold and cs is the material’s speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' There-  fore, materials with higher cs such as diamond will have  the best sensitivity to lower-mass DM given low detec-  tor thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We calculated the directional averaged  speed of sound of ice XIh to be 4376 ms−1 (details given  in the Supplemental Material), which is a factor of three  smaller than in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' However, for materials with  smaller cs the acoustic phonons can be kinematically in-  accessible, especially as the threshold increases to higher  energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' When this happens, the lowest DM sensitivity  is then set by the lowest optical phonon available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For  ice XIh we find that the minimum reachable DM mass  from kinematic considerations is ∼ 29 keV for a 1 meV  threshold, while the lowest optical phonon is 6 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' This  manifests as a change in the slope as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 3 where  optical phonons, rather than acoustic phonons, dominate  the low mass response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  In the high-mass,  high-threshold range (detector  threshold of 400 meV), we only include ice as it is the  only material with single phonons above ∼160 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We  note that the definition of ¯σn ∝ m−4  χ  leads to the gener-  ally better reach at high masses[38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We finally calculate the daily modulation rate assum-  ing that the z-axis of the target crystal is aligned with the  DM wind at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Each rate is normalised to the average  rate ¯R over one day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For a dark photon mediator we find  a strong modulation for a mass range of mχ = 104 − 105  eV (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 4 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We also find a significant daily modula-  tion for the scalar mediator case (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 4 (b)), especially  in the low 10 keV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' we extract the phonon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content='tw4Mms=">ACBHicbVDJSgNBEO1xjXGLesylMQgeQpiRoF6EgBePEcwCmWHo6VSJj0L3TViGHLw4q948aCI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='Vz/Cm39jZzlo4oOCx3tVNULEik02va3tbK6tr6xmdvKb+/s7u0XDg6bOk4VhwaPZazaAdMgRQNFCihnShgY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='SChFQyvJ37rHpQWcXSHowS8kPUj0ROcoZH8QjH0M5cPxJheUcembpm6CA+YDaE59gslu2JPQZeJMyclMkfdL3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='y53ZinIUTIJdO649gJehlTKLiEcd5NSMD1kfOoZGLATtZdMnxvTEKF3ai5WpCOlU/T2RsVDrURiYzpDhQC9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='6E/E/r5Ni79LRJSkCBGfLeqlkmJMJ4nQrlDAUY4MYVwJcyvlA6YR5Nb3oTgL68TJpnFe8Ur2tlmrleRw5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='UiTH5JQ45ILUyA2pkwbh5JE8k1fyZj1ZL9a79TFrXbHmM0fkD6zPH9eAluU=</latexit>m� = 10 keV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='<latexit sha1_base64="GCl674M+FiyTDYVAvji1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='6eR+lA=">ACBHicbVDJSgNBEO2JW4zbqMdcGoPgIYQZictFCHjxGMEskAmhp1NJmvQsdNeIYcjBi7/ixYMi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='Xv0Ib/6NneWgiQ8KHu9VUVXPj6XQ6DjfVmZldW19I7uZ29re2d2z9w/qOkoUhxqPZKSaPtMgRQg1FCihGStg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='S+h4Q+vJ37jHpQWUXiHoxjaAeuHoic4QyN17HzQST0+EGN6Rd0z6hWph/CA6RDq45dcErOFHSZuHNSIHNUO/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='aX1414EkCIXDKtW64TYztlCgWXM5iYaY8SHrQ8vQkAWg2+n0iTE9NkqX9iJlKkQ6VX9PpCzQehT4pjNgONC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='L3kT8z2sl2LtspyKME4SQzxb1EkxopNEaFco4ChHhjCuhLmV8gFTjKPJLWdCcBdfXib105J7XirflguV4jyO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='LMmTI3JCXHJBKuSGVEmNcPJInskrebOerBfr3fqYtWas+cwh+QPr8wfWpbq</latexit>m� = 15 keV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='<latexit sha1_base64="pqvUqeTIaDHUY9K24I0tJ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='IAtmJE=">ACBHicbVA9SwNBEN2LXzF+RS3TLAbBIoS7ENRGCNhYRjAfkAthbzNJlux9sDsnhiOFjX/FxkIR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='W3+Enf/GTXKFJj4YeLw3w8w8L5JCo21/W5m19Y3Nrex2bmd3b/8gf3jU1GsODR4KEPV9pgGKQJoEAJ7UgB8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='z0JLW98PfNb96C0CIM7nETQ9dkwEAPBGRqply/4vcTlIzGlV7RiU7dEXYQHTMbQnPbyRbtsz0FXiZOSIklR7+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='W/3H7IYx8C5Jp3XHsCLsJUyi4hGnOjTVEjI/ZEDqGBswH3U3mT0zpqVH6dBAqUwHSufp7ImG+1hPfM50+w5F  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='e9mbif14nxsFlNxFBFCMEfLFoEuKIZ0lQvtCAUc5MYRxJcytlI+YhxNbjkTgrP8ipVsrOebl6Wy3WSmkc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='WVIgJ+SMOSC1MgNqZMG4eSRPJNX8mY9WS/Wu/WxaM1Y6cwx+QPr8wfZE5bm</latexit>m� = 20 keV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='<latexit sha1_base64="IJklCWo5KhA1TtbB8H3du  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='kRHp6w=">ACBHicbVDJSgNBEO2JW4zbqMdcGoPgIYQZDepFCHjxGMEskAmhp1NJmvQsdNeIYcjBi7/ixYMi  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='Xv0Ib/6NneWgiQ8KHu9VUVXPj6XQ6DjfVmZldW19I7uZ29re2d2z9w/qOkoUhxqPZKSaPtMgRQg1FCihGStg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='S+h4Q+vJ37jHpQWUXiHoxjaAeuHoic4QyN17HzQST0+EGN6Rc8c6hWph/CA6RDq45dcErOFHSZuHNSIHNUO/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='aX1414EkCIXDKtW64TYztlCgWXM5iYaY8SHrQ8vQkAWg2+n0iTE9NkqX9iJlKkQ6VX9PpCzQehT4pjNgONC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='L3kT8z2sl2LtspyKME4SQzxb1EkxopNEaFco4ChHhjCuhLmV8gFTjKPJLWdCcBdfXib105J7XirflguV4jyO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='LMmTI3JCXHJBKuSGVEmNcPJInskrebOerBfr3fqYtWas+cwh+QPr8wfapbn</latexit>m� = 30 keV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='<latexit sha1_base64="rEF8SoY8PHwxr8kEtBA5r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='IzKoHE=">ACBHicbVDJSgNBEO1xjXGLesylMQgeQpiRoF6EgBePEcwCmWHo6VSJj0L3TViGHLw4q948aCI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='Vz/Cm39jZzlo4oOCx3tVNULEik02va3tbK6tr6xmdvKb+/s7u0XDg6bOk4VhwaPZazaAdMgRQNFCihnShgY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='SChFQyvJ37rHpQWcXSHowS8kPUj0ROcoZH8QjH0M5cPxJhe0apN3TJ1ER4wG0Jz7BdKdsWegi4TZ05KZI6X/  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='hyuzFPQ4iQS6Z1x7ET9DKmUHAJ47ybakgYH7I+dAyNWAjay6ZPjOmJUbq0FytTEdKp+nsiY6HWozAwnSHDgV7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='0JuJ/XifF3qWXiShJESI+W9RLJcWYThKhXaGAoxwZwrgS5lbKB0wxjia3vAnBWXx5mTPKs5pXpbLdXK8zhy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='pEiOySlxyAWpkRtSJw3CySN5Jq/kzXqyXqx362PWumLNZ47IH1ifP9w5lug=</latexit>m� = 40 keV  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' 4: Daily modulation for H2O XIh ice with (a) a light dark photon and (b) a light scalar mediator for various mχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  lifetimes τ from measured phonon linewidths Γph of ice  XI[31, 39] from τ = (πΓph)−1 giving a lifetime range of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='5 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='1 ps (see Supplemental Material), consistent with  other systems with high-frequency optical phonons[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Our results suggest that ice H2O is a competitive can-  didate target for single-phonon based light DM detection  both in the low-mass and high-mass range (here we con-  sidered ice XIh, but our conclusions are generally appli-  cable to other ice polymorphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Molecular units provide  ultra-low-mass optical phonons (down to 6 meV) giving  excellent reach in the low-mass DM range for dark pho-  ton mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' We expect similar reach to low masses  for polar semiconductors containing heavy ions (as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  CsI) or molecular crystals where the relative effective  masses of the ions/molecules result in low-frequency op-  tical phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' For near-term experiments, we find that  the broad range of single phonons in ice up to 400 meV  gives it enhanced reach for higher experimental thresh-  olds compared to alternative proposals such as multi-  phonon excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  However, the lower density of ice  due to its van-der-Waals bonding results in a reduced  reach compared to other solid-state targets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' this can be  addressed by the growth of large single crystals of ice  which is already well explored[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Finally, while ice H2O  is cheap, earth-abundant, and has been extensively char-  acterized, it remains an engineering challenge as to the  best way to incorporate a material that is a liquid at room  temperature into a (cryogenic) detector architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  We thank Dan Carney, Tanner Trickle, Kathryn Zurek,  Katherine Inzani and Thomas Harrelson for helpful dis-  cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  This work was supported by the US De-  partment of Energy under contract DEAC02-05CH11231  and Quantum Information Science Enabled Discovery  (QuantISED) for High Energy Physics grant KA2401032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  Computational resources were provided by the National  Energy Research Scientific Computing Center and the  Molecular Foundry, DOE Office of Science User Facilities  supported by the Office of Science, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Department of  Energy under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' DEAC02-05CH11231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' The  work performed at the Molecular Foundry was supported  by the Office of Science, Office of Basic Energy Sciences,  of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Department of Energy under the same con-  tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  ∗ sgriffin@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='gov  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Essig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Mardon, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Volansky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Hall, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content=' March-Russell,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  West, Journal of High Energy Physics 2010, 1 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content=' Heikinheimo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Tenkanen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Tuomi-  nen, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Vaskonen, International Journal of Modern  Physics A 32, 1730023 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Kahn and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Lin, Reports on Progress in Physics  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content='  [5] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Essig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Fernandez-Serra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
+page_content=' Mardon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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+page_content=' Volansky,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdE3T4oBgHgl3EQf5Au-/content/2301.04778v1.pdf'}
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diff --git a/jdFAT4oBgHgl3EQfah0B/content/tmp_files/2301.08551v1.pdf.txt b/jdFAT4oBgHgl3EQfah0B/content/tmp_files/2301.08551v1.pdf.txt
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+Transmission-based noise spectroscopy for quadratic qubit-resonator interactions
+Philipp M. Mutter∗ and Guido Burkard†
+Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
+We develop a theory describing the transient transmission through noisy qubit-resonator systems
+with quadratic interactions as are found in superconducting and nanomechanical resonators cou-
+pled to solid-state qubits. After generalizing the quantum Langevin equations to arbitrary qubit-
+resonator couplings, we show that only the cases of linear and quadratic couplings allow for an
+analytical treatment within standard input-output theory. Focussing for the first time on quadratic
+couplings and allowing for arbitrary initial qubit coherences, it is shown that noise characteristics
+can be extracted from input-output measurements by recording both the averaged fluctuations in
+the transmission probability and the averaged phase. Our results represent an extension to the field
+of transmission-based noise spectroscopy with immediate practical applications.
+I.
+INTRODUCTION
+We live in the age of noisy intermediate-scale quan-
+tum devices in which noise severely affects the coherence
+of qubits and thereby the performance of quantum com-
+puters [1]. The noise can have several origins but may
+roughly be categorized according to whether it arises as
+a consequence of imperfect system control or unwanted
+interactions with the environment.
+For instance, ran-
+dom electromagnetic fields due to fluctuating control gate
+voltages belong to the former class [2], while noise arising
+from various two-level fluctuators in the host material [3],
+the interaction with the nuclear spin bath in semiconduc-
+tor quantum dots [4–6], as well as magnetic flux noise and
+quasiparticles in superconducting circuits [7–10] can be
+assigned to the latter class.
+Common to all types of noise is their detrimental effect
+on the coherence of engineered quantum systems. To mit-
+igate noise-induced decoherence one requires knowledge
+of the spectral form of the fluctuations which is encoded
+in the frequency-dependent noise power spectral density
+S(ω) [11, 12]. Experiments assessing the effect of charge
+noise on a semiconductor double quantum dot by exam-
+ining the long-time resonator transmission have already
+been performed [13], and recently spectroscopy meth-
+ods for classical and quantum noise based on measure-
+ments of the transient transmission through a linearly
+coupled qubit-resonator system were developed [14, 15].
+Linear light-matter interactions involving the exchange
+of one photon are relevant in many fundamentally and
+technologically relevant systems, such as atoms in a cav-
+ity [16, 17] as well as charge qubits coupled to supercon-
+ducting transmission lines [18, 19]. Additionally, linear
+spin-photon couplings are made possible by the spin-orbit
+interaction [20–24] or the use of micromagnets [25, 26].
+Quadratic qubit-resonator couplings also allow for
+input-output measurements and are found in diverse
+physical systems. For instance, nanomechanical and op-
+tomechanical resonators have been shown to couple to
+∗ philipp.mutter@uni-konstanz.de
+† guido.burkard@uni-konstanz.de
+noisy qubit-
+resonator system
+⟨bin⟩
+⟨bout⟩
+HI ∝ a2σ+ + a†2σ−
+HI ∝ aσ+ + a†σ−
+κj
+|0⟩
+|1⟩
+ωq
+γ
+ωr
+port j
+g
+δωq = λδX
+FIG. 1. Potential systems for transmission-based noise spec-
+troscopy. A two-level system (qubit, blue) is affected by noise
+leading to fluctuations δωq in the energy splitting ωq between
+qubit states ∣0⟩ and ∣1⟩.
+The interaction with a single res-
+onator mode is described by the Hamiltonian HI containing
+the qubit and resonator operators σ± and a. We schematically
+show an optical resonator with linear qubit-mode couplings
+on the left and a nanomechanical resonator with quadratic
+qubit-mode couplings on the right. Both types of interactions
+allow for the characterization of noise features by studying
+the transmission A = ⟨bout⟩/⟨bin⟩ through the noisy qubit-
+resonator system. The bottom schematic introduces the pho-
+ton loss rate κj at the resonator port j, the qubit decay rate
+γ and the qubit-resonator coupling strength g.
+superconducting and spin qubits, and the characteristic
+feature is the quadratic qubit-resonator interaction in-
+cluding two-photon processes [27–31]. Quadratic qubit-
+resonator interactions can also be engineered in super-
+conducting circuits [32, 33], and earlier studies of such
+systems investigated thermal transport [34] and input-
+output theoretical transmission features in the noise-free
+steady-state case [35]. If one is interested in transmission-
+based noise spectroscopy, it is desirable to have a detailed
+knowledge about the input-output behaviour of both the
+linear and quadratic cases as the coupling of a given type
+of qubit to resonator modes may more easily be realized
+arXiv:2301.08551v1  [cond-mat.mes-hall]  20 Jan 2023
+
+2
+in one system or the other. In this paper we generalize
+the results in Ref. [14] to also include quadratic qubit-
+resonator interactions and show that noise characteristics
+can be inferred by measuring both the averaged fluctua-
+tions in the transmission probability δ⟪∣A∣2⟫/∣A∞∣2 and
+the averaged phase ⟪φ⟫. Moreover, as another extension
+of earlier results we allow for a general form of the initial
+qubit coherence ⟨σ0
+−⟩.
+The remainder of this paper is structured as follows: In
+Sec. II we consider a general n-photon-qubit interaction
+and derive the Langevin equations for this case. We then
+show that within standard input-output theory only the
+cases n = 1 and n = 2 allow for an analytical solution to
+lowest order in perturbation theory. Specializing on these
+cases we calculate the averaged transmission probability
+and phase in Sec. III, and propose a scheme for extracting
+noise characteristics in Sec. IV. Finally, Sec. V provides
+a conclusion.
+II.
+GENERAL MODEL AND LANGEVIN
+EQUATIONS
+We consider a resonator mode of frequency ωr that
+couples to a qubit with fluctuating energy separation
+ωq + δωq(t), see Fig. 1 (bottom). The noise δX affect-
+ing the control parameter X is related to the fluctuating
+energy separation by the first-order expansion δωq = λδX
+with the noise sensitivity λ = ∂Xωq∣δX=0. The number n
+of photons taking part in the interaction is left unspec-
+ified for now, and we assume a general qubit-resonator
+Hamiltonian of the form
+Hqr = ωq + δωq(t)
+2
+σz + ωra†a + g(a + a†)nσx,
+(1)
+where σx,z are Pauli matrices in the basis of the logi-
+cal qubit states {∣0⟩,∣1⟩}, and a is the mode annihilation
+operator. In the rotating frame defined by the transfor-
+mation Hqr → UHqrU † + i ˙UU † with the time-dependent
+unitary U = exp(iΩt[a†a/n+σz/2]) and within the rotat-
+ing wave approximation, the system is described by the
+Hamiltonian
+Hqr = ∆q + δωq(t)
+2
+σz + ∆(n)
+r
+a†a + g [anσ+ + a†nσ−], (2)
+where σ± are the qubit ladder operators, and the quan-
+tities ∆q = ωq − Ω and ∆(n)
+r
+= ωr − Ω/n are the shifted
+qubit and resonator frequencies.
+In addition, external bath modes b interact with the
+resonator, and the coupling is assumed to be linear. We
+take into account the effects of the corresponding input
+field ⟨bin(t)⟩ by including the injection Hamiltonian in
+the rotating frame,
+Hin = i√κ1 (⟨bin(t)⟩eiΩt/na† − H.c.),
+(3)
+The input field ⟨bin(t)⟩ may be used to probe the system,
+and it is assumed to be a classical wave of real amplitude
+⟨bin⟩ and frequency ωp, ⟨bin(t)⟩ = ⟨bin⟩e−iωpt. In the fol-
+lowing we work in the frame defined by U as above with
+Ω = nωp such that the the injection Hamiltonian (3) is
+constant in time. One then has ∆q ≡ ∆(n)
+q
+= ωq −nωp and
+∆(n)
+r
+≡ ∆r = ωr − ωp.
+Furthermore taking into account noise-independent
+qubit relaxation, absorption and dephasing at finite tem-
+perature as well as photon losses, the system may be
+described by a Lindblad master equation in the rotating
+frame,
+˙ρ = −i[Hqr + Hin,ρ] + L[ρ],
+L[ρ] = γ1
+2 ∑
+±
+1 ± 1 + 2nB
+2
+(2σ∓ρσ± − {ρ,σ±σ∓})
++ γϕ
+2 (σzρσz − ρ) + κ
+2 (2aρa† − {ρ,a†a}),
+(4)
+where γ1 is the relaxation rate at zero Kelvin, γϕ is the
+dephasing rate and nB = [exp(ωq/T) − 1]−1 is the Bose
+distribution, i.e., the mean number of bath photons at
+temperature T, evaluated at the qubit frequency where
+resonant energy exchange is possible. Additionally, we
+have introduced the total photon loss rate κ = ∑j κj +κint
+as the sum of the individual loss rates at port j ∈ {1,2}
+and the internal loss rate κint.
+By using the relation ⟨ ˙O⟩ = Tr( ˙ρO) for a given operator
+O and Eq. (4), one obtains the partial system of Langevin
+equations to leading order in g,
+˙⟨a⟩ = −[i∆r + κ
+2 ]⟨a⟩ − ign⟨σ−⟩⟨an−1⟩∗ + √κ1⟨bin⟩,
+˙
+⟨σz⟩ = −γ1(T)⟨σz⟩ − γ1(0) + 2ig (⟨an⟩⟨σ−⟩∗ − c.c.),
+˙
+⟨σ−⟩ = −[i∆(n)
+q
++ iλδX(t) + γ2]⟨σ−⟩ + ig⟨σz⟩⟨an⟩,
+(5)
+where γ1(T) = γ1 coth(ωq/2T), γ2 = γ1(T)/2 + γϕ, the
+star denotes complex conjugation, and we have used the
+relation ⟨QR⟩ = ⟨Q⟩⟨R⟩ + O(g) that is valid for all qubit
+(Q) and resonator (R) operators under the assumption
+of an initially separable state. In the following we define
+γ = 2γ2 such that the homogeneous parts of the Langevin
+equations for ⟨σ−⟩ and ⟨a⟩ have the same form, allowing
+us to obtain cleaner mathematical expressions later on.
+A.
+Analytical approach
+The system of Langevin equations (5) is not closed be-
+cause in general ⟨an⟩ ≠ ⟨a⟩n, and even if it the relation
+would hold true, an exact solution in the presence of arbi-
+trary time-dependent qubit noise could not be obtained.
+However, we may formally solve the equation for ⟨σ−⟩
+without any assumptions on the noise δX(t),
+⟨σ−(t)⟩ = ⟨σ0
+−⟩e−i∆(n)
+q
+t−γt/2e−iλX (t)
+(6)
++ ig ∫
+t
+0 ⟨σz(t′)⟩⟨an(t′)⟩e(i∆(n)
+q
++γ/2)(t′−t)eiλ[X (t′)−X (t)]dt′,
+
+3
+with the integrated noise X(t) = ∫
+t
+0 δX(t′)dt′ and the
+initial qubit coherence ⟨σ0
+−⟩, and subsequently insert the
+result into the equation for ⟨a⟩. By again formally solv-
+ing the resulting differential equation, we find an integral
+equation for the expectation value of the mode annihila-
+tion operator,
+⟨a(t)⟩ =
+√κ1⟨bin⟩
+i∆r + κ/2 (1 − e−i∆rt−κt/2) − ing⟨σ0
+−⟩e−i∆rt−κt/2 ∫
+t
+0 dt′⟨an−1(t′)⟩∗ei(∆r−∆(n)
+q
+)t′+(κ−γ)t′/2e−iλX (t′)
+(7)
++ ng2e−i∆rt−κt/2 ∫
+t
+0 dt′⟨an−1(t′)⟩∗ei(∆r−∆(n)
+q
+)t′+(κ−γ)t′/2e−iλX (t′) ∫
+t′
+0
+dt′′⟨σz(t′′)⟩⟨an(t′′)⟩ei∆(n)
+q
+t′′+γt′′/2e−iλX (t′′).
+where we assume ⟨a0⟩ = 0, e.g., the resonator may ini-
+tially be in an eigenstate of the mode number operator
+a†a. We first remark that if we wish to work to lead-
+ing order in g, we may neglect the second integral in (7),
+and hence the time-dependent populations ⟨σz(t)⟩ do not
+play a role. Secondly, if n ⩽ 2, then the ⟨an−1⟩ term in the
+remaining integral is equal to unity (n = 1) or ⟨a⟩ (n = 2),
+and the equation can be solved to first order in pertur-
+bation theory by substituting the zeroth-order value of
+⟨a⟩ into the first integral.
+Otherwise, the fact that in
+general ⟨an−1⟩ ≠ ⟨a⟩n−1 forces us to obtain an additional
+equation of motion for ⟨an−1⟩ which cannot be treated
+in a simple input-output model without further assump-
+tions due to additional terms of the form ∫ ⟨b(ω)an−2⟩dω
+which feature the bath modes b(ω) but cannot be related
+to the input fields. In the long-time limit the term linear
+in g vanishes, and the long-time transmission is deter-
+mined by the g2 term. Since this term features the quan-
+tity ⟨an⟩, the long-time solution can only be obtained for
+n = 1 due to the same arguments made above. A compre-
+hensive study of the long-time transmission for linearly
+coupled systems may be found in Ref. [36].
+Since the
+coupling of the resonator to the external bath modes is
+assumed to be linear, the standard input-output relation
+for the transmission amplitude A applies [37–39],
+A(t) = ⟨bout(t)⟩
+⟨bin(t)⟩ = −
+√κ2⟨a(t)⟩
+⟨bin(t)⟩ .
+(8)
+As a consequence, the transmission amplitude can be ob-
+tained by solving Eq. (7) for ⟨a(t)⟩.
+B.
+Numerical study
+To validate the truncation of the quantum Langevin
+equations (5) and the solution for the transmission at
+first order in the qubit-photon coupling g, we numerically
+solve the full Lindblad equation (4) and compare the re-
+sults to our approximate analytical expressions in Figs. 2
+and 3. We allow for up to 12 photons and assume an ini-
+tially empty resonator containing a qubit in a coherent
+superposition ∣ψ⟩ = (eiπ/8 ∣↑⟩+e−iπ/8 ∣↓⟩)/
+√
+2 such that the
+initial conditions ⟨a0⟩ = 0, Re⟨σ0
+−⟩ = Im⟨σ0
+−⟩ = 1/
+√
+8 and
+⟨σ0
+z⟩ = 0 are satisfied. The transmission amplitude is then
+calculated according to Eq. (8) as A = −√κ2Tr(ρa)/⟨bin⟩,
+and the averaged transmission and phase are obtained
+by averaging the solution over 103 distinct noise config-
+urations. We generally find excellent agreement in the
+regime considered, underlining the high accuracy of our
+analytical results.
+III.
+OBSERVABLE FIGURES OF MERIT
+We now focus on the cases n ⩽ 2 that can be treated
+perturbatively.
+The case n = 1 has been studied in
+Ref. [14], and we include it here to be able to make
+contact to these previous considerations.
+In addition,
+we complement earlier results by including the averaged
+phase ⟪φ⟫ as an observable figure of merit and allowing
+for arbitrary complex initial qubit coherences ⟨σ0
+−⟩. The
+case n = 2 constitutes a completely new contribution to
+the theory of transmission-based noise spectroscopy, and
+it is of immediate interest for nanomechanical and super-
+conducting resonators coupled to solid-state qubits.
+The complex transmission amplitude can be written as
+A = ∣A∣eiφ,
+(9)
+where both the transmission ∣A∣ and the phase φ are ex-
+perimentally observable and thus allow for a noise aver-
+age ⟪...⟫ over many measurements. Solving Eq. (7) to
+first order in the small parameter ε(n) ≡ ng/max{∣δ(n)
+0
+∣ ≡
+∣∆r − ∆(n)
+q
+∣,∣κ − γ∣}, we find according to Eq. (8),
+An(t)
+A∞
+= 1 − e−i∆rt−κt/2
+(10)
+× [1 + ign⟨σ0
+−⟩κ/2 + i∆r
+κ/2 − i∆r
+(κ/2 − i∆r
+√κ1⟨bin⟩ )
+2−n
+In(t)],
+with the zeroth-order long-time transmission amplitude
+A∞ = −√κ1κ2/(i∆r + κ/2) and the noise integrals
+In(t) =∫
+t
+0 ei(ωr−ωq)t′+(κ−γ)t′/2
+× (eiωpt′ − eiωrt′−κt′/2)
+n−1
+e−iλX (t′)dt′.
+(11)
+
+4
+Δr = − 0.4δ(n)
+0
+δ(n)
+0 t
+Δr = − 0.2δ(n)
+0
+Δr = 0
+λ ≠ 0
+λ = 0
+analytical
+numerical
+0
+50
+0
+−4
+4
+0
+2
+−2
+0
+2
+−2
+(a)
+(b)
+(c)
+ng = 0.002δ(n)
+0
+alsoforphase
+�����
+��
+�
+��
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+��
+�
+��
+�
+��
+�
+��
+�
+���
+���������������
+0
+10
+20
+30
+40
+50
+-0.0003
+-0.0002
+-0.0001
+0.0000
+0.0001
+0.0002
+0.0003
+����
+�
+��
+�
+��
+�
+��
+�
+��
+�
+��
+�
+��
+�����������������������������
+0
+10
+20
+30
+40
+50
+-0.003
+-0.002
+-0.001
+0.000
+0.001
+0.002
+����
+�
+��
+�
+��
+�
+�
+�
+��������������������������������������
+0
+10
+20
+30
+40
+50
+-0.006
+-0.004
+-0.002
+0.000
+0.002
+0.004
+6
+In this case the ANI is described by the same functional
+form as in Eq. (25) with the substitution �X2
+rms = P￿⇡,
+where P = ∫ S(!)d! is the noise power in the positive
+frequency band [!ir,!uv] under consideration. The in-
+formation on the noise attainable by using the above ex-
+pressions hence includes the noise amplitude S0 for white
+noise, the root mean square of the noise �Xrms for qua-
+sistatic noise and the noise power P for low-frequency
+noise.
+C.
+Relating the noise integrals
+Finally, we wish to relate the n = 1 and n = 2 ANIs
+since we already know how to extract the power spec-
+tral density S from ￿I1￿ = ￿I￿ from the discussion in
+Ref. [32]. Comparing the expressions for I1 and I2 in
+Eq. (11), one finds the relation
+I2(t) = I1(t) − ￿
+t
+0 ei�rt′−t′￿2 ˙I1(t′)dt′,
+(26)
+where ˙I1(t′) = @tI1(t)￿t=t′. Eq. (26) can be inverted, and
+after averaging over the noise configurations, we obtain
+￿I1(t)￿ = ￿
+t
+0
+￿ ˙I2(t′)￿
+ei!pt − ei!rt′−t′￿2 dt′ =
+￿I2(t)￿
+ei!pt − ei!rt−t￿2
++ ￿
+t
+0
+￿I2(t′)￿￿i!pei!pt′ − ￿i!r − 
+2 ￿ei!rt′−t′￿2￿
+(ei!pt′ − ei!rt′−t′￿2)2
+dt′,
+(27)
+where the second equality is obtained via partial inte-
+gration.
+After converting the ANI ￿I2(t)￿ to ￿I1(t)￿
+via Eq. (27), one may follow the approach outlined in
+Ref. [32] to obtain noise characteristics from the ANI
+in the linearly coupled case.
+Hence, the power spec-
+tral density can also be obtained in quadratically cou-
+pled qubit-resonator systems. Noise features for common
+noise types can be obtained by reducing ￿I2￿ to ￿I1￿ as
+well, even though it may be easier to directly use the an-
+alytical expression for ￿I2￿ given Eqs. (23) and (25) in
+practice.
+V.
+CONCLUSION
+In this paper we have extended the theory of noise
+spectroscopy based on the transient transmission to in-
+clude quadratic qubit-resonator interactions, which oc-
+cur in nanonmechanical and superconducting resonators
+coupled to solid-state qubits. Allowing for arbitrary ini-
+tial qubit coherences, we derived and solved the quan-
+tum Langevin equations in the presence of a noisy qubit
+in first-order perturbation theory and showed that noise
+features are imprinted in the averaged resonator trans-
+mission and phase. Measuring both of these quantities
+permits the extraction of the ANI ￿I2￿ for quadrati-
+cally coupled systems. By converting ￿I2￿ to the ANI
+for linearly coupled systems, one may extract the noise
+power spectral density from transmission measurements
+using established methods. Our results expand the scope
+of applications of transmission based noise spectroscopy
+to a wide variety of systems including superconducting,
+charge and spin qubits coupled to electromagnetic cavi-
+ties and nanomechanical resonators.
+￿�2￿￿⇡
+￿�1￿￿⇡
+�￿￿A2￿2￿￿￿A∞￿2
+Appendix A: Real and imaginary parts of A
+In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-
+photon coupling (n = 1,2). Introducing the quantity ⇣ = arctan(2�r￿), they read
+Re(An) = −
+√12
+�2r + 2￿4￿
+2 − e−t￿2 ￿cos(�rt)
+2 − sin(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿cos([n − 1]⇣ − �rt) + ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿sin([n − 1]⇣ − �rt)￿￿,
+Im(An) = −
+√12
+�2r + 2￿4￿ − �r + e−t￿2 ￿sin(�rt)
+2 + cos(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿sin([n − 1]⇣ − �rt) − ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿cos([n − 1]⇣ − �rt)￿￿.
+(A1)
+[1] J. Preskill, Quantum Computing in the NISQ era and
+beyond, Quantum 2, 79 (2018).
+[2] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-
+shuler, 1/f noise: Implications for solid-state quantum
+103
+6
+In this case the ANI is described by the same functional
+form as in Eq. (25) with the substitution �X2
+rms = P￿⇡,
+where P = ∫ S(!)d! is the noise power in the positive
+frequency band [!ir,!uv] under consideration. The in-
+formation on the noise attainable by using the above ex-
+pressions hence includes the noise amplitude S0 for white
+noise, the root mean square of the noise �Xrms for qua-
+sistatic noise and the noise power P for low-frequency
+noise.
+C.
+Relating the noise integrals
+Finally, we wish to relate the n = 1 and n = 2 ANIs
+since we already know how to extract the power spec-
+tral density S from ￿I1￿ = ￿I￿ from the discussion in
+Ref. [32]. Comparing the expressions for I1 and I2 in
+Eq. (11), one finds the relation
+I2(t) = I1(t) − ￿
+t
+0 ei�rt′−t′￿2 ˙I1(t′)dt′,
+(26)
+where ˙I1(t′) = @tI1(t)￿t=t′. Eq. (26) can be inverted, and
+after averaging over the noise configurations, we obtain
+￿I1(t)￿ = ￿
+t
+0
+￿ ˙I2(t′)￿
+ei!pt − ei!rt′−t′￿2 dt′ =
+￿I2(t)￿
+ei!pt − ei!rt−t￿2
++ ￿
+t
+0
+￿I2(t′)￿￿i!pei!pt′ − ￿i!r − 
+2 ￿ei!rt′−t′￿2￿
+(ei!pt′ − ei!rt′−t′￿2)2
+dt′,
+(27)
+where the second equality is obtained via partial inte-
+gration.
+After converting the ANI ￿I2(t)￿ to ￿I1(t)￿
+via Eq. (27), one may follow the approach outlined in
+Ref. [32] to obtain noise characteristics from the ANI
+in the linearly coupled case.
+Hence, the power spec-
+tral density can also be obtained in quadratically cou-
+pled qubit-resonator systems. Noise features for common
+noise types can be obtained by reducing ￿I2￿ to ￿I1￿ as
+well, even though it may be easier to directly use the an-
+alytical expression for ￿I2￿ given Eqs. (23) and (25) in
+practice.
+V.
+CONCLUSION
+In this paper we have extended the theory of noise
+spectroscopy based on the transient transmission to in-
+clude quadratic qubit-resonator interactions, which oc-
+cur in nanonmechanical and superconducting resonators
+coupled to solid-state qubits. Allowing for arbitrary ini-
+tial qubit coherences, we derived and solved the quan-
+tum Langevin equations in the presence of a noisy qubit
+in first-order perturbation theory and showed that noise
+features are imprinted in the averaged resonator trans-
+mission and phase. Measuring both of these quantities
+permits the extraction of the ANI ￿I2￿ for quadrati-
+cally coupled systems. By converting ￿I2￿ to the ANI
+for linearly coupled systems, one may extract the noise
+power spectral density from transmission measurements
+using established methods. Our results expand the scope
+of applications of transmission based noise spectroscopy
+to a wide variety of systems including superconducting,
+charge and spin qubits coupled to electromagnetic cavi-
+ties and nanomechanical resonators.
+￿�2￿￿⇡
+￿�1￿￿⇡
+�￿￿A2￿2￿￿￿A∞￿2
+Appendix A: Real and imaginary parts of A
+In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-
+photon coupling (n = 1,2). Introducing the quantity ⇣ = arctan(2�r￿), they read
+Re(An) = −
+√12
+�2r + 2￿4￿
+2 − e−t￿2 ￿cos(�rt)
+2 − sin(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿cos([n − 1]⇣ − �rt) + ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿sin([n − 1]⇣ − �rt)￿￿,
+Im(An) = −
+√12
+�2r + 2￿4￿ − �r + e−t￿2 ￿sin(�rt)
+2 + cos(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿sin([n − 1]⇣ − �rt) − ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿cos([n − 1]⇣ − �rt)￿￿.
+(A1)
+[1] J. Preskill, Quantum Computing in the NISQ era and
+beyond, Quantum 2, 79 (2018).
+[2] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-
+shuler, 1/f noise: Implications for solid-state quantum
+103
+6
+In this case the ANI is described by the same functional
+form as in Eq. (25) with the substitution �X2
+rms = P￿⇡,
+where P = ∫ S(!)d! is the noise power in the positive
+frequency band [!ir,!uv] under consideration. The in-
+formation on the noise attainable by using the above ex-
+pressions hence includes the noise amplitude S0 for white
+noise, the root mean square of the noise �Xrms for qua-
+sistatic noise and the noise power P for low-frequency
+noise.
+C.
+Relating the noise integrals
+Finally, we wish to relate the n = 1 and n = 2 ANIs
+since we already know how to extract the power spec-
+tral density S from ￿I1￿ = ￿I￿ from the discussion in
+Ref. [32]. Comparing the expressions for I1 and I2 in
+Eq. (11), one finds the relation
+I2(t) = I1(t) − ￿
+t
+0 ei�rt′−t′￿2 ˙I1(t′)dt′,
+(26)
+where ˙I1(t′) = @tI1(t)￿t=t′. Eq. (26) can be inverted, and
+after averaging over the noise configurations, we obtain
+￿I1(t)￿ = ￿
+t
+0
+￿ ˙I2(t′)￿
+ei!pt − ei!rt′−t′￿2 dt′ =
+￿I2(t)￿
+ei!pt − ei!rt−t￿2
++ ￿
+t
+0
+￿I2(t′)￿￿i!pei!pt′ − ￿i!r − 
+2 ￿ei!rt′−t′￿2￿
+(ei!pt′ − ei!rt′−t′￿2)2
+dt′,
+(27)
+where the second equality is obtained via partial inte-
+gration.
+After converting the ANI ￿I2(t)￿ to ￿I1(t)￿
+via Eq. (27), one may follow the approach outlined in
+Ref. [32] to obtain noise characteristics from the ANI
+in the linearly coupled case.
+Hence, the power spec-
+tral density can also be obtained in quadratically cou-
+pled qubit-resonator systems. Noise features for common
+noise types can be obtained by reducing ￿I2￿ to ￿I1￿ as
+well, even though it may be easier to directly use the an-
+alytical expression for ￿I2￿ given Eqs. (23) and (25) in
+practice.
+V.
+CONCLUSION
+In this paper we have extended the theory of noise
+spectroscopy based on the transient transmission to in-
+clude quadratic qubit-resonator interactions, which oc-
+cur in nanonmechanical and superconducting resonators
+coupled to solid-state qubits. Allowing for arbitrary ini-
+tial qubit coherences, we derived and solved the quan-
+tum Langevin equations in the presence of a noisy qubit
+in first-order perturbation theory and showed that noise
+features are imprinted in the averaged resonator trans-
+mission and phase. Measuring both of these quantities
+permits the extraction of the ANI ￿I2￿ for quadrati-
+cally coupled systems. By converting ￿I2￿ to the ANI
+for linearly coupled systems, one may extract the noise
+power spectral density from transmission measurements
+using established methods. Our results expand the scope
+of applications of transmission based noise spectroscopy
+to a wide variety of systems including superconducting,
+charge and spin qubits coupled to electromagnetic cavi-
+ties and nanomechanical resonators.
+￿�2￿￿⇡
+￿�1￿￿⇡
+�￿￿A2￿2￿￿￿A∞￿2
+Appendix A: Real and imaginary parts of A
+In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-
+photon coupling (n = 1,2). Introducing the quantity ⇣ = arctan(2�r￿), they read
+Re(An) = −
+√12
+�2r + 2￿4￿
+2 − e−t￿2 ￿cos(�rt)
+2 − sin(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿cos([n − 1]⇣ − �rt) + ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿sin([n − 1]⇣ − �rt)￿￿,
+Im(An) = −
+√12
+�2r + 2￿4￿ − �r + e−t￿2 ￿sin(�rt)
+2 + cos(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿sin([n − 1]⇣ − �rt) − ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿cos([n − 1]⇣ − �rt)￿￿.
+(A1)
+[1] J. Preskill, Quantum Computing in the NISQ era and
+beyond, Quantum 2, 79 (2018).
+[2] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-
+shuler, 1/f noise: Implications for solid-state quantum
+104
+FIG. 2. The normalized fluctuations in the averaged transmis-
+sion probability δ⟪∣An∣2⟫/∣A∞∣2 for quadratic qubit-resonator
+couplings (n = 2) as a function of time t.
+Solid lines are
+drawn according to Eq. (13) with the ANI for quasistatic noise
+[Eq. (24)] with root-mean-square δXrms = 0.05δ(n)
+0
+, while the
+numerical data points (squares) are obtained as described
+in Sec. (II B). The parameter values set to ng = 0.002δ(n)
+0
+,
+κ = 0.1δ(n)
+0
+, κ1 = κ2 = κ/2, κint = 0, γ1 = γϕ = 0.025δ(n)
+0
+,
+T = δ(n)
+0
+, λ = 0.9, and the resonator-probe detunings are cho-
+sen as indicated in the figure.
+Interestingly, the corrections to the empty resonator
+transmission come with a factor ⟨bin⟩n−2, and hence the
+transient transmission for linear interactions depends on
+the input field while for quadratic interactions it does not,
+a characteristic feature of this type of qubit-resonator
+coupling. To extract noise characteristics, we now pro-
+ceed to study the averaged transient transmission and
+phase.
+A.
+Transmission
+Taking the absolute value of (10) and averaging over
+distinct noise configurations yields
+⟪∣An∣⟫ = ∣A∞∣
+√
+ξ0 + ξ1,n,
+(12)
+where ∣A∞∣ =
+√
+κ1κ2/(∆2r + κ2/4) is the long-time empty
+cavity transmission,
+ξ0(t) = 1 + e−κt − 2e−κt/2 cos(∆rt),
+ξ1,n(t) = 2nge−κt/2 ⎛
+⎝
+√
+∆2r + κ2/4
+√κ1⟨bin⟩
+⎞
+⎠
+2−n
+(13)
+× (Yn(t)[Re⟨σ0
+−⟩Re⟪In⟫ − Im⟨σ0
+−⟩Im⟪In⟫]
++ Zn(t)[Im⟨σ0
+−⟩Re⟪In⟫ + Re⟨σ0
+−⟩Im⟪In⟫]),
+and we have introduced the functions
+Yn(t) = sin(nζ − ∆rt) − e−κt/2 sin(nζ),
+Zn(t) = cos(nζ − ∆rt) − e−κt/2 cos(nζ),
+(14)
+with ζ = arctan(2∆r/κ).
+In obtaining (12) we have
+used the fact that the variance of ∣A∣ is non-zero only
+at quadratic order in ε(n) or higher [14]. The averaged
+noise integrals (ANIs) ⟪In⟫ are as given in Eq. (11) but
+with the factor exp(−iλX(t′)) in the integral replaced by
+the averaged random phase ⟪exp(−iλX(t′))⟫. One may
+rewrite Eq. (12) in terms of the normalized fluctuations
+of the averaged transmission probability,
+δ⟪∣An∣2⟫
+∣A∞∣2
+= ⟪∣An∣2⟫ − ξ0∣A∞∣2
+∣A∞∣2
+= ξ1,n.
+(15)
+The effect of the noise on the averaged transmission can
+be assessed in a clear and disentangled fashion by using
+Eq. (15) as can be seen from Fig. 2.
+Often the resonator is probed on resonance, ∆r = 0,
+since one may expect the most pronounced response at
+this point in experiments. In this case, one finds
+δ⟪∣An∣2⟫
+∣A∞∣2
+= 2ng (
+κ
+2√κ1⟨bin⟩)
+2−n
+e−κt/2 (1 − e−κt/2)
+× [Im⟨σ0
+−⟩Re⟪In⟫ + Re⟨σ0
+−⟩Im⟪In⟫],
+(16)
+where ∣A∞∣ = 1 for symmetric mirrors.
+The transient
+signal for the resonant case is shown in Fig. 2(a).
+B.
+Phase
+We now turn to the averaged phase of the transmission
+signal, which has not been investigated in the context of
+noise spectroscopy so far. One has
+⟪φn⟫ = ⟪arctan(Im(An)
+Re(An))⟫,
+(17)
+where the general expressions for the real and imagi-
+nary parts of the transmission amplitude to leading or-
+der in g are straightforward to calculate from Eq. (10)
+but lengthy, and we display them in Appendix A. In the
+
+《Φ2》/π
+《Φ1》/π
+8《IA212》//Al5
+δ(n)
+0 t
+0
+50
+0
+2
+−2
+λ ≠ 0
+λ = 0
+λ ≠ 0
+analytical
+numerical
+0
+2
+−2
+4
+ng = 0.002δ(n)
+0
+6
+C.
+Relating the noise integrals
+Finally, we wish to relate the n = 1 and n = 2 ANIs
+since we already know how to extract the power spec-
+tral density S from ￿I1￿ = ￿I￿ from the discussion in
+Ref. [32]. Comparing the expressions for I1 and I2 in
+Eq. (11), one finds the relation
+I2(t) = I1(t) − ￿
+t
+0 ei�rt′−t′￿2 ˙I1(t′)dt′,
+(26)
+where ˙I1(t′) = @tI1(t)￿t=t′. Eq. (26) can be inverted, and
+after averaging over the noise configurations, we obtain
+￿I1(t)￿ = ￿
+t
+0
+￿ ˙I2(t′)￿
+ei!pt − ei!rt′−t′￿2 dt′ =
+￿I2(t)￿
+ei!pt − ei!rt−t￿2
++ ￿
+t
+0
+￿I2(t′)￿￿i!pei!pt′ − ￿i!r − 
+2 ￿ei!rt′−t′￿2￿
+(ei!pt′ − ei!rt′−t′￿2)2
+dt′,
+(27)
+where the second equality is obtained via partial inte-
+gration.
+After converting the ANI ￿I2(t)￿ to ￿I1(t)￿
+via Eq. (27), one may follow the approach outlined in
+Ref. [32] to obtain noise characteristics from the ANI
+in the linearly coupled case.
+Hence, the power spec-
+tral density can also be obtained in quadratically cou-
+pled qubit-resonator systems. Noise features for common
+noise types can be obtained by reducing ￿I2￿ to ￿I1￿ as
+well, even though it may be easier to directly use the an-
+alytical expression for ￿I2￿ given Eqs. (23) and (25) in
+practice.
+V.
+CONCLUSION
+In this paper we have extended the theory of noise
+spectroscopy based on the transient transmission to in-
+clude quadratic qubit-resonator interactions, which oc-
+cur in nanonmechanical and superconducting resonators
+coupled to solid-state qubits. Allowing for arbitrary ini-
+tial qubit coherences, we derived and solved the quan-
+tum Langevin equations in the presence of a noisy qubit
+in first-order perturbation theory and showed that noise
+features are imprinted in the averaged resonator trans-
+mission and phase. Measuring both of these quantities
+permits the extraction of the ANI ￿I2￿ for quadrati-
+cally coupled systems. By converting ￿I2￿ to the ANI
+for linearly coupled systems, one may extract the noise
+power spectral density from transmission measurements
+using established methods. Our results expand the scope
+of applications of transmission based noise spectroscopy
+to a wide variety of systems including superconducting,
+charge and spin qubits coupled to electromagnetic cavi-
+ties and nanomechanical resonators.
+￿�2￿￿⇡
+￿�1￿￿⇡
+�￿￿A2￿2￿￿￿A∞￿2
+Appendix A: Real and imaginary parts of A
+In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-
+photon coupling (n = 1,2). Introducing the quantity ⇣ = arctan(2�r￿), they read
+Re(An) = −
+√12
+�2r + 2￿4￿
+2 − e−t￿2 ￿cos(�rt)
+2 − sin(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿cos([n − 1]⇣ − �rt) + ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿sin([n − 1]⇣ − �rt)￿￿,
+Im(An) = −
+√12
+�2r + 2￿4￿ − �r + e−t￿2 ￿sin(�rt)
+2 + cos(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿sin([n − 1]⇣ − �rt) − ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿cos([n − 1]⇣ − �rt)￿￿.
+(A1)
+[1] J. Preskill, Quantum Computing in the NISQ era and
+beyond, Quantum 2, 79 (2018).
+[2] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-
+shuler, 1/f noise: Implications for solid-state quantum
+information, Rev. Mod. Phys. 86, 361 (2014).
+[3] C. Kloe�el and D. Loss, Prospects for spin-based quan-
+tum computing in quantum dots, Annual Review of Con-
+densed Matter Physics 4, 51 (2013).
+[4] X. Zhang, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, and G.-
+P. Guo, Semiconductor quantum computation, National
+Science Review 6, 32 (2019).
+[5] G. Burkard,
+T. D. Ladd,
+J. M. Nichol,
+A. Pan,
+and J. R. Petta, Semiconductor spin qubits (2021),
+arXiv:2112.08863 [cond-mat.mes-hall].
+[6] J. Clarke and F. K. Wilhelm, Superconducting quantum
+bits, Nature 453, 1031 (2008).
+104
+6
+C.
+Relating the noise integrals
+Finally, we wish to relate the n = 1 and n = 2 ANIs
+since we already know how to extract the power spec-
+tral density S from ￿I1￿ = ￿I￿ from the discussion in
+Ref. [32]. Comparing the expressions for I1 and I2 in
+Eq. (11), one finds the relation
+I2(t) = I1(t) − ￿
+t
+0 ei�rt′−t′￿2 ˙I1(t′)dt′,
+(26)
+where ˙I1(t′) = @tI1(t)￿t=t′. Eq. (26) can be inverted, and
+after averaging over the noise configurations, we obtain
+￿I1(t)￿ = ￿
+t
+0
+￿ ˙I2(t′)￿
+ei!pt − ei!rt′−t′￿2 dt′ =
+￿I2(t)￿
+ei!pt − ei!rt−t￿2
++ ￿
+t
+0
+￿I2(t′)￿￿i!pei!pt′ − ￿i!r − 
+2 ￿ei!rt′−t′￿2￿
+(ei!pt′ − ei!rt′−t′￿2)2
+dt′,
+(27)
+where the second equality is obtained via partial inte-
+gration.
+After converting the ANI ￿I2(t)￿ to ￿I1(t)￿
+via Eq. (27), one may follow the approach outlined in
+Ref. [32] to obtain noise characteristics from the ANI
+in the linearly coupled case.
+Hence, the power spec-
+tral density can also be obtained in quadratically cou-
+pled qubit-resonator systems. Noise features for common
+noise types can be obtained by reducing ￿I2￿ to ￿I1￿ as
+well, even though it may be easier to directly use the an-
+alytical expression for ￿I2￿ given Eqs. (23) and (25) in
+practice.
+V.
+CONCLUSION
+In this paper we have extended the theory of noise
+spectroscopy based on the transient transmission to in-
+clude quadratic qubit-resonator interactions, which oc-
+cur in nanonmechanical and superconducting resonators
+coupled to solid-state qubits. Allowing for arbitrary ini-
+tial qubit coherences, we derived and solved the quan-
+tum Langevin equations in the presence of a noisy qubit
+in first-order perturbation theory and showed that noise
+features are imprinted in the averaged resonator trans-
+mission and phase. Measuring both of these quantities
+permits the extraction of the ANI ￿I2￿ for quadrati-
+cally coupled systems. By converting ￿I2￿ to the ANI
+for linearly coupled systems, one may extract the noise
+power spectral density from transmission measurements
+using established methods. Our results expand the scope
+of applications of transmission based noise spectroscopy
+to a wide variety of systems including superconducting,
+charge and spin qubits coupled to electromagnetic cavi-
+ties and nanomechanical resonators.
+￿�2￿￿⇡
+￿�1￿￿⇡
+�￿￿A2￿2￿￿￿A∞￿2
+Appendix A: Real and imaginary parts of A
+In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-
+photon coupling (n = 1,2). Introducing the quantity ⇣ = arctan(2�r￿), they read
+Re(An) = −
+√12
+�2r + 2￿4￿
+2 − e−t￿2 ￿cos(�rt)
+2 − sin(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿cos([n − 1]⇣ − �rt) + ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿sin([n − 1]⇣ − �rt)￿￿,
+Im(An) = −
+√12
+�2r + 2￿4￿ − �r + e−t￿2 ￿sin(�rt)
+2 + cos(�rt)�r￿ + gn
+￿
+�2r + 2
+4
+￿
+￿
+√1￿bin￿
+￿
+�2r + 2￿4
+￿
+￿
+n−2
+e−t￿2
+× ￿￿Re￿�0
+−￿ImIn + Im￿�0
+−￿ReIn￿sin([n − 1]⇣ − �rt) − ￿Re￿�0
+−￿ReIn − Im￿�0
+−￿ImIn￿cos([n − 1]⇣ − �rt)￿￿.
+(A1)
+[1] J. Preskill, Quantum Computing in the NISQ era and
+beyond, Quantum 2, 79 (2018).
+[2] E. Paladino, Y. M. Galperin, G. Falci, and B. L. Alt-
+shuler, 1/f noise: Implications for solid-state quantum
+information, Rev. Mod. Phys. 86, 361 (2014).
+[3] C. Kloe�el and D. Loss, Prospects for spin-based quan-
+tum computing in quantum dots, Annual Review of Con-
+densed Matter Physics 4, 51 (2013).
+[4] X. Zhang, H.-O. Li, G. Cao, M. Xiao, G.-C. Guo, and G.-
+P. Guo, Semiconductor quantum computation, National
+Science Review 6, 32 (2019).
+[5] G. Burkard,
+T. D. Ladd,
+J. M. Nichol,
+A. Pan,
+and J. R. Petta, Semiconductor spin qubits (2021),
+arXiv:2112.08863 [cond-mat.mes-hall].
+[6] J. Clarke and F. K. Wilhelm, Superconducting quantum
+bits, Nature 453, 1031 (2008).
+104
+�
+�
+��
+�
+��
+�
+�
+�
+�
+�
+�
+�
+�
+��
+�
+��
+�
+���
+���������������������������
+0
+10
+20
+30
+40
+50
+-0.0002
+-0.0001
+0.0000
+0.0001
+0.0002
+⟨bin⟩ = 0.5
+κ
+⟨bin⟩ = 0.7
+κ
+⟨bin⟩ =
+κ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+� �
+�
+� �
+�
+� �
+�
+� � �
+� � � � � � � � � � � � � � � � � � � � � � � �
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+� �
+�
+� �
+�
+� �
+�
+� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
+�
+�
+�
+�
+�
+� �
+�
+� �
+�
+�
+� �
+�
+� �
+�
+� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �
+0
+10
+20
+30
+40
+50
+-0.0002
+0.0000
+0.0002
+0.0004
+analytical
+numerical
+FIG. 3. The transient averaged phase ⟪φn⟫ as a function of
+the measurement time t for quasistatic noise at ∆r = 0 for
+the cases of linear qubit-resonator couplings (n = 1, top) and
+quadratic qubit-resonator couplings (n = 2, bottom).
+The
+solid lines are drawn according to the approximate analytical
+result (18), while the numerical data is obtained as described
+in Sec. II B using the exact expression (17). For the case n = 1
+we show the averaged phase for several values of the input
+field ⟨bin⟩ as indicated in the figure. We set κ = δ(n)
+0
+and the
+remaining parameter values are chosen as in Fig. 2(a).
+above form the noise average of the phase cannot be per-
+formed analytically due to the arctan function. However,
+when the resonator is driven on resonance, ∆r = 0, the
+dominant term in the expansion of Im(An) in orders of
+ε(n) vanishes, while it is non-zero in the expansion of
+Re(An), and one has ∣Im(An)∣ ≪ ∣Re(An)∣ for g ≪ κ. In
+this case, one may keep only the leading-order term in
+the Taylor expansion of the arctan function, whereby it
+becomes possible to average the phase analytically,
+⟪φn⟫ ≈
+ng
+1 − eκt/2 (
+κ
+2√κ1⟨bin⟩)
+2−n
+× [Re⟨σ0
+−⟩Re⟪In⟫ − Im⟨σ0
+−⟩Im⟪In⟫].
+(18)
+As was the case for the corrections to the empty resonator
+transmission probability, the phase is also inversely pro-
+portional to the input field ⟨bin⟩ for linear couplings but
+independent of the input field for quadratic couplings. To
+underline the validity of the approximation made when
+deriving (18), we show a comparison between numerical
+and analytical results for the transient averaged phase in
+Fig. 3. We find excellent agreement, suggesting that the
+averaged phase measured at the resonance ∆r = 0 can be
+used to infer noise characteristics according to Eq. (18).
+IV.
+NOISE SPECTROSCOPY
+The aim of noise spectroscopy is the determination of
+spectral features of the fluctuations affecting the system
+under consideration [40]. In this section we show that in-
+formation on the noise power spectral density S(ω) of the
+qubit noise can be obtained from the averaged transmis-
+sion and phase by extracting the ANI and investigating
+its properties.
+A.
+Extracting the averaged noise integral
+Having obtained expressions for the averaged transmis-
+sion and averaged phase, we now turn to investigate how
+noise features can be extracted from the measurement of
+these quantities. For n = 1, the extraction of the ANI
+and the extraction of noise features from the ANI is de-
+tailed in Ref. [14]. The former relies on the fact that the
+ANI does not depend on the probe frequency ωp, allowing
+for an extraction scheme based on measurements of the
+transmission at two distinct resonator-probe detunings.
+For n = 2, however, the noise integral I2 does depend on
+ωp, and hence it is impossible to extract it by recording
+the transmission at two distinct values of the probe fre-
+quency. However, if in addition the phase is measured,
+it is possible to obtain the ANI for both n = 1 and n = 2.
+Since we aim to use the analytical result for the averaged
+phase in Eq. (18), we work in the resonant regime where
+∆r is fixed to zero. One may then infer the following set
+of equations from Eqs. (12) and (18),
+Im⟨σ0
+−⟩Re⟪In⟫ + Re⟨σ0
+−⟩Im⟪In⟫ = Υ∣A∣,
+Re⟨σ0
+−⟩Re⟪In⟫ − Im⟨σ0
+−⟩Im⟪In⟫ = Υφ,
+(19)
+where the functions on the right-hand side are obtained
+from the averaged transmission and phase,
+Υ∣A∣ = ⟪∣An∣⟫2/∣A∞∣2 − (1 − e−κt/2)
+2
+2nge−κt/2(1 − e−κt/2)
+(
+κ
+2√κ1⟨bin⟩)
+n−2
+,
+Υφ = ⟪φn⟫1 − eκt/2
+ng
+(
+κ
+2√κ1⟨bin⟩)
+n−2
+.
+(20)
+The inhomogeneous linear system of equations (19) can
+be uniquely solved for the real and imaginary parts of
+⟪In⟫ if
+det
+⎛
+⎜
+⎝
+Im⟨σ0
+−⟩
+Re⟨σ0
+−⟩
+Re⟨σ0
+−⟩ −Im⟨σ0
+−⟩
+⎞
+⎟
+⎠
+= −∣⟨σ0
+−⟩∣
+2 ≠ 0.
+(21)
+Hence, the qubit must be initialized in a coherent super-
+position to permit the extraction of the ANI from trans-
+mission measurements. This condition, however, is not
+restrictive as it is required to be fulfilled for the noisy
+part of the transmission signal to be non-vanishing in
+the transient phase in the first place [see Eq. (10)] and
+hence does not introduce additional constraints.
+B.
+Common types of noise
+Since we have shown that the ANI can be extracted
+from transmission measurements even for quadratic
+
+《Φ2》/π
+《Φ1》/π
+8《IA212》//A
+126
+qubit-resonator interactions, it is possible to extract noise
+features by comparing the measured data to our the-
+ory.
+For this purpose, we now aim to obtain analyti-
+cal expressions for the ANI for the case n = 2. To pro-
+ceed, we assume zero-mean noise and express the factor
+⟪exp(−iλX(t′))⟫ that appears in the ANI in the Gaus-
+sian approximation using the noise power spectral density
+S(ω) [41]. We may then evaluate the ANI for three noise
+types that are common in engineered quantum systems:
+white noise, quasistatic noise and low-frequency noise.
+White noise is characterized by a constant noise am-
+plitude, S(ω) = S0, and the corresponding ANI takes the
+form
+⟪I2⟫ =
+2
+∑
+j=1
+(−1)j ecjt−λ2S0t/2 − 1
+cj − λ2S0/2
+,
+(22)
+where we introduce the shorthand notation
+c1 = i(2ωr − ωq) − γ
+2 ,
+c2 = i(ωr − ωq + ωp) + κ − γ
+2
+. (23)
+Fluctuations are quasistatic if they are constant during
+a given measurement but change between two measure-
+ments. In this case, the spectral density may be written
+as S(ω) = 2πδX2
+rmsδ(ω), where δXrms =
+√
+⟪δX2⟫ is the
+root-mean-square of the noise and δ(ω) is the Dirac delta
+distribution, and one finds
+⟪I2⟫ =
+2
+∑
+j=1
+(−1)jeC2
+j
+λδXrms
+[E (Cj) + E (λδXrms
+√
+2
+t − Cj)], (24)
+where Cj = cj/(
+√
+2λδXrms) and E(z) =
+√
+π/2 erf(z) with
+error function erf(z).
+Finally, one may consider low-
+frequency noise such as 1/f noise in a frequency band
+[ωir,ωuv].
+If large frequencies are sufficiently strongly
+suppressed such that the ultraviolet cutoff frequency ωuv
+satisfies ωuvt ≪ 1 in the transient phase, the ANI is in
+good approximation described by the same functional
+form as in Eq. (24) with the substitution δXrms →
+√
+P/π,
+where P = ∫ S(ω)dω is the noise power in the frequency
+band under consideration. The information on the noise
+attainable by fitting the experimental data to the above
+expressions hence includes the noise amplitude S0 for
+white noise, the root mean square of the noise δXrms for
+quasistatic noise and the noise power P for low-frequency
+noise. While useful when investigating common material
+platforms, the approach requires a priori knowledge on
+the type of noise. We now proceed to introduce a formal-
+ism that allows the extraction of noise features in terms
+of the power spectral density S(ω) without the need to
+make any assumptions on the form of the noise.
+C.
+Relating the noise integrals
+In a final step, we wish to relate the n = 1 and n =
+2 ANIs since we already know how to extract S from
+⟪I1⟫ via the convolution theorem from the discussion in
+Ref. [14]. Comparing the expressions for I1 and I2 in
+Eq. (11), one straightforwardly finds the relation
+I2(t) = ∫
+t
+0 (eiωpt′ − eiωrt′−κt′/2) ˙I1(t′)dt′,
+(25)
+where ˙I1(t′) = ∂tI1(t)∣t=t′. Eq. (25) can be inverted, and
+after averaging over the noise configurations, we find
+⟪I1(t)⟫ = ∫
+t
+0
+⟪ ˙I2(t′)⟫
+eiωpt′ − eiωrt′−κt′/2 dt′ =
+⟪I2(t)⟫
+eiωpt − eiωrt−κt/2
++ ∫
+t
+0 ⟪I2(t′)⟫iωpeiωpt′ − [iωr − κ
+2 ]eiωrt′−κt′/2
+[eiωpt′ − eiωrt′−κt′/2]
+2
+dt′,
+(26)
+where the second equality is obtained by partial inte-
+gration.
+After converting the ANI ⟪I2(t)⟫ to ⟪I1(t)⟫
+via Eq. (26), one may follow the approach outlined in
+Ref. [14] to obtain noise characteristics from the ANI in
+the linearly coupled case. Hence, the power spectral den-
+sity can also be obtained in quadratically coupled qubit-
+resonator systems. Noise features of common noise types
+can be obtained by reducing ⟪I2⟫ to ⟪I1⟫ as well, even
+though it may often be easier to directly use the analyt-
+ical expression for ⟪I2⟫ given in Eqs. (22) and (24) in
+practice.
+V.
+CONCLUSION
+In this paper we extended the theory of noise spec-
+troscopy based on the transient transmission to include
+quadratic qubit-resonator interactions, which occur in
+nanonmechanical and superconducting resonators cou-
+pled to solid-state qubits.
+Allowing for arbitrary ini-
+tial qubit coherences, we derived and solved the quan-
+tum Langevin equations in the presence of a noisy qubit
+in first-order perturbation theory and showed that noise
+features are imprinted in the averaged resonator trans-
+mission and phase. Measuring both of these quantities
+permits the extraction of the ANI ⟪I2⟫ for quadrati-
+cally coupled systems. By converting ⟪I2⟫ to the ANI
+for linearly coupled systems, one may extract the noise
+power spectral density from transmission measurements
+using established methods. Our results expand the scope
+of applications of transmission-based noise spectroscopy
+to a wide variety of systems including superconducting,
+charge and spin qubits coupled to electromagnetic cavi-
+ties and nanomechanical resonators.
+VI.
+ACKNOWLEDGMENTS
+This research is supported by the German Research
+Foundation [Deutsche Forschungsgemeinschaft (DFG)]
+under Project No. 450396347 and No. 425217212 - SFB
+1432.
+
+7
+Appendix A: Real and imaginary parts of the transmission amplitude
+In this Appendix we display the real and imaginary parts of the transmission amplitude in Eq. (10) for both linear
+(n = 1) and quadratic (n = 2) qubit-resonator couplings. They read
+Re(An) = −
+√κ1κ2
+∆2r + κ2/4[κ
+2 − e−κt/2 (cos(∆rt)κ
+2 − sin(∆rt)∆r) + gn
+√
+∆2r + κ2
+4
+⎛
+⎝
+√κ1⟨bin⟩
+√
+∆2r + κ2/4
+⎞
+⎠
+n−2
+e−κt/2
+× {[Re⟨σ0
+−⟩ImIn + Im⟨σ0
+−⟩ReIn]cos([n − 1]ζ − ∆rt) + [Re⟨σ0
+−⟩ReIn − Im⟨σ0
+−⟩ImIn]sin([n − 1]ζ − ∆rt)}],
+(A1)
+Im(An) = −
+√κ1κ2
+∆2r + κ2/4[ − ∆r + e−κt/2 (sin(∆rt)κ
+2 + cos(∆rt)∆r) + gn
+√
+∆2r + κ2
+4
+⎛
+⎝
+√κ1⟨bin⟩
+√
+∆2r + κ2/4
+⎞
+⎠
+n−2
+e−κt/2
+× {[Re⟨σ0
+−⟩ImIn + Im⟨σ0
+−⟩ReIn]sin([n − 1]ζ − ∆rt) − [Re⟨σ0
+−⟩ReIn − Im⟨σ0
+−⟩ImIn]cos([n − 1]ζ − ∆rt)}],
+(A2)
+where as in the main text we use the notation ζ = arctan(2∆r/κ).
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+
diff --git a/jdFAT4oBgHgl3EQfah0B/content/tmp_files/load_file.txt b/jdFAT4oBgHgl3EQfah0B/content/tmp_files/load_file.txt
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@@ -0,0 +1,908 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf,len=907
+page_content='Transmission-based noise spectroscopy for quadratic qubit-resonator interactions  Philipp M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Mutter∗ and Guido Burkard†  Department of Physics, University of Konstanz, D-78457 Konstanz, Germany  We develop a theory describing the transient transmission through noisy qubit-resonator systems  with quadratic interactions as are found in superconducting and nanomechanical resonators cou-  pled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' After generalizing the quantum Langevin equations to arbitrary qubit-  resonator couplings, we show that only the cases of linear and quadratic couplings allow for an  analytical treatment within standard input-output theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Focussing for the first time on quadratic  couplings and allowing for arbitrary initial qubit coherences, it is shown that noise characteristics  can be extracted from input-output measurements by recording both the averaged fluctuations in  the transmission probability and the averaged phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Our results represent an extension to the field  of transmission-based noise spectroscopy with immediate practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  INTRODUCTION  We live in the age of noisy intermediate-scale quan-  tum devices in which noise severely affects the coherence  of qubits and thereby the performance of quantum com-  puters [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The noise can have several origins but may  roughly be categorized according to whether it arises as  a consequence of imperfect system control or unwanted  interactions with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  For instance, ran-  dom electromagnetic fields due to fluctuating control gate  voltages belong to the former class [2], while noise arising  from various two-level fluctuators in the host material [3],  the interaction with the nuclear spin bath in semiconduc-  tor quantum dots [4–6], as well as magnetic flux noise and  quasiparticles in superconducting circuits [7–10] can be  assigned to the latter class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Common to all types of noise is their detrimental effect  on the coherence of engineered quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' To mit-  igate noise-induced decoherence one requires knowledge  of the spectral form of the fluctuations which is encoded  in the frequency-dependent noise power spectral density  S(ω) [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Experiments assessing the effect of charge  noise on a semiconductor double quantum dot by exam-  ining the long-time resonator transmission have already  been performed [13], and recently spectroscopy meth-  ods for classical and quantum noise based on measure-  ments of the transient transmission through a linearly  coupled qubit-resonator system were developed [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Linear light-matter interactions involving the exchange  of one photon are relevant in many fundamentally and  technologically relevant systems, such as atoms in a cav-  ity [16, 17] as well as charge qubits coupled to supercon-  ducting transmission lines [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Additionally, linear  spin-photon couplings are made possible by the spin-orbit  interaction [20–24] or the use of micromagnets [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Quadratic qubit-resonator couplings also allow for  input-output measurements and are found in diverse  physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' For instance, nanomechanical and op-  tomechanical resonators have been shown to couple to  ∗ philipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='mutter@uni-konstanz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='de  † guido.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='burkard@uni-konstanz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='de  noisy qubit-  resonator system  ⟨bin⟩  ⟨bout⟩  HI ∝ a2σ+ + a†2σ−  HI ∝ aσ+ + a†σ−  κj  |0⟩  |1⟩  ωq  γ  ωr  port j  g  δωq = λδX  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Potential systems for transmission-based noise spec-  troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' A two-level system (qubit, blue) is affected by noise  leading to fluctuations δωq in the energy splitting ωq between  qubit states ∣0⟩ and ∣1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  The interaction with a single res-  onator mode is described by the Hamiltonian HI containing  the qubit and resonator operators σ± and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We schematically  show an optical resonator with linear qubit-mode couplings  on the left and a nanomechanical resonator with quadratic  qubit-mode couplings on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Both types of interactions  allow for the characterization of noise features by studying  the transmission A = ⟨bout⟩/⟨bin⟩ through the noisy qubit-  resonator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The bottom schematic introduces the pho-  ton loss rate κj at the resonator port j, the qubit decay rate  γ and the qubit-resonator coupling strength g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  superconducting and spin qubits, and the characteristic  feature is the quadratic qubit-resonator interaction in-  cluding two-photon processes [27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Quadratic qubit-  resonator interactions can also be engineered in super-  conducting circuits [32, 33], and earlier studies of such  systems investigated thermal transport [34] and input-  output theoretical transmission features in the noise-free  steady-state case [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' If one is interested in transmission-  based noise spectroscopy, it is desirable to have a detailed  knowledge about the input-output behaviour of both the  linear and quadratic cases as the coupling of a given type  of qubit to resonator modes may more easily be realized  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='08551v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='mes-hall]  20 Jan 2023  2  in one system or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In this paper we generalize  the results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [14] to also include quadratic qubit-  resonator interactions and show that noise characteristics  can be inferred by measuring both the averaged fluctua-  tions in the transmission probability δ⟪∣A∣2⟫/∣A∞∣2 and  the averaged phase ⟪φ⟫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Moreover, as another extension  of earlier results we allow for a general form of the initial  qubit coherence ⟨σ0  −⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  The remainder of this paper is structured as follows: In  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' II we consider a general n-photon-qubit interaction  and derive the Langevin equations for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We then  show that within standard input-output theory only the  cases n = 1 and n = 2 allow for an analytical solution to  lowest order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Specializing on these  cases we calculate the averaged transmission probability  and phase in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' III, and propose a scheme for extracting  noise characteristics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' V provides  a conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  GENERAL MODEL AND LANGEVIN  EQUATIONS  We consider a resonator mode of frequency ωr that  couples to a qubit with fluctuating energy separation  ωq + δωq(t), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 1 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The noise δX affect-  ing the control parameter X is related to the fluctuating  energy separation by the first-order expansion δωq = λδX  with the noise sensitivity λ = ∂Xωq∣δX=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The number n  of photons taking part in the interaction is left unspec-  ified for now, and we assume a general qubit-resonator  Hamiltonian of the form  Hqr = ωq + δωq(t)  2  σz + ωra†a + g(a + a†)nσx,  (1)  where σx,z are Pauli matrices in the basis of the logi-  cal qubit states {∣0⟩,∣1⟩}, and a is the mode annihilation  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In the rotating frame defined by the transfor-  mation Hqr → UHqrU † + i ˙UU † with the time-dependent  unitary U = exp(iΩt[a†a/n+σz/2]) and within the rotat-  ing wave approximation, the system is described by the  Hamiltonian  Hqr = ∆q + δωq(t)  2  σz + ∆(n)  r  a†a + g [anσ+ + a†nσ−], (2)  where σ± are the qubit ladder operators, and the quan-  tities ∆q = ωq − Ω and ∆(n)  r  = ωr − Ω/n are the shifted  qubit and resonator frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  In addition, external bath modes b interact with the  resonator, and the coupling is assumed to be linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We  take into account the effects of the corresponding input  field ⟨bin(t)⟩ by including the injection Hamiltonian in  the rotating frame,  Hin = i√κ1 (⟨bin(t)⟩eiΩt/na† − H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='),  (3)  The input field ⟨bin(t)⟩ may be used to probe the system,  and it is assumed to be a classical wave of real amplitude  ⟨bin⟩ and frequency ωp, ⟨bin(t)⟩ = ⟨bin⟩e−iωpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In the fol-  lowing we work in the frame defined by U as above with  Ω = nωp such that the the injection Hamiltonian (3) is  constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' One then has ∆q ≡ ∆(n)  q  = ωq −nωp and  ∆(n)  r  ≡ ∆r = ωr − ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Furthermore taking into account noise-independent  qubit relaxation, absorption and dephasing at finite tem-  perature as well as photon losses, the system may be  described by a Lindblad master equation in the rotating  frame,  ˙ρ = −i[Hqr + Hin,ρ] + L[ρ],  L[ρ] = γ1  2 ∑  ±  1 ± 1 + 2nB  2  (2σ∓ρσ± − {ρ,σ±σ∓})  + γϕ  2 (σzρσz − ρ) + κ  2 (2aρa† − {ρ,a†a}),  (4)  where γ1 is the relaxation rate at zero Kelvin, γϕ is the  dephasing rate and nB = [exp(ωq/T) − 1]−1 is the Bose  distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=', the mean number of bath photons at  temperature T, evaluated at the qubit frequency where  resonant energy exchange is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Additionally, we  have introduced the total photon loss rate κ = ∑j κj +κint  as the sum of the individual loss rates at port j ∈ {1,2}  and the internal loss rate κint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  By using the relation ⟨ ˙O⟩ = Tr( ˙ρO) for a given operator  O and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (4), one obtains the partial system of Langevin  equations to leading order in g,  ˙⟨a⟩ = −[i∆r + κ  2 ]⟨a⟩ − ign⟨σ−⟩⟨an−1⟩∗ + √κ1⟨bin⟩,  ˙  ⟨σz⟩ = −γ1(T)⟨σz⟩ − γ1(0) + 2ig (⟨an⟩⟨σ−⟩∗ − c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' ),  ˙  ⟨σ−⟩ = −[i∆(n)  q  + iλδX(t) + γ2]⟨σ−⟩ + ig⟨σz⟩⟨an⟩,  (5)  where γ1(T) = γ1 coth(ωq/2T), γ2 = γ1(T)/2 + γϕ, the  star denotes complex conjugation, and we have used the  relation ⟨QR⟩ = ⟨Q⟩⟨R⟩ + O(g) that is valid for all qubit  (Q) and resonator (R) operators under the assumption  of an initially separable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In the following we define  γ = 2γ2 such that the homogeneous parts of the Langevin  equations for ⟨σ−⟩ and ⟨a⟩ have the same form, allowing  us to obtain cleaner mathematical expressions later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Analytical approach  The system of Langevin equations (5) is not closed be-  cause in general ⟨an⟩ ≠ ⟨a⟩n, and even if it the relation  would hold true, an exact solution in the presence of arbi-  trary time-dependent qubit noise could not be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  However, we may formally solve the equation for ⟨σ−⟩  without any assumptions on the noise δX(t),  ⟨σ−(t)⟩ = ⟨σ0  −⟩e−i∆(n)  q  t−γt/2e−iλX (t)  (6)  + ig ∫  t  0 ⟨σz(t′)⟩⟨an(t′)⟩e(i∆(n)  q  +γ/2)(t′−t)eiλ[X (t′)−X (t)]dt′,  3  with the integrated noise X(t) = ∫  t  0 δX(t′)dt′ and the  initial qubit coherence ⟨σ0  −⟩, and subsequently insert the  result into the equation for ⟨a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' By again formally solv-  ing the resulting differential equation, we find an integral  equation for the expectation value of the mode annihila-  tion operator,  ⟨a(t)⟩ =  √κ1⟨bin⟩  i∆r + κ/2 (1 − e−i∆rt−κt/2) − ing⟨σ0  −⟩e−i∆rt−κt/2 ∫  t  0 dt′⟨an−1(t′)⟩∗ei(∆r−∆(n)  q  )t′+(κ−γ)t′/2e−iλX (t′)  (7)  + ng2e−i∆rt−κt/2 ∫  t  0 dt′⟨an−1(t′)⟩∗ei(∆r−∆(n)  q  )t′+(κ−γ)t′/2e−iλX (t′) ∫  t′  0  dt′′⟨σz(t′′)⟩⟨an(t′′)⟩ei∆(n)  q  t′′+γt′′/2e−iλX (t′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  where we assume ⟨a0⟩ = 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=', the resonator may ini-  tially be in an eigenstate of the mode number operator  a†a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We first remark that if we wish to work to lead-  ing order in g, we may neglect the second integral in (7),  and hence the time-dependent populations ⟨σz(t)⟩ do not  play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Secondly, if n ⩽ 2, then the ⟨an−1⟩ term in the  remaining integral is equal to unity (n = 1) or ⟨a⟩ (n = 2),  and the equation can be solved to first order in pertur-  bation theory by substituting the zeroth-order value of  ⟨a⟩ into the first integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Otherwise, the fact that in  general ⟨an−1⟩ ≠ ⟨a⟩n−1 forces us to obtain an additional  equation of motion for ⟨an−1⟩ which cannot be treated  in a simple input-output model without further assump-  tions due to additional terms of the form ∫ ⟨b(ω)an−2⟩dω  which feature the bath modes b(ω) but cannot be related  to the input fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In the long-time limit the term linear  in g vanishes, and the long-time transmission is deter-  mined by the g2 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Since this term features the quan-  tity ⟨an⟩, the long-time solution can only be obtained for  n = 1 due to the same arguments made above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' A compre-  hensive study of the long-time transmission for linearly  coupled systems may be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Since the  coupling of the resonator to the external bath modes is  assumed to be linear, the standard input-output relation  for the transmission amplitude A applies [37–39],  A(t) = ⟨bout(t)⟩  ⟨bin(t)⟩ = −  √κ2⟨a(t)⟩  ⟨bin(t)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (8)  As a consequence, the transmission amplitude can be ob-  tained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (7) for ⟨a(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Numerical study  To validate the truncation of the quantum Langevin  equations (5) and the solution for the transmission at  first order in the qubit-photon coupling g, we numerically  solve the full Lindblad equation (4) and compare the re-  sults to our approximate analytical expressions in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 2  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We allow for up to 12 photons and assume an ini-  tially empty resonator containing a qubit in a coherent  superposition ∣ψ⟩ = (eiπ/8 ∣↑⟩+e−iπ/8 ∣↓⟩)/  √  2 such that the  initial conditions ⟨a0⟩ = 0, Re⟨σ0  −⟩ = Im⟨σ0  −⟩ = 1/  √  8 and  ⟨σ0  z⟩ = 0 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The transmission amplitude is then  calculated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (8) as A = −√κ2Tr(ρa)/⟨bin⟩,  and the averaged transmission and phase are obtained  by averaging the solution over 103 distinct noise config-  urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We generally find excellent agreement in the  regime considered, underlining the high accuracy of our  analytical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  OBSERVABLE FIGURES OF MERIT  We now focus on the cases n ⩽ 2 that can be treated  perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  The case n = 1 has been studied in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [14], and we include it here to be able to make  contact to these previous considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  In addition,  we complement earlier results by including the averaged  phase ⟪φ⟫ as an observable figure of merit and allowing  for arbitrary complex initial qubit coherences ⟨σ0  −⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The  case n = 2 constitutes a completely new contribution to  the theory of transmission-based noise spectroscopy, and  it is of immediate interest for nanomechanical and super-  conducting resonators coupled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  The complex transmission amplitude can be written as  A = ∣A∣eiφ,  (9)  where both the transmission ∣A∣ and the phase φ are ex-  perimentally observable and thus allow for a noise aver-  age ⟪.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='⟫ over many measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (7) to  first order in the small parameter ε(n) ≡ ng/max{∣δ(n)  0  ∣ ≡  ∣∆r − ∆(n)  q  ∣,∣κ − γ∣}, we find according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (8),  An(t)  A∞  = 1 − e−i∆rt−κt/2  (10)  × [1 + ign⟨σ0  −⟩κ/2 + i∆r  κ/2 − i∆r  (κ/2 − i∆r  √κ1⟨bin⟩ )  2−n  In(t)],  with the zeroth-order long-time transmission amplitude  A∞ = −√κ1κ2/(i∆r + κ/2) and the noise integrals  In(t) =∫  t  0 ei(ωr−ωq)t′+(κ−γ)t′/2  × (eiωpt′ − eiωrt′−κt′/2)  n−1  e−iλX (t′)dt′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (11)  4  Δr = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='4δ(n)  0  δ(n)  0 t  Δr = − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='2δ(n)  0  Δr = 0  λ ≠ 0  λ = 0  analytical  numerical  0  50  0  −4  4  0  2  −2  0  2  −2  (a)  (b)  (c)  ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='002δ(n)  0  alsoforphase  �����  ��  �  ��  �  �  �  �  �  �  �  �  �  �  �  ��  �  ��  �  ��  �  ��  �  ���  ���������������  0  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0003  ����  �  ��  �  ��  �  ��  �  ��  �  ��  �  ��  �����������������������������  0  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='002  ����  �  ��  �  ��  �  �  �  ��������������������������������������  0  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='004  6  In this case the ANI is described by the same functional  form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (25) with the substitution �X2  rms = P\uffff⇡,  where P = ∫ S(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=')d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' is the noise power in the positive  frequency band [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='ir,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='uv] under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The in-  formation on the noise attainable by using the above ex-  pressions hence includes the noise amplitude S0 for white  noise, the root mean square of the noise �Xrms for qua-  sistatic noise and the noise power P for low-frequency  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Relating the noise integrals  Finally, we wish to relate the n = 1 and n = 2 ANIs  since we already know how to extract the power spec-  tral density S from \uffffI1\uffff = \uffffI\uffff from the discussion in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Comparing the expressions for I1 and I2 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (11), one finds the relation  I2(t) = I1(t) − \uffff  t  0 ei�rt′−\uf8fft′\uffff2 ˙I1(t′)dt′,  (26)  where ˙I1(t′) = @tI1(t)\ufffft=t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (26) can be inverted, and  after averaging over the noise configurations, we obtain  \uffffI1(t)\uffff = \uffff  t  0  \uffff ˙I2(t′)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2 dt′ =  \uffffI2(t)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt−\uf8fft\uffff2  + \uffff  t  0  \uffffI2(t′)\uffff\uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − \uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='r − \uf8ff  2 \uffffei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2\uffff  (ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2)2  dt′,  (27)  where the second equality is obtained via partial inte-  gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  After converting the ANI \uffffI2(t)\uffff to \uffffI1(t)\uffff  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (27), one may follow the approach outlined in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32] to obtain noise characteristics from the ANI  in the linearly coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Hence, the power spec-  tral density can also be obtained in quadratically cou-  pled qubit-resonator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Noise features for common  noise types can be obtained by reducing \uffffI2\uffff to \uffffI1\uffff as  well, even though it may be easier to directly use the an-  alytical expression for \uffffI2\uffff given Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (23) and (25) in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  CONCLUSION  In this paper we have extended the theory of noise  spectroscopy based on the transient transmission to in-  clude quadratic qubit-resonator interactions, which oc-  cur in nanonmechanical and superconducting resonators  coupled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Allowing for arbitrary ini-  tial qubit coherences, we derived and solved the quan-  tum Langevin equations in the presence of a noisy qubit  in first-order perturbation theory and showed that noise  features are imprinted in the averaged resonator trans-  mission and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Measuring both of these quantities  permits the extraction of the ANI \uffffI2\uffff for quadrati-  cally coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' By converting \uffffI2\uffff to the ANI  for linearly coupled systems, one may extract the noise  power spectral density from transmission measurements  using established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Our results expand the scope  of applications of transmission based noise spectroscopy  to a wide variety of systems including superconducting,  charge and spin qubits coupled to electromagnetic cavi-  ties and nanomechanical resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  \uffff�2\uffff\uffff⇡  \uffff�1\uffff\uffff⇡  �\uffff\uffffA2\uffff2\uffff\uffff\uffffA∞\uffff2  Appendix A: Real and imaginary parts of A  In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-  photon coupling (n = 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Introducing the quantity ⇣ = arctan(2�r\uffff\uf8ff),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' they read  Re(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff\uf8ff  2 − e−\uf8fft\uffff2 \uffffcos(�rt)\uf8ff  2 − sin(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffcos([n − 1]⇣ − �rt) + \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffsin([n − 1]⇣ − �rt)\uffff\uffff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Im(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff − �r + e−\uf8fft\uffff2 \uffffsin(�rt)\uf8ff  2 + cos(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffsin([n − 1]⇣ − �rt) − \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffcos([n − 1]⇣ − �rt)\uffff\uffff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (A1)  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Preskill, Quantum Computing in the NISQ era and  beyond, Quantum 2, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Paladino, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Galperin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Falci, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Alt-  shuler, 1/f noise: Implications for solid-state quantum  103  6  In this case the ANI is described by the same functional  form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (25) with the substitution �X2  rms = P\uffff⇡,  where P = ∫ S(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=')d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' is the noise power in the positive  frequency band [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='ir,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='uv] under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The in-  formation on the noise attainable by using the above ex-  pressions hence includes the noise amplitude S0 for white  noise, the root mean square of the noise �Xrms for qua-  sistatic noise and the noise power P for low-frequency  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Relating the noise integrals  Finally, we wish to relate the n = 1 and n = 2 ANIs  since we already know how to extract the power spec-  tral density S from \uffffI1\uffff = \uffffI\uffff from the discussion in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Comparing the expressions for I1 and I2 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (11), one finds the relation  I2(t) = I1(t) − \uffff  t  0 ei�rt′−\uf8fft′\uffff2 ˙I1(t′)dt′,  (26)  where ˙I1(t′) = @tI1(t)\ufffft=t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (26) can be inverted, and  after averaging over the noise configurations, we obtain  \uffffI1(t)\uffff = \uffff  t  0  \uffff ˙I2(t′)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2 dt′ =  \uffffI2(t)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt−\uf8fft\uffff2  + \uffff  t  0  \uffffI2(t′)\uffff\uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − \uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='r − \uf8ff  2 \uffffei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2\uffff  (ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2)2  dt′,  (27)  where the second equality is obtained via partial inte-  gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  After converting the ANI \uffffI2(t)\uffff to \uffffI1(t)\uffff  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (27), one may follow the approach outlined in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32] to obtain noise characteristics from the ANI  in the linearly coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Hence, the power spec-  tral density can also be obtained in quadratically cou-  pled qubit-resonator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Noise features for common  noise types can be obtained by reducing \uffffI2\uffff to \uffffI1\uffff as  well, even though it may be easier to directly use the an-  alytical expression for \uffffI2\uffff given Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (23) and (25) in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  CONCLUSION  In this paper we have extended the theory of noise  spectroscopy based on the transient transmission to in-  clude quadratic qubit-resonator interactions, which oc-  cur in nanonmechanical and superconducting resonators  coupled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Allowing for arbitrary ini-  tial qubit coherences, we derived and solved the quan-  tum Langevin equations in the presence of a noisy qubit  in first-order perturbation theory and showed that noise  features are imprinted in the averaged resonator trans-  mission and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Measuring both of these quantities  permits the extraction of the ANI \uffffI2\uffff for quadrati-  cally coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' By converting \uffffI2\uffff to the ANI  for linearly coupled systems, one may extract the noise  power spectral density from transmission measurements  using established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Our results expand the scope  of applications of transmission based noise spectroscopy  to a wide variety of systems including superconducting,  charge and spin qubits coupled to electromagnetic cavi-  ties and nanomechanical resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  \uffff�2\uffff\uffff⇡  \uffff�1\uffff\uffff⇡  �\uffff\uffffA2\uffff2\uffff\uffff\uffffA∞\uffff2  Appendix A: Real and imaginary parts of A  In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-  photon coupling (n = 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Introducing the quantity ⇣ = arctan(2�r\uffff\uf8ff),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' they read  Re(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff\uf8ff  2 − e−\uf8fft\uffff2 \uffffcos(�rt)\uf8ff  2 − sin(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffcos([n − 1]⇣ − �rt) + \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffsin([n − 1]⇣ − �rt)\uffff\uffff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Im(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff − �r + e−\uf8fft\uffff2 \uffffsin(�rt)\uf8ff  2 + cos(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffsin([n − 1]⇣ − �rt) − \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffcos([n − 1]⇣ − �rt)\uffff\uffff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (A1)  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Preskill, Quantum Computing in the NISQ era and  beyond, Quantum 2, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Paladino, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Galperin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Falci, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Alt-  shuler, 1/f noise: Implications for solid-state quantum  103  6  In this case the ANI is described by the same functional  form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (25) with the substitution �X2  rms = P\uffff⇡,  where P = ∫ S(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=')d!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' is the noise power in the positive  frequency band [!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='ir,!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='uv] under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The in-  formation on the noise attainable by using the above ex-  pressions hence includes the noise amplitude S0 for white  noise, the root mean square of the noise �Xrms for qua-  sistatic noise and the noise power P for low-frequency  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Relating the noise integrals  Finally, we wish to relate the n = 1 and n = 2 ANIs  since we already know how to extract the power spec-  tral density S from \uffffI1\uffff = \uffffI\uffff from the discussion in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Comparing the expressions for I1 and I2 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (11), one finds the relation  I2(t) = I1(t) − \uffff  t  0 ei�rt′−\uf8fft′\uffff2 ˙I1(t′)dt′,  (26)  where ˙I1(t′) = @tI1(t)\ufffft=t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (26) can be inverted, and  after averaging over the noise configurations, we obtain  \uffffI1(t)\uffff = \uffff  t  0  \uffff ˙I2(t′)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2 dt′ =  \uffffI2(t)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt−\uf8fft\uffff2  + \uffff  t  0  \uffffI2(t′)\uffff\uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − \uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='r − \uf8ff  2 \uffffei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2\uffff  (ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2)2  dt′,  (27)  where the second equality is obtained via partial inte-  gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  After converting the ANI \uffffI2(t)\uffff to \uffffI1(t)\uffff  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (27), one may follow the approach outlined in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32] to obtain noise characteristics from the ANI  in the linearly coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Hence, the power spec-  tral density can also be obtained in quadratically cou-  pled qubit-resonator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Noise features for common  noise types can be obtained by reducing \uffffI2\uffff to \uffffI1\uffff as  well, even though it may be easier to directly use the an-  alytical expression for \uffffI2\uffff given Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (23) and (25) in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  CONCLUSION  In this paper we have extended the theory of noise  spectroscopy based on the transient transmission to in-  clude quadratic qubit-resonator interactions, which oc-  cur in nanonmechanical and superconducting resonators  coupled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Allowing for arbitrary ini-  tial qubit coherences, we derived and solved the quan-  tum Langevin equations in the presence of a noisy qubit  in first-order perturbation theory and showed that noise  features are imprinted in the averaged resonator trans-  mission and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Measuring both of these quantities  permits the extraction of the ANI \uffffI2\uffff for quadrati-  cally coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' By converting \uffffI2\uffff to the ANI  for linearly coupled systems, one may extract the noise  power spectral density from transmission measurements  using established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Our results expand the scope  of applications of transmission based noise spectroscopy  to a wide variety of systems including superconducting,  charge and spin qubits coupled to electromagnetic cavi-  ties and nanomechanical resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  \uffff�2\uffff\uffff⇡  \uffff�1\uffff\uffff⇡  �\uffff\uffffA2\uffff2\uffff\uffff\uffffA∞\uffff2  Appendix A: Real and imaginary parts of A  In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-  photon coupling (n = 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Introducing the quantity ⇣ = arctan(2�r\uffff\uf8ff),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' they read  Re(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff\uf8ff  2 − e−\uf8fft\uffff2 \uffffcos(�rt)\uf8ff  2 − sin(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffcos([n − 1]⇣ − �rt) + \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffsin([n − 1]⇣ − �rt)\uffff\uffff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Im(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff − �r + e−\uf8fft\uffff2 \uffffsin(�rt)\uf8ff  2 + cos(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffsin([n − 1]⇣ − �rt) − \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffcos([n − 1]⇣ − �rt)\uffff\uffff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (A1)  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Preskill, Quantum Computing in the NISQ era and  beyond, Quantum 2, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Paladino, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Galperin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Falci, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Alt-  shuler, 1/f noise: Implications for solid-state quantum  104  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The normalized fluctuations in the averaged transmis-  sion probability δ⟪∣An∣2⟫/∣A∞∣2 for quadratic qubit-resonator  couplings (n = 2) as a function of time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Solid lines are  drawn according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (13) with the ANI for quasistatic noise  [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (24)] with root-mean-square δXrms = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='05δ(n)  0  , while the  numerical data points (squares) are obtained as described  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (II B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The parameter values set to ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='002δ(n)  0  ,  κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='1δ(n)  0  , κ1 = κ2 = κ/2, κint = 0, γ1 = γϕ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='025δ(n)  0  ,  T = δ(n)  0  , λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='9, and the resonator-probe detunings are cho-  sen as indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Interestingly, the corrections to the empty resonator  transmission come with a factor ⟨bin⟩n−2, and hence the  transient transmission for linear interactions depends on  the input field while for quadratic interactions it does not,  a characteristic feature of this type of qubit-resonator  coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' To extract noise characteristics, we now pro-  ceed to study the averaged transient transmission and  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Transmission  Taking the absolute value of (10) and averaging over  distinct noise configurations yields  ⟪∣An∣⟫ = ∣A∞∣  √  ξ0 + ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (12)  where ∣A∞∣ =  √  κ1κ2/(∆2r + κ2/4) is the long-time empty  cavity transmission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  ξ0(t) = 1 + e−κt − 2e−κt/2 cos(∆rt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  ξ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='n(t) = 2nge−κt/2 ⎛  ⎝  √  ∆2r + κ2/4  √κ1⟨bin⟩  ⎞  ⎠  2−n  (13)  × (Yn(t)[Re⟨σ0  −⟩Re⟪In⟫ − Im⟨σ0  −⟩Im⟪In⟫]  + Zn(t)[Im⟨σ0  −⟩Re⟪In⟫ + Re⟨σ0  −⟩Im⟪In⟫]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  and we have introduced the functions  Yn(t) = sin(nζ − ∆rt) − e−κt/2 sin(nζ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Zn(t) = cos(nζ − ∆rt) − e−κt/2 cos(nζ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (14)  with ζ = arctan(2∆r/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  In obtaining (12) we have  used the fact that the variance of ∣A∣ is non-zero only  at quadratic order in ε(n) or higher [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The averaged  noise integrals (ANIs) ⟪In⟫ are as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (11) but  with the factor exp(−iλX(t′)) in the integral replaced by  the averaged random phase ⟪exp(−iλX(t′))⟫.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' One may  rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (12) in terms of the normalized fluctuations  of the averaged transmission probability,  δ⟪∣An∣2⟫  ∣A∞∣2  = ⟪∣An∣2⟫ − ξ0∣A∞∣2  ∣A∞∣2  = ξ1,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (15)  The effect of the noise on the averaged transmission can  be assessed in a clear and disentangled fashion by using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (15) as can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Often the resonator is probed on resonance, ∆r = 0,  since one may expect the most pronounced response at  this point in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In this case, one finds  δ⟪∣An∣2⟫  ∣A∞∣2  = 2ng (  κ  2√κ1⟨bin⟩)  2−n  e−κt/2 (1 − e−κt/2)  × [Im⟨σ0  −⟩Re⟪In⟫ + Re⟨σ0  −⟩Im⟪In⟫],  (16)  where ∣A∞∣ = 1 for symmetric mirrors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  The transient  signal for the resonant case is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Phase  We now turn to the averaged phase of the transmission  signal, which has not been investigated in the context of  noise spectroscopy so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' One has  ⟪φn⟫ = ⟪arctan(Im(An)  Re(An))⟫,  (17)  where the general expressions for the real and imagi-  nary parts of the transmission amplitude to leading or-  der in g are straightforward to calculate from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (10)  but lengthy, and we display them in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In the  《Φ2》/π  《Φ1》/π  8《IA212》//Al5  δ(n)  0 t  0  50  0  2  −2  λ ≠ 0  λ = 0  λ ≠ 0  analytical  numerical  0  2  −2  4  ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='002δ(n)  0  6  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Relating the noise integrals  Finally, we wish to relate the n = 1 and n = 2 ANIs  since we already know how to extract the power spec-  tral density S from \uffffI1\uffff = \uffffI\uffff from the discussion in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Comparing the expressions for I1 and I2 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (11), one finds the relation  I2(t) = I1(t) − \uffff  t  0 ei�rt′−\uf8fft′\uffff2 ˙I1(t′)dt′,  (26)  where ˙I1(t′) = @tI1(t)\ufffft=t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (26) can be inverted, and  after averaging over the noise configurations, we obtain  \uffffI1(t)\uffff = \uffff  t  0  \uffff ˙I2(t′)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2 dt′ =  \uffffI2(t)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt−\uf8fft\uffff2  + \uffff  t  0  \uffffI2(t′)\uffff\uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − \uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='r − \uf8ff  2 \uffffei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2\uffff  (ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2)2  dt′,  (27)  where the second equality is obtained via partial inte-  gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  After converting the ANI \uffffI2(t)\uffff to \uffffI1(t)\uffff  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (27), one may follow the approach outlined in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32] to obtain noise characteristics from the ANI  in the linearly coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Hence, the power spec-  tral density can also be obtained in quadratically cou-  pled qubit-resonator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Noise features for common  noise types can be obtained by reducing \uffffI2\uffff to \uffffI1\uffff as  well, even though it may be easier to directly use the an-  alytical expression for \uffffI2\uffff given Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (23) and (25) in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  CONCLUSION  In this paper we have extended the theory of noise  spectroscopy based on the transient transmission to in-  clude quadratic qubit-resonator interactions, which oc-  cur in nanonmechanical and superconducting resonators  coupled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Allowing for arbitrary ini-  tial qubit coherences, we derived and solved the quan-  tum Langevin equations in the presence of a noisy qubit  in first-order perturbation theory and showed that noise  features are imprinted in the averaged resonator trans-  mission and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Measuring both of these quantities  permits the extraction of the ANI \uffffI2\uffff for quadrati-  cally coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' By converting \uffffI2\uffff to the ANI  for linearly coupled systems, one may extract the noise  power spectral density from transmission measurements  using established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Our results expand the scope  of applications of transmission based noise spectroscopy  to a wide variety of systems including superconducting,  charge and spin qubits coupled to electromagnetic cavi-  ties and nanomechanical resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  \uffff�2\uffff\uffff⇡  \uffff�1\uffff\uffff⇡  �\uffff\uffffA2\uffff2\uffff\uffff\uffffA∞\uffff2  Appendix A: Real and imaginary parts of A  In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-  photon coupling (n = 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Introducing the quantity ⇣ = arctan(2�r\uffff\uf8ff),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' they read  Re(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff\uf8ff  2 − e−\uf8fft\uffff2 \uffffcos(�rt)\uf8ff  2 − sin(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffcos([n − 1]⇣ − �rt) + \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffsin([n − 1]⇣ − �rt)\uffff\uffff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Im(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff − �r + e−\uf8fft\uffff2 \uffffsin(�rt)\uf8ff  2 + cos(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffsin([n − 1]⇣ − �rt) − \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffcos([n − 1]⇣ − �rt)\uffff\uffff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (A1)  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Preskill, Quantum Computing in the NISQ era and  beyond, Quantum 2, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Paladino, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Galperin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Falci, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Alt-  shuler, 1/f noise: Implications for solid-state quantum  information, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 86, 361 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Kloe�el and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Loss, Prospects for spin-based quan-  tum computing in quantum dots, Annual Review of Con-  densed Matter Physics 4, 51 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [4] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Cao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Xiao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Guo, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='-  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Guo, Semiconductor quantum computation, National  Science Review 6, 32 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Burkard,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Ladd,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Nichol,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Pan,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Petta, Semiconductor spin qubits (2021),  arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='08863 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='mes-hall].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Clarke and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Wilhelm, Superconducting quantum  bits, Nature 453, 1031 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  104  6  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Relating the noise integrals  Finally, we wish to relate the n = 1 and n = 2 ANIs  since we already know how to extract the power spec-  tral density S from \uffffI1\uffff = \uffffI\uffff from the discussion in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Comparing the expressions for I1 and I2 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (11), one finds the relation  I2(t) = I1(t) − \uffff  t  0 ei�rt′−\uf8fft′\uffff2 ˙I1(t′)dt′,  (26)  where ˙I1(t′) = @tI1(t)\ufffft=t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (26) can be inverted, and  after averaging over the noise configurations, we obtain  \uffffI1(t)\uffff = \uffff  t  0  \uffff ˙I2(t′)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2 dt′ =  \uffffI2(t)\uffff  ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt−\uf8fft\uffff2  + \uffff  t  0  \uffffI2(t′)\uffff\uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − \uffffi!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='r − \uf8ff  2 \uffffei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2\uffff  (ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='pt′ − ei!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='rt′−\uf8fft′\uffff2)2  dt′,  (27)  where the second equality is obtained via partial inte-  gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  After converting the ANI \uffffI2(t)\uffff to \uffffI1(t)\uffff  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (27), one may follow the approach outlined in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [32] to obtain noise characteristics from the ANI  in the linearly coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Hence, the power spec-  tral density can also be obtained in quadratically cou-  pled qubit-resonator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Noise features for common  noise types can be obtained by reducing \uffffI2\uffff to \uffffI1\uffff as  well, even though it may be easier to directly use the an-  alytical expression for \uffffI2\uffff given Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (23) and (25) in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  CONCLUSION  In this paper we have extended the theory of noise  spectroscopy based on the transient transmission to in-  clude quadratic qubit-resonator interactions, which oc-  cur in nanonmechanical and superconducting resonators  coupled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Allowing for arbitrary ini-  tial qubit coherences, we derived and solved the quan-  tum Langevin equations in the presence of a noisy qubit  in first-order perturbation theory and showed that noise  features are imprinted in the averaged resonator trans-  mission and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Measuring both of these quantities  permits the extraction of the ANI \uffffI2\uffff for quadrati-  cally coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' By converting \uffffI2\uffff to the ANI  for linearly coupled systems, one may extract the noise  power spectral density from transmission measurements  using established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Our results expand the scope  of applications of transmission based noise spectroscopy  to a wide variety of systems including superconducting,  charge and spin qubits coupled to electromagnetic cavi-  ties and nanomechanical resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  \uffff�2\uffff\uffff⇡  \uffff�1\uffff\uffff⇡  �\uffff\uffffA2\uffff2\uffff\uffff\uffffA∞\uffff2  Appendix A: Real and imaginary parts of A  In this Appendix we display the real and imaginary parts of the transmission amplitude (10) for a n-type qubit-  photon coupling (n = 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Introducing the quantity ⇣ = arctan(2�r\uffff\uf8ff),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' they read  Re(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff\uf8ff  2 − e−\uf8fft\uffff2 \uffffcos(�rt)\uf8ff  2 − sin(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffcos([n − 1]⇣ − �rt) + \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffsin([n − 1]⇣ − �rt)\uffff\uffff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Im(An) = −  √\uf8ff1\uf8ff2  �2r + \uf8ff2\uffff4\uffff − �r + e−\uf8fft\uffff2 \uffffsin(�rt)\uf8ff  2 + cos(�rt)�r\uffff + gn  \uffff  �2r + \uf8ff2  4  \uffff  \uffff  √\uf8ff1\uffffbin\uffff  \uffff  �2r + \uf8ff2\uffff4  \uffff  \uffff  n−2  e−\uf8fft\uffff2  × \uffff\uffffRe\uffff�0  −\uffffImIn + Im\uffff�0  −\uffffReIn\uffffsin([n − 1]⇣ − �rt) − \uffffRe\uffff�0  −\uffffReIn − Im\uffff�0  −\uffffImIn\uffffcos([n − 1]⇣ − �rt)\uffff\uffff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (A1)  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Preskill, Quantum Computing in the NISQ era and  beyond, Quantum 2, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Paladino, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Galperin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Falci, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Alt-  shuler, 1/f noise: Implications for solid-state quantum  information, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 86, 361 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Kloe�el and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Loss, Prospects for spin-based quan-  tum computing in quantum dots, Annual Review of Con-  densed Matter Physics 4, 51 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [4] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='-O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Cao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Xiao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Guo, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='-  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Guo, Semiconductor quantum computation, National  Science Review 6, 32 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Burkard,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Ladd,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Nichol,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Pan,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Petta, Semiconductor spin qubits (2021),  arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='08863 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='mes-hall].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Clarke and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Wilhelm, Superconducting quantum  bits, Nature 453, 1031 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  104  �  �  ��  �  ��  �  �  �  �  �  �  �  �  ��  �  ��  �  ���  ���������������������������  0  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0002  ⟨bin⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='5  κ  ⟨bin⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='7  κ  ⟨bin⟩ =  κ  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  � �  �  � �  �  � �  �  � � �  � � � � � � � � � � � � � � � � � � � � � � � �  �  �  �  �  �  �  �  �  �  �  �  �  � �  �  � �  �  � �  �  � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �  �  �  �  �  �  � �  �  � �  �  �  � �  �  � �  �  � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �  0  10  20  30  40  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='0004  analytical  numerical  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The transient averaged phase ⟪φn⟫ as a function of  the measurement time t for quasistatic noise at ∆r = 0 for  the cases of linear qubit-resonator couplings (n = 1, top) and  quadratic qubit-resonator couplings (n = 2, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  The  solid lines are drawn according to the approximate analytical  result (18), while the numerical data is obtained as described  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' II B using the exact expression (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' For the case n = 1  we show the averaged phase for several values of the input  field ⟨bin⟩ as indicated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We set κ = δ(n)  0  and the  remaining parameter values are chosen as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  above form the noise average of the phase cannot be per-  formed analytically due to the arctan function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' However,  when the resonator is driven on resonance, ∆r = 0, the  dominant term in the expansion of Im(An) in orders of  ε(n) vanishes, while it is non-zero in the expansion of  Re(An), and one has ∣Im(An)∣ ≪ ∣Re(An)∣ for g ≪ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In  this case, one may keep only the leading-order term in  the Taylor expansion of the arctan function, whereby it  becomes possible to average the phase analytically,  ⟪φn⟫ ≈  ng  1 − eκt/2 (  κ  2√κ1⟨bin⟩)  2−n  × [Re⟨σ0  −⟩Re⟪In⟫ − Im⟨σ0  −⟩Im⟪In⟫].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (18)  As was the case for the corrections to the empty resonator  transmission probability, the phase is also inversely pro-  portional to the input field ⟨bin⟩ for linear couplings but  independent of the input field for quadratic couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' To  underline the validity of the approximation made when  deriving (18), we show a comparison between numerical  and analytical results for the transient averaged phase in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We find excellent agreement, suggesting that the  averaged phase measured at the resonance ∆r = 0 can be  used to infer noise characteristics according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  NOISE SPECTROSCOPY  The aim of noise spectroscopy is the determination of  spectral features of the fluctuations affecting the system  under consideration [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In this section we show that in-  formation on the noise power spectral density S(ω) of the  qubit noise can be obtained from the averaged transmis-  sion and phase by extracting the ANI and investigating  its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Extracting the averaged noise integral  Having obtained expressions for the averaged transmis-  sion and averaged phase, we now turn to investigate how  noise features can be extracted from the measurement of  these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' For n = 1, the extraction of the ANI  and the extraction of noise features from the ANI is de-  tailed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The former relies on the fact that the  ANI does not depend on the probe frequency ωp, allowing  for an extraction scheme based on measurements of the  transmission at two distinct resonator-probe detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  For n = 2, however, the noise integral I2 does depend on  ωp, and hence it is impossible to extract it by recording  the transmission at two distinct values of the probe fre-  quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' However, if in addition the phase is measured,  it is possible to obtain the ANI for both n = 1 and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Since we aim to use the analytical result for the averaged  phase in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (18), we work in the resonant regime where  ∆r is fixed to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' One may then infer the following set  of equations from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (12) and (18),  Im⟨σ0  −⟩Re⟪In⟫ + Re⟨σ0  −⟩Im⟪In⟫ = Υ∣A∣,  Re⟨σ0  −⟩Re⟪In⟫ − Im⟨σ0  −⟩Im⟪In⟫ = Υφ,  (19)  where the functions on the right-hand side are obtained  from the averaged transmission and phase,  Υ∣A∣ = ⟪∣An∣⟫2/∣A∞∣2 − (1 − e−κt/2)  2  2nge−κt/2(1 − e−κt/2)  (  κ  2√κ1⟨bin⟩)  n−2  ,  Υφ = ⟪φn⟫1 − eκt/2  ng  (  κ  2√κ1⟨bin⟩)  n−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (20)  The inhomogeneous linear system of equations (19) can  be uniquely solved for the real and imaginary parts of  ⟪In⟫ if  det  ⎛  ⎜  ⎝  Im⟨σ0  −⟩  Re⟨σ0  −⟩  Re⟨σ0  −⟩ −Im⟨σ0  −⟩  ⎞  ⎟  ⎠  = −∣⟨σ0  −⟩∣  2 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (21)  Hence, the qubit must be initialized in a coherent super-  position to permit the extraction of the ANI from trans-  mission measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' This condition, however, is not  restrictive as it is required to be fulfilled for the noisy  part of the transmission signal to be non-vanishing in  the transient phase in the first place [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (10)] and  hence does not introduce additional constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Common types of noise  Since we have shown that the ANI can be extracted  from transmission measurements even for quadratic  《Φ2》/π  《Φ1》/π  8《IA212》//A  126  qubit-resonator interactions, it is possible to extract noise  features by comparing the measured data to our the-  ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  For this purpose, we now aim to obtain analyti-  cal expressions for the ANI for the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' To pro-  ceed, we assume zero-mean noise and express the factor  ⟪exp(−iλX(t′))⟫ that appears in the ANI in the Gaus-  sian approximation using the noise power spectral density  S(ω) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We may then evaluate the ANI for three noise  types that are common in engineered quantum systems:  white noise, quasistatic noise and low-frequency noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  White noise is characterized by a constant noise am-  plitude, S(ω) = S0, and the corresponding ANI takes the  form  ⟪I2⟫ =  2  ∑  j=1  (−1)j ecjt−λ2S0t/2 − 1  cj − λ2S0/2  ,  (22)  where we introduce the shorthand notation  c1 = i(2ωr − ωq) − γ  2 ,  c2 = i(ωr − ωq + ωp) + κ − γ  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (23)  Fluctuations are quasistatic if they are constant during  a given measurement but change between two measure-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' In this case, the spectral density may be written  as S(ω) = 2πδX2  rmsδ(ω), where δXrms =  √  ⟪δX2⟫ is the  root-mean-square of the noise and δ(ω) is the Dirac delta  distribution, and one finds  ⟪I2⟫ =  2  ∑  j=1  (−1)jeC2  j  λδXrms  [E (Cj) + E (λδXrms  √  2  t − Cj)], (24)  where Cj = cj/(  √  2λδXrms) and E(z) =  √  π/2 erf(z) with  error function erf(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Finally, one may consider low-  frequency noise such as 1/f noise in a frequency band  [ωir,ωuv].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  If large frequencies are sufficiently strongly  suppressed such that the ultraviolet cutoff frequency ωuv  satisfies ωuvt ≪ 1 in the transient phase, the ANI is in  good approximation described by the same functional  form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (24) with the substitution δXrms →  √  P/π,  where P = ∫ S(ω)dω is the noise power in the frequency  band under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' The information on the noise  attainable by fitting the experimental data to the above  expressions hence includes the noise amplitude S0 for  white noise, the root mean square of the noise δXrms for  quasistatic noise and the noise power P for low-frequency  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' While useful when investigating common material  platforms, the approach requires a priori knowledge on  the type of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' We now proceed to introduce a formal-  ism that allows the extraction of noise features in terms  of the power spectral density S(ω) without the need to  make any assumptions on the form of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Relating the noise integrals  In a final step, we wish to relate the n = 1 and n =  2 ANIs since we already know how to extract S from  ⟪I1⟫ via the convolution theorem from the discussion in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Comparing the expressions for I1 and I2 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (11), one straightforwardly finds the relation  I2(t) = ∫  t  0 (eiωpt′ − eiωrt′−κt′/2) ˙I1(t′)dt′,  (25)  where ˙I1(t′) = ∂tI1(t)∣t=t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (25) can be inverted, and  after averaging over the noise configurations, we find  ⟪I1(t)⟫ = ∫  t  0  ⟪ ˙I2(t′)⟫  eiωpt′ − eiωrt′−κt′/2 dt′ =  ⟪I2(t)⟫  eiωpt − eiωrt−κt/2  + ∫  t  0 ⟪I2(t′)⟫iωpeiωpt′ − [iωr − κ  2 ]eiωrt′−κt′/2  [eiωpt′ − eiωrt′−κt′/2]  2  dt′,  (26)  where the second equality is obtained by partial inte-  gration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  After converting the ANI ⟪I2(t)⟫ to ⟪I1(t)⟫  via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (26), one may follow the approach outlined in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' [14] to obtain noise characteristics from the ANI in  the linearly coupled case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Hence, the power spectral den-  sity can also be obtained in quadratically coupled qubit-  resonator systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Noise features of common noise types  can be obtained by reducing ⟪I2⟫ to ⟪I1⟫ as well, even  though it may often be easier to directly use the analyt-  ical expression for ⟪I2⟫ given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (22) and (24) in  practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  CONCLUSION  In this paper we extended the theory of noise spec-  troscopy based on the transient transmission to include  quadratic qubit-resonator interactions, which occur in  nanonmechanical and superconducting resonators cou-  pled to solid-state qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  Allowing for arbitrary ini-  tial qubit coherences, we derived and solved the quan-  tum Langevin equations in the presence of a noisy qubit  in first-order perturbation theory and showed that noise  features are imprinted in the averaged resonator trans-  mission and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Measuring both of these quantities  permits the extraction of the ANI ⟪I2⟫ for quadrati-  cally coupled systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' By converting ⟪I2⟫ to the ANI  for linearly coupled systems, one may extract the noise  power spectral density from transmission measurements  using established methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Our results expand the scope  of applications of transmission-based noise spectroscopy  to a wide variety of systems including superconducting,  charge and spin qubits coupled to electromagnetic cavi-  ties and nanomechanical resonators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This research is supported by the German Research  Foundation [Deutsche Forschungsgemeinschaft (DFG)]  under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 450396347 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' 425217212 - SFB  1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  7  Appendix A: Real and imaginary parts of the transmission amplitude  In this Appendix we display the real and imaginary parts of the transmission amplitude in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' (10) for both linear  (n = 1) and quadratic (n = 2) qubit-resonator couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' They read  Re(An) = −  √κ1κ2  ∆2r + κ2/4[κ  2 − e−κt/2 (cos(∆rt)κ  2 − sin(∆rt)∆r) + gn  √  ∆2r + κ2  4  ⎛  ⎝  √κ1⟨bin⟩  √  ∆2r + κ2/4  ⎞  ⎠  n−2  e−κt/2  × {[Re⟨σ0  −⟩ImIn + Im⟨σ0  −⟩ReIn]cos([n − 1]ζ − ∆rt) + [Re⟨σ0  −⟩ReIn − Im⟨σ0  −⟩ImIn]sin([n − 1]ζ − ∆rt)}],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (A1)  Im(An) = −  √κ1κ2  ∆2r + κ2/4[ − ∆r + e−κt/2 (sin(∆rt)κ  2 + cos(∆rt)∆r) + gn  √  ∆2r + κ2  4  ⎛  ⎝  √κ1⟨bin⟩  √  ∆2r + κ2/4  ⎞  ⎠  n−2  e−κt/2  × {[Re⟨σ0  −⟩ImIn + Im⟨σ0  −⟩ReIn]sin([n − 1]ζ − ∆rt) − [Re⟨σ0  −⟩ReIn − Im⟨σ0  −⟩ImIn]cos([n − 1]ζ − ∆rt)}],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  (A2)  where as in the main text we use the notation ζ = arctan(2∆r/κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Preskill, Quantum Computing in the NISQ era and  beyond, Quantum 2, 79 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
+page_content=' Jirovec, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFAT4oBgHgl3EQfah0B/content/2301.08551v1.pdf'}
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf,len=915
+page_content='The passive symmetries of machine learning  Soledad Villar * 1 2 David W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Hogg * 3 4 5 Weichi Yao 6 George A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Kevrekidis 1 7 Bernhard Sch¨olkopf 8  Abstract  Any representation of data involves arbitrary in-  vestigator choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Because those choices are ex-  ternal to the data-generating process, each choice  leads to an exact symmetry, corresponding to the  group of transformations that takes one possi-  ble representation to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These are the pas-  sive symmetries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' they include coordinate free-  dom, gauge symmetry and units covariance, all of  which have led to important results in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Our  goal is to understand the implications of passive  symmetries for machine learning: Which passive  symmetries play a role (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', permutation sym-  metry in graph neural networks)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' What are dos  and don’ts in machine learning practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We as-  say conditions under which passive symmetries  can be implemented as group equivariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We  also discuss links to causal modeling, and argue  that the implementation of passive symmetries is  particularly valuable when the goal of the learn-  ing problem is to generalize out of sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' While  this paper is purely conceptual, we believe that it  can have a significant impact on helping machine  learning make the transition that took place for  modern physics in the first half of the Twentieth  century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Introduction  Many important ideas in machine learning (ML) have come  from—or been inspired by—mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These  include the kernel trick (Courant & Hilbert, 1953;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Sch¨olkopf  Equal contribution  1Department of Applied Mathematics  and Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Baltimore MD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' USA  2Mathematical Institute for Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Johns Hopkins Univer-  sity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Baltimore MD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' USA 3Center for Cosmology and Particle  Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' New York  NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' USA 4Max-Planck-Insitut f¨ur Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Heideilberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Ger-  many 5Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' a division of the Simons Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' New  York NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' USA 6Department of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Operations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' and Statis-  tics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Stern School of Business,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' New York NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  USA 7Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Los Alamos NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' USA  8Max Planck Institute for Intelligent Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' T¨ubingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Correspondence to: David W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Hogg <david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='hogg@nyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='edu>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Copyright 2023 the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  & Smola, 2002) and the use of statistical mechanics tech-  niques to solve probabilistic problems (Hastings, 1970;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Gelfand, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Here we suggest another connection be-  tween physics and ML, which relates to representation of  observables: When features and labels are represented in a  mathematical form that involves investigator choices, the  ML method (or any relevant function) ought to be written in  a form that is exactly equivariant to changes in those investi-  gator choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These ideas appear in the physics literature as  early as Einstein (1915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' They are given in the introduction  of The Classical Groups by Weyl (1946) as a motivation to  study group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The first sentences of Modern Classical  Physics by Thorne & Blandford (2017) say  [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='] a central theme will be a Geometric Principle:  The laws of physics must all be expressible as  geometric (coordinate-independent and reference-  frame-independent) relationships between geo-  metric objects (scalars, vectors, tensors, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=') that  represent physical entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  This principle leads to the important physical symmetries  of coordinate freedom and gauge symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' a small gen-  eralization would include units covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Each of these  symmetries has led to fundamental results in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Some  of these ideas are also exploited in ML, in particular in the  geometric deep learning literature (Bronstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Weiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We argue—in this purely conceptual  contribution—that analogs of these symmetries could have  big impacts in ML, and enormously increase the scope of  group-equivariant methods in ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  In natural science there are two types of symmetries (see, for  example, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='1 of Rovelli & Gaul 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The first kind  is passive, arising from the arbitrariness of the mathemati-  cal representation described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' An example familiar to  machine learners is the equivariance of functions on graphs  to the relabeling of the graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This is an exact, pas-  sive symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' graph neural network architectures (GNNs)  build this passive symmetry in by design (Bruna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Duvenaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' An example fa-  miliar to physicists is what we call units covariance, which  is the requirement that any correct description of the world  have inputs and outputs with the correct units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This passive  symmetry leads to remarkable results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' we discuss these in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='13724v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='ML]  31 Jan 2023  The passive symmetries of machine learning  2  The second kind of symmetries is active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These are the ones  that must be established by observations and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The fundamental laws of physics do not seem (at current  precision) to depend on position, orientation, or time, which  in turn imply conservation of momentum, angular momen-  tum and energy (the celebrated theorem of Noether 1918).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Of course the motion of a particle depends on time and  position!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' But the fundamental laws governing the motion  don’t themselves appear to depend on the absolute value of  the time, nor the position of the experimental apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Ac-  tive symmetries like these are empirical and can be falsified  by experimental tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Both active and passive symmetries  can be expressed in terms of group or groupoid actions and  equivariances, but their epistemological content and range  of applicability are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  In this contribution, we argue that passive symmetries apply  to essentially all data analysis problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' They have implica-  tions for how we structure ML methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' While we provide  some examples, most of our contributions are conceptual:  Our contributions:  We introduce the concept of passive symmetries to ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  While this is an old concept in classical physics, it has  not been applied widely in ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  We give a formal definition of passive symmetries in  terms of group actions and explain how passive sym-  metries are always in play in data problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  We illustrate with toy examples how enforcing passive  symmetries can improve regressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  We demonstrate that imposing passive symmetries can  lead to the discovery of important hidden objects in a  data problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  We draw connections with causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' One is  that all causal graphs and mechanistic models are con-  strained to be consistent with the passive symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Another is that the determination that a data problem  has all the inputs necessary to express the symmetry  exactly looks like a causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  We provide guidance on how to structure ML models  so that they respect the passive symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We call out  some current standard practices that prevent models  from obeying symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Some of the ideas seem deceptively simple, yet they have  profound implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Since ML has not yet absorbed these  notions as naturally as physics has, we need to discuss analo-  gies to physics in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We have tried to make the  discussion accessible to an ML audience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' the Appendices in-  cludes a glossary that can be used to translate ideas between  physics and ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Figure depicting the difference between active and pas-  sive transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' (Left panel): The passive or alias transforma-  tion corresponding to a rotation of the coordinates through an angle  θ in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The equivariance of the dynamics with respect to  this tranformation is a passive symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' (Right panel): The active  or alibi transformation corresponding to a rotation of the double  pendulum state through an angle −θ in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Equivariance  with respect to this transformation is an active symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Passive symmetries  Passive symmetries arise from redundancies or free param-  eters or investigator choices in the representation of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  They are to be contrasted with the active symmetries, which  arise from observed or empirical invariances of the laws of  physics with respect to parameters, like position, velocity,  particle labeling, or angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Passive symmetries can be es-  tablished with no need of observations, as they arise solely  from the principle that the physical world is independent  of the mathematical choices we make to describe it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The  groups involved in coordinate freedom can be large and  complicated (for example, groups of reparameterizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  A big part of the literature on equivariant ML is implicitly  or explicitly looking at active symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This is possibly  because in most problems the coordinate system is fixed  before the problem is posed, and both training and test data  are expressed in those fixed coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' If a data set is  made with a fixed coordinate system, but still exhibits an  observable invariance or equivariance with respect to (say)  rotations, then that represents an active symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' However,  cases of exact active symmetries are rare;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' they only really  appear in natural-science contexts like protein folding or  cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' For example, in a protein folding problem, the  way the protein folds may not depend on its orientation  in space (rendering the problem actively O(3) equivariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  This finding relies on the (empirical) observation that the lo-  cal gravitational field (on Earth) does not affect the folding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  This may be approximately true or assumed or experimen-  tally established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' it is an active symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In contrast, the  fact that the protein folds in a way that doesn’t depend on  the coordinate system chosen to describe it is absolutely  and always true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' it is not experimentally established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' it is a  passive symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The relationship between active and passive symmetries is  reflected in the relationship between what are sometimes  called active and passive transformations, or alibi and alias  Z  x  x  yThe passive symmetries of machine learning  3  transformations, depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' An active or alibi  transformation is one in which the objects of study are  moved (rotated, translated, interchanged, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' A passive or  alias transformation is one in which the coordinate system in  which the objects of study are described is changed (rotated,  translated, relabeled, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Mathematically, the two kinds  of transformations seem very similar: For example, how do  we know whether we rotated all the vectors in our problem  by 30 deg, or else rotated the coordinate system used by  −30 deg?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The answer is that if you rotated absolutely all  the vectors (and tensors) in your problem, including possi-  bly many latent physical vectors, then there would be no  mathematical difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' But in real problems, where some  vectors can’t be actively rotated (think, for example of the  local gravitational-field vector, or the vector pointing to-  wards the Sun), or some may not be known or measurable,  the two kinds of transformations are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The protein-folding example suggests that it is hard to im-  plement or enforce a true passive symmetry in a real data-  analysis problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' It requires us to incorporate all relevant  contextual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' How do we know if all relevant  features are part of our data set?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We could perform the  protein-folding experiment in a closed, isolated environ-  ment to make sure no external forces are in play;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' this is  impossible for many practical applications, and furthermore  there could still exist fundamental constants that are not  part of our model or knowledge (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Another  approach is to perform the experiment multiple times after  actively putting the molecules into different orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  If the protein folds differently, we learn that the problem  is not symmetric with respect to the 3d coordinates of the  molecule, and therefore when a rotation is performed there  must be at least one more vector that needs to be rotated  as well (for instance, the gravity vector or the eigenvectors  of a stress tensor, say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This identification of all necessary  inputs to establish the passive symmetry is similar to the  problem of performing interventions to learn the existence  of confounding factors in causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We will come  back to the connections to causality in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Once a passive symmetry—and all relevant contextual  information—is identified, we want to write the data analy-  sis problem or learned function such that it is exactly equiv-  ariant with respect to the relevant group: If the representa-  tion or coordinate system of the inputs is changed, the repre-  sentation or coordinate system of the output should change  correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' ML methods that are not constrained to  respect passive symmetries are doomed to make certain  kinds of mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We will provide some examples thereof  in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The most restrictive form of the Geometric Principle quoted  in Section 1 states that physical law must be written in  terms of vectors, tensors, and coordinate-invariant scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  These objects can only be combined by rules set out in  the Ricci calculus (Ricci & Levi-Civita 1900;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' sometimes  Einstein summation notation, Einstein 1916).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This calculus  was introduced to make objects equivariant to coordinate  diffeomorphisms on curved manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In the Ricci calculus,  objects are written in index notation (a scalar has no indices,  a vector has one index, and a k-tensor has k indices), outer  products are formed, and only certain kinds of sums over  pairs of indices are permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' When the inputs to a function  are scalars, vectors, and tensors, and the function conforms  to the rules of the Ricci calculus, the function will produce  a geometric output (a scalar, vector, or tensor,1 depending  on the number of unsummed indices), and the function  will be precisely equivariant to rotations and reflections of  the coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This is how a large class of passive  symmetries are enforced in physics contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  There are many other passive symmetries, including co-  ordinate diffeomorphisms, reparameterizations (including  canonical transformations), units covariance (see Section 3),  and gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Some of these are easy to implement and some  are difficult;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' not all passive symmetries have practical im-  plementations available at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The passive coordinate diffeomorphism symmetry on curved  manifolds has been tremendously important in the devel-  opment of contemporary physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Einstein, for example,  found the equations of general relativity by looking at every  expression to some polynomial degree consistent with the  passive symmetry enforced by the Ricci calculus (he called  this family of polynomials covariant2 ) until he found the  one that reduced to Newtonian gravity in the weak-field  limit (Einstein, 1915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Even before that, while not expressed  in terms of passive symmetry, the discovery of special rela-  tivity (Einstein, 1905) can be thought of as the decision to  move an active symmetry (the observation that the speed of  light is the same for all observers) into a passive symmetry  (Lorentz invariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We stress that these insights were truly  profound (Earman & Glymour, 1978);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' they revolutionized  physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' not too long before Einstein, such considerations  would have been highly unusual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Similarly, passive symme-  tries do not feature in today’s ML practice—if the develop-  ment of physics is any indication, their potential could be  very significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  1It should be noted here that with the word “vector” and “tensor”  here we are making specific technical reference to true vectors  and tensors in 3-space, subject to passive O(3) symmetries, like  (physical) velocities, accelerations, and stress tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We are not  including arbitrary lists or tables of data or coefficients, which are  sometimes called “vectors” and “tensors” in ML contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  2Following the lead of physics, we could in principle call ML  methods equivariant to passive symmetries “covariant”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The passive symmetries of machine learning  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Example: Units covariance  Perhaps the most universal passive symmetry is units  covariance—the behavior of a system doesn’t depend on the  units system in which we write the measured quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' It  is a passive symmetry with extremely useful consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Consider a mass m near the surface of the Earth, close  enough to the surface such that the gravitational field can  be considered to be determined by a constant (not spatially  varying) vector with magnitude g and direction downwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Question 1: If this mass m is dropped (released at rest) from  a height h from above the ground, how much time T does  it take to fall to the ground?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Question 2: If this mass m is  launched from the surface at a velocity of magnitude v at an  angle θ to the horizontal, how much horizontal distance L  will it fly before it hits the surface again?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Assume that only  m, g, h come in to the solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='3 assume that the height h  and the velocity v are both small enough that air resistance,  say, can be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The answers to these questions are almost completely de-  termined by dimensional (or units-covariance) arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The mass m has units of kg, the gravitational acceleration  magnitude g has units of m s−2, the velocity magnitude  v has units of m s−1, the time T has units of s, and the  lengths h and L have units of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The angle θ is dimension-  less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The only possible combination of m, g, h that has units  of time is α  �  h/g, where α is a dimensionless constant,  which doesn’t depend on any of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The only pos-  sible combination of m, g, v, θ that has units of length is  β(θ) v2/g, where β(θ) is a dimensionless function of only  one dimensionless input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' That is, both Questions 1 and 2  can be answered up to a dimensionless prefactor without  any considerations beyond those of the units of the inputs  and outputs, and without any training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' And both of  those answers don’t depend in any way on the input mass  m, which is a fundamental observation (Einstein, 1915).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  This shows that a function can sometimes be inferred from  units covariance only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', from a purely passive symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Units covariance has been introduced to ML methods pre-  viously (Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Bakarji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=',  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' It can help with training, predictive accuracy, and  out-of-sample generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In particular, out-of-sample  generalization improves because the enforcement of the  symmetry enforces scaling properties of the learned func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These, in turn, help make predictions when the test  data are outside the range of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Formal definition  Consider X to be the state space of a specific physical sys-  tem (for instance x ∈ X could be the positions, velocities,  3We will return to this seemingly innocuous point below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  masses, spins, and charges of a set of particles at a given  time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We consider a family of maps {Φi : X → Hi}i∈I  where the Hi are spaces of observations or encodings in-  dexed by i ∈ I (for instance Φi(x) could be the total me-  chanical energy of the system x at a set of times measured  in joules, and Φj(x) could be in kcal, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The family of  maps {Φi}i∈I not only describes the possible observables  but also a way to express those observables in a specific  coordinate system and with specific units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  We say that two encodings Φi and Φj are compatible if there  exists an invertible morphism βi  j : Φi(X) → Φj(X) that  makes the diagram (1) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  X  X  Hi  Hj  id  Φi  Φj  βi  j  (1)  Note that not all observables are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We expect  that observables of different types (for instance, masses  and positions) are not compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Which observables are  compatible depend on the definition of the spaces Hi and  the invertible morphisms β between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The passive symmetries are the groupoid of invertible mor-  phisms β between compatible encodings that make the di-  agram commute P = {β : Φi(X) → Φj(X) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' β ◦ Φi =  Φj, i, j ∈ I}, where the groupoid operator is the com-  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We note that this is a groupoid and not a group  because not every pair of transformations are composable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The groupoid of passive symmetries P consists of all the  possible changes of coordinates or automorphisms between  encodings or observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Critical to the definition is that we establish beforehand  the spaces of possible observables or encodings Hi, the  observables or encodings Φi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' and the class of invertible mor-  phisms β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' For instance, Hi can be in the category of smooth  manifolds with smooth diffeomorphisms, or in the category  of vector spaces with invertible linear transformations, or  orthogonal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  For example, take X to be a protein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Each map Φi could  encode a list of positions of each of its atoms in some coor-  dinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The passive symmetries include reordering  of the atoms, and changes of the coordinate system by any  invertible morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We may restrict Hi to the category of  smooth manifolds where the β’s are diffeomorphisms, or  we may only allow the β’s to be orthogonal transformations  that fix the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Restricting the space of valid coordi-  nates naturally restricts the valid changes of coordinates and  therefore the space of passive symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Active symmetries, on the other hand, can be thought as  transformations of the world that preserve an observable  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' They involve interventions in the physical system,  and therefore they are typically empirical and approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The passive symmetries of machine learning  5  Not every passive symmetry corresponds to an active sym-  metry, nor vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Imposing a passive symmetry on the structure of an ML  model can permit the discovery of scalings, structures, or  missing elements in the physical description of, or predic-  tions about, the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We illustrate these ideas with some  toy examples below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The passive symmetries are seemingly  trivial statements about the world, but they lead to strong  constraints on the laws of physics, and deliver scaling ar-  guments that solve real problems in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' They can also  constrain ML in valuable ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We conjecture that enforc-  ing passive symmetries in ML and data-analysis tasks will  lead to generalization improvements in a wide range of  circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In particular, we make this conjecture even  for problems in which no (or few) active symmetries are  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' After all, most problems (like reading handwriting  or predicting gravitational trajectories near the surface of  the Earth) are not equivariant to rotations, reflections, and  translations, but they are all, in their data-generating pro-  cesses, exactly coordinate free, and exactly agnostic to the  choices made in data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Experiments and examples  Black-body radiation: An important moment in the his-  tory of physics was the discovery that the electromagnetic  radiation intensity Bλ (energy per time per area per solid  angle per wavelength) of thermal black-body radiation can  be described with a simple equation (Planck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 1901)  Bλ(λ) = 2 h c2  λ5  1  exp  h c  λ k T − 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  (2)  where h is Planck’s constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' c is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' λ is  the wavelength of the electromagnetic radiation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' k is Boltz-  mann’s constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' and T is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In finding this  formula, Planck had to posit the existence (and units) of  the constant h = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='62607015 × 10−34 kg m2 s−1 (Planck’s  original value was presented in erg s, which are different  units but the same dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Prior to the introduction  of h, the only dimensionally acceptable expression for the  black-body radiation intensity was Bλ(λ) = 2 c k T/λ4,  which is the long-wavelength (infrared) or high-temperature  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Planck’s discovery solved the “ultraviolet catastrophe”  of classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This is the problem that, classically,  the black-body spectrum, or any thermal object, ought to  contain infinite numbers of excited modes at short wave-  lengths, or high frequencies, and thus infinite energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Planck’s solution seeded the development of quantum me-  chanics, which governs the behavior of all matter at small  scales, and which cuts off the ultraviolet modes through  quantization of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Planck’s problem can be solved almost directly with the  passive symmetry of units covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' That is, the exponen-  tial cut-off of the intensity appears at a wavelength set by  the temperature and a new constant, that must have units  of action (or action times c, or action divided by k, or one  equivalent in terms of the lattice of dimensional features,  see Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  In Figure 2 (left) we perform the following toy experiment:  We generate noisy samples of intensities as a function of  wavelength and temperature according to (2), and the learn-  ing task is to predict the intensity for different values of  wavelengths and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We perform a units covariant  regression (employing the approach of Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2022) us-  ing only λ, T, c, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' a units covariant regression with an extra  dimensional constant found by cross-validation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' and a stan-  dard MLP with no units constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Our results show that  no units-covariant regression for the intensity as a function  of λ, T, c, k can reproduce accurately the intensity Bλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' How-  ever when the regression is permitted to introduce a new  dimensional constant (and remains units covariant given  the new constant), it finds a constant with units (and, less  precisely, magnitude) that is consistent with h (or h times a  combination of c and k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The units covariant model outper-  forms the baseline MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Again, this shows that the passive  symmetry leads to powerful capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Springy double pendulum: The double pendulum con-  nected by springs is a toy example often used in equivariant  ML demonstrations (Finzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The final conditions (position and veloc-  ities of both masses after elapsed time T) are related to the  initial conditions (position and velocities of the masses at  the initial time), and the dynamics is classically chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The system is subject to a passive O(3) symmetry (equivari-  ance with respect to orthogonal coordinate transformations),  and an active O(2) symmetry (equivariance with respect  to rotations and reflections in the 2-d plane normal to the  gravity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The O(3) symmetry is passive, because it is guar-  anteed by the fact that all vectors must be described in a  coordinate system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' nothing physical can change as the vec-  tors undergo passive transformations because of coordinate-  system changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The O(2) symmetry is active, because it is  an experimental fact that if the initial conditions are changed  by an active or alibi rotation in the plane perpendicular to  gravity, the dynamics and final state rotate accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Here  we can see that the O(2) active symmetry corresponds to  the set of transformations in O(3) that fix the gravity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The O(3) passive symmetry requires that the coordinates  of all relevant vectors are transformed identically, the posi-  tions and momenta of both masses and the gravity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  If the model doesn’t contain all relevant vectors as inputs  then the predictions will not be O(3) equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We per-  form an experiment in which we predict the dynamics of the  double pendulum using O(3)-equivariant models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The sym-  metries are implemented by converting the network inputs  The passive symmetries of machine learning  6  10  8  10  7  10  6  10  5  10  4  wavelength  106  109  1012  1015  1018  1021  intensity  test labels  units covariant without extra dimensional constant, RMSE=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='35  units covariant with extra dimensional constant, RMSE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='39  standard MLP (no covariance), RMSE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  ex q1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  ex p1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  ey q1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='6  ey p1  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='4  ez q1  1  0  1  ez p1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  ex q2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='6  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='6  ex p2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  ey q2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  ey p2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='2  ez q2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='8  ez p2  0  30  60  90  120  time  0  30  60  90  120  time  0  30  60  90  120  time  0  30  60  90  120  time  Ground truth  Known-g  Learned-g  Ground truth  Known-g  Learned-g  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' (Left panel) We predict the intensity of black body radiation as a function of wavelength and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' For all experiments  we use an MLP consisting of 3 layers with 20 hidden units each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The standard MLP uses wavelength and temperature as features and it  doesn’t require the output to be dimensionally correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The units covariant without extra constant learns a scaling of the only dimensionally  correct object one can construct with inputs λ, T, c, k (see description main in text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The units covariant with extra dimensional constant  incorporates a constant with units [kg, m, s, K] ∈ [−1, 0, 1]4 as an input feature, it performs a units covariant regression with the original  features λ, T, c, k and the extra constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' It then selects a constant with low validation error and reports the results on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The  constant learned for the depicted plot is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='61e52 kg−1m−1s−1K−1, which is the same units and similar magnitude to the valid physical  combination c k h−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' (Right panel) Performance of learning the dynamics of the springy double pendulum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We consider the three models  (described in main text): (Known-g) an O(3)-equivariant model where the gravity is an input to the model, (No-g) an O(3)-equivariant  model where the gravity is not given, and (learned-g) an O(3)-equivariant model that uses the position and momenta as well as an  unknown vector that the model learns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The results show that O(3)-equivariance permits the learning of the gravity vector from data with  only minimal impact in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' See Appendix A for a more detailed description of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  (scalars and components of vectors) into invariant scalar  quantities according to the Ricci calculus (which explic-  itly encodes O(3)), building the model in the space of the  invariant scalars (as per Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The models im-  plemented are (Known-g)—an O(3)-equivariant model that  has the positions, momenta and gravity vector all as features  (similar to the models in Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  (No-g)—an O(3)-equivariant model that that is missing  the gravity vector as an input feature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' and (Learned-g)–an  O(3)-equivariant model that has the position, momenta and  an extra unknown vector as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The latter model op-  timizes model weights along with the unknown vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In  the right panel of Figure 2 we show the performance of the  three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We remark that in (Learned-g), the learned  vector in the performed experiments was nearly parallel to  the true (but unknown) gravity vector g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' the angle between  the learned and true gravity vector was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='00016 radians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Connections with causality  There is nothing statistical about the notion of passive sym-  metries, and thus everything we have said above also applies  to causal models (Peters et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' There are, however, a  few comments specific to causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The passive symmetry discussed in Section 3—and indeed  all passive symmetries—can also deliver information per-  taining to the (hard problem of) inference of causal structure:  treating g as a constant, we can construct a structural causal  model with the following vertices: (a) an initial value of v,  (b) a value of m, chosen independently, and (c) a final value  of L, affected by a noise term θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Time ordering implies that  possible causal arrows are from v, m, θ to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' As argued  above, dimensional analysis rules out the arrow m → L,  leaving us with the non-trivial result that in the causal graph,  only v, θ cause L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' As in Section 3, this conclusion can be  reached without any training data or interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  That said, dimensional analysis makes a strong assumption,  which is that all relevant quantities for predicting L have  been specified in the list m, g, v, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' For example, if the pro-  jectile is large enough or the speed v is high enough, air  resistance will come into play, and the size of the object and  the density of air will enter, bringing new variables and new  combinations of variables that matter to the answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This  difficulty is related to the problem in causal inference of  knowing or specifying all possible confounding variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  This can also be linked to the notion of experimental inter-  ventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Suppose we assume that only certain quantities  come into the solution (say, m, g, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' How would we con-  firm this in practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In essence, this is not a probabilistic  statement, but one about the behavior of a system under  interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' A set of experiments can indicate that a cer-  Scenario  Known-g  Learned-g  No-g  State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='RelErr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='0052 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='0004  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='0054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='0011  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='3160 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='0014The passive symmetries of machine learning  7  tain outcome (or effect variable) depends on a certain set of  input (cause) variables but is independent of certain other  potential cause variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In this case, the physical law is  not inferred from dimensional arguments alone, but from a  combination of dimensional and causal arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Even if interventions are not available (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', for g), physicists  trying to infer a law will not do so based (purely) on input-  output data: they will have prior knowledge from related  problems informing them as to which variables are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', we may know from having previously solved a related  problem that we expect a problem to depend on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This  is a form of qualitative transfer that we expect will also  become relevant for model transfer in ML (Rojas-Carulla  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Connections to current ML practice  Most ML implementations don’t impose exact symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Sometimes they approximate equivariances by means of  data augmentation (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  In the present work we focus on exact symmetries: Given  data spaces X and Y and a group G acting on X and Y ,  equivariant ML restricts the function space to those satis-  fying f(g · x) = g · f(x) for all f ∈ F, g ∈ G, x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  There are two main approaches to perform optimization in  the space of equivariant functions:  Explicitly parameterizing the space of equivariant func-  tions via equivariant layers or weight sharing (Cohen &  Welling, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Kondor & Trivedi, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=',  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Geiger & Smidt, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Finzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Finding a set of invariant features and expressing the  equivariant functions in terms of those features (Villar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Blum-Smith & Villar, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The two approaches are theoretically equivalent, but their  practical implementation may be dramatically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' For  example, efficiently computing a complete generating set  of invariant/equivariant features may be prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' On the  other hand, in some cases it may hard to construct a perfectly  invariant/equivariant layer (or family of approximating func-  tions);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' more often, it may be possible to construct such a  family, but we may lack proof that they are universal approx-  imators of all invariant/equivariant functions within some  well-defined context, even in a limiting sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Aside from the previously mentioned results in Convolu-  tional/Graph Neural Networks, another example of success-  ful exact universal parametrization of a family of functions  is the implementation of symplectic networks, which ex-  actly preserve a (not-necessarily-known) differential 2-form  on a manifold by acting on ambient space (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Burby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Even the non-trivial diffeomorphism  symmetries of general relativity have been considered for  ML (Weiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Equivariant ML models can predict the properties and be-  haviour of physical systems (see Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2019), and  have plenty of scientific applications (Batzner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Musaelian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' St¨ark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The implicit bias, generalization error,  and sample complexity of equivariant ML models have been  recently studied (Lawrence et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Bietti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Elesedy & Zaidi, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Elesedy, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Mei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Dos and Don’ts  MacKay famously wrote (see Muldoon 2021)  Principal Component Analysis is a dimensionally  invalid method that gives people a delusion that  they are doing something useful with their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  If you change the units that one of the variables  is measured in, it will change all the “principal  components”  This comment is aligned with our mission, but also mis-  leading: If a rectangular data set contains only data with  identical units (that is, all features of all records have the  same units), then PCA does excactly the right thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' That  said, if a rectangular data set has features with different units  (for example, if every record contains a position, a tempera-  ture, a voltage, and a few intensities), then indeed the output  of PCA will be extremely sensitive to the units system in  which the features are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' If PCA is run on such a  data set, the subsequent data model or data manipulations  will be, by construction, asymmetric or not consistent with  the passive symmetry of units covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The choice to use  PCA on such a data set is the choice to be wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Consider a kernel function with inputs that are lists of fea-  tures with different units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' If the kernel function involves,  say, an exponential of a sum of squares of differences of the  input features, the output of the kernel function cannot obey  the passive symmetry of units covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Quantities with  different units cannot be summed, and dimensional quanti-  ties cannot be exponentiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' On the other hand, if a kernel  function can be chosen that is units covariant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', if all  features have the same units, or if the kernel is constructed  from tensor products of kernels, each of which uses only one  type of input), then the result of a kernel algorithm can be  covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These considerations are relevant for maximum  margin hyperplane (in SVMs), eigenvectors in kernel PCA  (Sch¨olkopf & Smola, 2002), or Gaussian processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Learning involves optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Optimization is of a scalar  cost function (a number, which is a function of many param-  eters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' If passive geometric groups are in play, like O(3),  the parameters that are explicitly or implicitly components  of vectors can only be combined into the scalar objective  through the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Otherwise the scalar objective  isn’t scalar in the geometric sense of “invariant to O(3)”,  The passive symmetries of machine learning  8  and the optimization won’t return a result that is invariant  (or equivariant) to O(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Similarly, if the components of the  vector are normalized differently before they are summed  in quadrature, the objective won’t be invariant to O(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' And  similarly, if all the different contributions to the objective  aren’t converted to the same units before being combined  into the objective, then the model won’t be units covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The common practices of making objectives with functional  forms other than Euclidean norm, normalizing features with  data ranges, and combining features with different units, all  make common ML methods, by construction, inconsistent  with the passive symmetries in play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Neural nets, in their current form, violate many rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  For example: Transcendental functions like exp() and  arctanh() and most other nonlinear functions can only  be applied to scalars—that is, not components of vectors  or tensors but only scalars—and only dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' That  means that the nonlinearities in neural networks are predi-  cated on the weights removing the units of the input features,  and the linear combinations performing some kind of dot  products on the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' That, in turn, means that the internal  weights in the bottom and top layers of a neural network  implicitly have geometric properties and units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' They have  the geometric properties and units such that the latent vari-  ables passed into the nonlinear functions are dimensionless  scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Because they have these properties, a trained neural  network cannot be covariant in the end, unless the inputs  and outputs are already covariant scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  There are exceptions to the restrictions on nonlinear func-  tions: If nonlinearities are mathematically homogeneous, as  it is for a pure monomial, or for the RELU function, dimen-  sional scalars (but not vector or tensor components) can be  taken as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' It’s interesting to ask whether the success  of RELU in ML might be related to its homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  L1 and L∞ norms are almost always inconsistent with the  passive symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This is because the sum of absolute  values of input components, and the maximum of inputs,  are rarely either geometrically, or from a units perspective,  covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' There is a rare exception if all features have the  same units, and none of the features are components of  geometric objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Similarly, regularizers such as those favoring flat loss min-  ima (Hochreiter & Schmidhuber, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Dinh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Petzka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2021) are often not units covariant, changing  their values under certain weight transformations that leave  the overall function invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' If reformulated as a regular-  izer that is a covariant function of the training points, this  problem vanishes (von Luxburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Finally, we mention that passive symmetries play a cru-  cial role also when it comes to latent variable models and  ICA, since unobserved latent factors usually come with a  large class of allowed gauge transformations (permutations,  coordinate-wise nonlinear transformations) which should  be incorporated correctly when studying notions of identifi-  ability (Khemakhem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Buchholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Discussion  In this conceptual contribution, we argue that passive sym-  metries are in play in essentially all ML or data-analysis  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' They are exact, and true by definition, since they  emerge from the redundancies or freedom in coordinate  systems, units, or data representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Enforcement of these  symmetries should improve enormously the generalization  capabilities of ML methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We demonstrate this with toy  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  In practice, implementation of the passive symmetries in an  ML problem might be very difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' One reason is that the  symmetries are only exact when all relevant problem param-  eters (including often fundamental, unvaried constants) are  known and included in the learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' If the prob-  lem has a passive symmetry by a group G, but there are  missing elements K in the problem formulation (such as  Planck’s constant or the gravity vector in Section 5), then  the symmetry that is actually in play is the subgroup H  of G that fixes K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Naively there should be no difference  in the in-distribution performance between enforcing the  symmetry by H, or including K to the inputs and enforcing  the symmetry induced by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' However, using the full group  equivariance is conceptually more elegant and it allows for  out-of-distribution generalization (the model can generalize  to settings where K has changed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These unknown constants  or features K are pieces of essential contextual information  and can be hard to find or learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In our toy examples we  show that with sufficient knowledge of the problem (rich  training data and knowledge of the group of passive sym-  metries) the relevant constant K can be learned from data,  including the Planck constant (for the blackbody-radiation  problem) and the gravitational acceleration vector (for the  double-pendulum example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Identifiability issues may arise  when more constants or non-constant features are missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Another difficulty is that some kinds of symmetries are  hard to enforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' For example, complete coordinate diffeo-  morphisms and problem reparameterizations involve enor-  mous groups which are hard to implement in a realistic  ML method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' That said, many groups have been imple-  mented usefully, including translations, rotations, permuta-  tions, changes of units, and some coordinate transformations  (see Weiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2021 for a review of the latter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  In addition to the exact (and true by definition) passive  symmetries, and the observed active symmetries, there are  other kinds of approximate or weakly broken symmetries  we might call observer symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These arise from the  The passive symmetries of machine learning  9  point that the content of a data record (an image, say) is  independent of the minor choices made by the observer in  taking that data record (shooting the image, say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The details  of the six-axis location and orientation of the camera, and  of the exposure time and focus, can be changed without  changing the semantic or label content of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These  symmetries are approximate, because these changes don’t  lead to invertible changes in the recorded data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' there is no  group or groupoid in the space of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' However, the  success of convolutional structure in image models might  have to do with the importance of these observer symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  There is much more to do in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Acknowledgement: It is a pleasure to thank Ben Blum-  Smith (JHU) for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This project was  started at the meeting Machine Learning for Science at  Schloss Dagstuhl, 2022 September 18–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' SV was partially  supported by ONR N00014-22-1-2126, the NSF–Simons  Research Collaboration on the Mathematical and Scien-  tific Foundations of Deep Learning (MoDL) (NSF DMS  2031985), NSF CISE 2212457, and an AI2AI Amazon re-  search award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
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+page_content='  ISBN 978-0-262-03731-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
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+page_content='  The passive symmetries of machine learning  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Springy double pendulum  We consider the dissipationless spherical double pendulum with springs, with a pivot o and two masses connected by springs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The kinetic energy T and potential energy U of the system are given by  KE = |p1|2  2m1  + |p2|2  2m2  ,  (3)  PE = 1  2k1(|q1 − qo| − l1)2 + 1  2k2(|q2 − q1| − l2)2 − m1 g · (q1 − qo) − m2 g · (q2 − qo),  (4)  where q1, p1 are the position and momentum vectors for mass m1, similarly q2, p2 for mass m2, and a position qo for the  pivot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The springs have scalar spring constants k1, k2, and natural lengths l1, l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The gravitational acceleration vector is g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  In this work, we fix qo with values (0, 0, 0) in base length units and g with (0, 0, −1) in base acceleration units, as well as  (m1, m2, k1, k2, l1, l2) set to (1, 1, 1, 1, 1, 1), but with each element of that list having appropriate base units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The prediction task is to learn the positions and momenta over a set of T later times t given the initializations of the  pendulum positions and momenta at t0,  z(t) = (q1(t), q2(t), p1(t), p2(t)),  t ∈ {t0, t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' , tT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  (5)  The training inputs consist of N = 500 different initializations of the pendulum positions and momenta {z(i)(t(i)  0 )}N  i=1, and  the labels are the set of positions and momenta {z(i)(t(i)  1 ), z(i)(t(i)  2 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' , z(i)(t(i)  T )}N  i=1 with T = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The model is evaluated  on a test data set with T = 150 and t0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  For the same prediction task, we consider three different O(3)-equivariant models, fKnown-g, fLearned-g and fNo-g, depending  how the gravitational acceleration vector g is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Known-g  The model fKnown-g is a function that predicts the dynamics:  fKnown-g : (R3)4 × R3 × R3 × R → (R3)4  (z(0), qo, g, ∆t) �→ ˆz(∆t)  (6)  where g is known as (0, 0, −1) in the base acceleration units and used with positions and momenta as input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Learned-g  The model fLearned-g is a function that predicts the dynamics:  fLearned-g : (R3)4 × R3 × R → (R3)4  (z(0), qo, ∆t) �→ ˆz(∆t)  (7)  where g is unknown but set as an learnable variable and used with positions and momenta as input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  No-g  The model fNo-g is a function that predicts the dynamics:  fNo-g : (R3)4 × R3 × R → (R3)4  (z(0), qo, ∆t) �→ ˆz(∆t)  (8)  where g is unknown and not used as an input feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  We evaluate the performance of the three predictive models based on the state relative error at a given time t in terms of the  positions and momenta of the masses,  State.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='RelErr(t) =  �  (ˆz(t) − z(t))⊤(ˆz(t) − z(t))  �  ˆz(t)⊤ˆz(t) +  �  z(t)⊤z(t)  ,  t ∈ {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' , tT },  (9)  where ˆz(t) denotes the predicted positions and momenta at time t and z(t) the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  The passive symmetries of machine learning  13  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Glossary  active symmetry: A symmetry is active when it is an observed or empirical regularity of the laws of physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Examples  include the observation that the fundamental laws don’t depend on the location or time at which the experiment takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  conservation law: We say that a quantity obeys a conservation law if changes in that quantity (with time) inside some closed  volume can are quantitatively explained by fluxes of that quantity through the surface of that volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Active symmetries can  lead to conservation laws in dynamical systems (Noether, 1918).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  coordinate freedom: When physical quantities are measured, or represented in a computer, they must be expressed in some  coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The redundancy of this representation—the fact that the investigator had many choices for the coordinate  system—leads to the passive symmetry coordinate freedom: If the inputs to a physics problem are moved to a different  coordinate system (because of a change in the origin or orientation), the outputs of the problem must be correspondingly  moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' In much of the literature “coordinate freedom” is only used in relationship to general covariance, but it applies in all  contexts (including non-physics contexts) in which a coordinate system has been chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  covariance: When a physical law is written in a way that is equivariant with respect to all (or some) passive symmetries,  then the law is sometimes said to be covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  dimensional analysis: The technique in physics of deducing scalings by consideration of units covariance is dimensional  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  equivariance: Let G be a group that acts on vector spaces X and Y as ρX and ρY respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' We say that a function  f : X → Y is equivariant if for any group element g ∈ G and any possible input x, the function obeys f(ρX(g)x) =  ρY (g) · f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The actions of G in X and Y induce an action on the space of maps from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' If f ∈ Maps(X,Y) then  g · f = ρY (g) ◦ f ◦ ρX(g)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The equivariant maps are the fixed points of this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Equivariances define symmetries in  the space of maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  gauge freedom: Some physical quantities in field theories (for example the vector potential in electromagnetism) have  additional degrees of freedom that go beyond the choice of coordinate system and units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These freedoms lead to additional  passive symmetries that are known as gauge freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  general covariance: The covariance of relevance in general relativity (Einstein, 1916) is known as general covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  Because general relativity is a metric theory in 3 + 1 spacetime dimensions with invariance with respect to arbitrary  diffeomorphisms, this is a very strong symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' General covariance is sometimes called “coordinate freedom”, but it is a  special case thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  invariance: An equivariance in which the action in the output space is trivial is called an invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Physicists sometimes  use invariant (gauge invariant, for example) for things we would call covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  passive symmetry: A symmetry is passive when it arises from a choice in the representation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Examples include  coordinate freedom, gauge freedom, and units covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' These symmetries are exact and true by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  scalar: A number (with or without units), whose value does not depend on the coordinate system in which it is represented,  is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Thus, say, the charge of a particle is a scalar, but the x coordinate of its velocity is not a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  symmetry: Given a mathematical object X of any sort, (like a manifold, metric space, equation, etc), any mapping of the  object onto itself that preserves the corresponding structure is a symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  tensor: A linear function of k − 1 vectors that outputs a vector, or a linear function of k vectors that outputs a scalar, is a  k-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' A rectangular array of data is not usually a tensor according to this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' A vector can be seen as a 1-tensor,  and a scalar can be seen as a 0-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  units: All physical quantities are measured with a system of what we call units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' A quantity can be transformed from one  unit system to another by multiplication with a dimensionless number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' Almost all quantities—including almost all scalars,  vectors, and tensors—have units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  units covariance: The left-hand side and the right-hand side of any equation must have the same units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' This symmetry is  called (by us) units covariance (contra Villar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' 2022 where it is called “units equivariance”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content='  vector: An ordered list of d numbers, all of which have the same units, that is subject to the passive O(d) symmetry  corresponding to coordinate-system rotations, is a vector in d dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' A generic list of d features is not usually a vector  The passive symmetries of machine learning  14  according to this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' The inner (or dot) product of two vectors produces a scalar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
+page_content=' for this reason a vector can be seen  as a 1-tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFST4oBgHgl3EQfHDjE/content/2301.13724v1.pdf'}
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+arXiv:2301.08554v1  [physics.bio-ph]  18 Jan 2023
+Magnetic isotope effects: a potential testing ground
+for quantum biology
+Hadi Zadeh-Haghighi1,2,3,* and Christoph Simon1,2,3,*
+1Department of Physics and Astronomy, University of Calgary, Calgary, AB, T2N 1N4, Canada
+2Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada
+3Hotchkiss Brain Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada
+*hadi.zadehhaghighi@ucalgary.ca & csimo@ucalgary.ca
+ABSTRACT
+One possible explanation for magnetosensing in biology, such as avian magnetoreception, is based
+on the spin dynamics of certain chemical reactions that involve radical pairs. Radical pairs have been
+suggested to also play a role in anesthesia, hyperactivity, neurogenesis, circadian clock rhythm, micro-
+tubule assembly, etc. It thus seems critical to probe the credibility of such models. One way to do so
+is through isotope effects with different nuclear spins. Here we briefly review the papers involving spin-
+related isotope effects in biology. We suggest studying isotope effects can be an interesting avenue for
+quantum biology.
+Keywords: magnetic isotope effects in biology, radical pair mechanism, quantum spin, spin chemistry
+I
+n atoms, the number of protons determines the element (e.g. carbon, oxygen, etc.), and the number of
+neutrons determines the isotope of the desired element. It has been observed that different isotopes
+of the element in certain chemical reactions can influence the outcomes differently. This has been shown
+in many chemical reactions1–8 including biological systems9–14. Not only do different isotopes of an
+element have different masses, but they can also possess different spin angular momentum, which has a
+magnetic property. Thus, one can consider isotope effects in (bio)chemical reactions from two distinct
+points of view: mass- and spin-dependency.
+Isotope effects have been reported for numerous (bio)chemical reactions13–22. Sechzer and et al.
+observed that administering different lithium isotopes resulted in different parenting behaviors and po-
+tentially delayed offspring development in rats23. In 2020, Ettenberg co-workers24 reported that lithium
+isotope effect on rat’s hyperactivity, where 6Li produced a longer suppression of hyperactivity in an ani-
+mal model of mania compared to 7Li. Buchachenko et al. reported that ATP production was more than
+twofold in the presence of 25Mg compared to 24Mg. They suggested that the different nuclear spin of
+these isotopes was the key to these observations. The same group, in multiple studies, also observed that
+25Mg reduced enzymatic activity in DNA synthesis compared to 24Mg, where the rate of DNA synthe-
+sis was suggested to be magnetic field-dependent25–27. They also observed isotope effects by replacing
+magnesium with calcium and zinc ions28,29. Li et al. observed that different xenon isotopes induced
+anesthesia in mice differently. In that experiment, four different xenon isotopes were used, 129Xe, 131Xe,
+132Xe, and 134Xe with nuclear spins of 1/2, 3/2, 0, and 0, respectively30. They reported that isotopes of
+xenon with non-zero nuclear spin had lower anesthetic potency than isotopes without nuclear spin.
+The first mass-independent isotope effect was detected by Buchachenko and co-workers in 197631, in
+which applied magnetic fields discriminated isotope effects by their nuclear spins and nuclear magnetic
+moments. Since then, the term "magnetic isotope effect" was introduced for such phenomena as they are
+1
+
+controlled by electron-nuclear hyperfine coupling in the paramagnetic species.
+The sensitivity of biological systems to weak magnetic fields is an intriguing phenomenon32, yet
+incompletely understood. It is challenging to understand because the corresponding energies for such
+low fields are far smaller than the energies for thermal fluctuations and motions. So from a classical point
+of view, these effects should be washed out. But that is not the case.
+One possible explanation for such effects is based on the spin dynamics of naturally occurring radical
+pairs, namely the radical pair mechanism33. Spin has a magnetic property, and thus for every spin, any
+surrounding magnetic field from either other spins or applied magnetic field influences its state. On the
+other hand, spin states can determine which chemical reactions are possible, providing a mechanism for
+magnetic fields to influence chemical reaction products. A considerable amount of studies suggest that
+isotope effects in biology can be due to the spin dynamics of radical pairs in biochemical reactions.
+In the context of avian magnetoreception34, it was suggested that substituting 17O2 for 17O2 would
+strongly attenuate magnetosensing and also accelerate the generation of the fully oxidized state of flavin
+adenine dinucleotide (FADox)35. Recent studies have proposed that radical pair models help explain
+isotope effects in xenon anesthesia36 and lithium treatment for hyperactivity37. In these models, it is
+proposed that anesthesia and hyperactivity involve spin-selective electron transfer, and different isotopes
+of xenon and lithium influence the electron transfer process differently due to the hyperfine interaction
+between the xenon or lithium nuclear spin and the electron spin of the radicals, and hence possess differ-
+ent potency. Based on similar models, it has also been suggested that isotope effects can be tested in the
+role of superoxide in neurogenesis38, the effect of lithium on the circadian clock39, and the effect of zinc
+on microtubule assembly40.
+It is also worth mentioning that non-mass-dependent effects or mass-independent fractionation in
+isotope effects have been observed with oxygen, sulfur, mercury, lead, and thallium41–46, which are
+based on non-magnetic mechanisms. However, it is reported that biomolecules susceptible to oxidation
+by reactive oxygen species (ROS) can be protected using heavier isotopes such as 2H (D, deuterium) and
+13C (carbon-13)47. Moreover, in numerous studies, magnetic field effects in biology are accompanied
+by modulation in the ROS levels32. This suggests radical pairs might be involved in such ROS-related
+effects.
+Exploring isotope effects may thus be a potential avenue to probe the radial pair mechanism hy-
+pothesis and ultimately to see whether Nature harnesses quantum physics in biology. We hope that this
+short article will encourage experimental experts in the field of quantum biology to test isotope effects.
+Furthermore, this could pave new paths for discovering new medicine and treatments.
+Acknowledgment
+The authors would like to thank Michael Wieser for his valuable input. This work was supported by
+NSERC Discovery Grant RGPIN-2020-03945 and National Research Council CSTIP Grant QSP 022.
+References
+1. Bigeleisen, J. Chemistry of isotopes. Science 147, 463–471, DOI: 10.1126/science.147.3657.463
+(1965).
+2. Zel'dovich, Y. B., Buchachenko, A. L. & Frankevich, E. L. Magnetic-spin effects in chemistry and
+molecular physics. Sov. Phys. Uspekhi 31, 385–408, DOI: 10.1070/pu1988v031n05abeh003544
+(1988).
+2/5
+
+3. Wolfsberg, M., Hook, W. A., Paneth, P. & Rebelo, L. P. N. Isotope Effects (Springer Netherlands,
+2009).
+4. Faure, G. Principles of isotope geology (John Wiley and Sons, Inc., New York, NY, 1977).
+5. Hoefs, J. & Hoefs, J. Stable isotope geochemistry, vol. 285 (Springer, 2009).
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf,len=520
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='08554v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='bio-ph]  18 Jan 2023  Magnetic isotope effects: a potential testing ground  for quantum biology  Hadi Zadeh-Haghighi1,2,3,* and Christoph Simon1,2,3,*  1Department of Physics and Astronomy, University of Calgary, Calgary, AB, T2N 1N4, Canada  2Institute for Quantum Science and Technology, University of Calgary, Calgary, AB, T2N 1N4, Canada  3Hotchkiss Brain Institute, University of Calgary, Calgary, AB, T2N 1N4, Canada  hadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='zadehhaghighi@ucalgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='ca & csimo@ucalgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='ca  ABSTRACT  One possible explanation for magnetosensing in biology, such as avian magnetoreception, is based  on the spin dynamics of certain chemical reactions that involve radical pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Radical pairs have been  suggested to also play a role in anesthesia, hyperactivity, neurogenesis, circadian clock rhythm, micro-  tubule assembly, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' It thus seems critical to probe the credibility of such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' One way to do so  is through isotope effects with different nuclear spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Here we briefly review the papers involving spin-  related isotope effects in biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' We suggest studying isotope effects can be an interesting avenue for  quantum biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Keywords: magnetic isotope effects in biology, radical pair mechanism, quantum spin, spin chemistry  I  n atoms, the number of protons determines the element (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' carbon, oxygen, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' ), and the number of  neutrons determines the isotope of the desired element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' It has been observed that different isotopes  of the element in certain chemical reactions can influence the outcomes differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' This has been shown  in many chemical reactions1–8 including biological systems9–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Not only do different isotopes of an  element have different masses, but they can also possess different spin angular momentum, which has a  magnetic property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Thus, one can consider isotope effects in (bio)chemical reactions from two distinct  points of view: mass- and spin-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Isotope effects have been reported for numerous (bio)chemical reactions13–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Sechzer and et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  observed that administering different lithium isotopes resulted in different parenting behaviors and po-  tentially delayed offspring development in rats23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' In 2020, Ettenberg co-workers24 reported that lithium  isotope effect on rat’s hyperactivity, where 6Li produced a longer suppression of hyperactivity in an ani-  mal model of mania compared to 7Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Buchachenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' reported that ATP production was more than  twofold in the presence of 25Mg compared to 24Mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' They suggested that the different nuclear spin of  these isotopes was the key to these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' The same group, in multiple studies, also observed that  25Mg reduced enzymatic activity in DNA synthesis compared to 24Mg, where the rate of DNA synthe-  sis was suggested to be magnetic field-dependent25–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' They also observed isotope effects by replacing  magnesium with calcium and zinc ions28,29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' observed that different xenon isotopes induced  anesthesia in mice differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' In that experiment, four different xenon isotopes were used, 129Xe, 131Xe,  132Xe, and 134Xe with nuclear spins of 1/2, 3/2, 0, and 0, respectively30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' They reported that isotopes of  xenon with non-zero nuclear spin had lower anesthetic potency than isotopes without nuclear spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  The first mass-independent isotope effect was detected by Buchachenko and co-workers in 197631, in  which applied magnetic fields discriminated isotope effects by their nuclear spins and nuclear magnetic  moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Since then, the term "magnetic isotope effect" was introduced for such phenomena as they are  1  controlled by electron-nuclear hyperfine coupling in the paramagnetic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  The sensitivity of biological systems to weak magnetic fields is an intriguing phenomenon32, yet  incompletely understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' It is challenging to understand because the corresponding energies for such  low fields are far smaller than the energies for thermal fluctuations and motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' So from a classical point  of view, these effects should be washed out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' But that is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  One possible explanation for such effects is based on the spin dynamics of naturally occurring radical  pairs, namely the radical pair mechanism33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Spin has a magnetic property, and thus for every spin, any  surrounding magnetic field from either other spins or applied magnetic field influences its state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' On the  other hand, spin states can determine which chemical reactions are possible, providing a mechanism for  magnetic fields to influence chemical reaction products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' A considerable amount of studies suggest that  isotope effects in biology can be due to the spin dynamics of radical pairs in biochemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  In the context of avian magnetoreception34, it was suggested that substituting 17O2 for 17O2 would  strongly attenuate magnetosensing and also accelerate the generation of the fully oxidized state of flavin  adenine dinucleotide (FADox)35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Recent studies have proposed that radical pair models help explain  isotope effects in xenon anesthesia36 and lithium treatment for hyperactivity37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' In these models, it is  proposed that anesthesia and hyperactivity involve spin-selective electron transfer, and different isotopes  of xenon and lithium influence the electron transfer process differently due to the hyperfine interaction  between the xenon or lithium nuclear spin and the electron spin of the radicals, and hence possess differ-  ent potency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Based on similar models, it has also been suggested that isotope effects can be tested in the  role of superoxide in neurogenesis38, the effect of lithium on the circadian clock39, and the effect of zinc  on microtubule assembly40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  It is also worth mentioning that non-mass-dependent effects or mass-independent fractionation in  isotope effects have been observed with oxygen, sulfur, mercury, lead, and thallium41–46, which are  based on non-magnetic mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' However, it is reported that biomolecules susceptible to oxidation  by reactive oxygen species (ROS) can be protected using heavier isotopes such as 2H (D, deuterium) and  13C (carbon-13)47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Moreover, in numerous studies, magnetic field effects in biology are accompanied  by modulation in the ROS levels32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' This suggests radical pairs might be involved in such ROS-related  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Exploring isotope effects may thus be a potential avenue to probe the radial pair mechanism hy-  pothesis and ultimately to see whether Nature harnesses quantum physics in biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' We hope that this  short article will encourage experimental experts in the field of quantum biology to test isotope effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Furthermore, this could pave new paths for discovering new medicine and treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Acknowledgment  The authors would like to thank Michael Wieser for his valuable input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' This work was supported by  NSERC Discovery Grant RGPIN-2020-03945 and National Research Council CSTIP Grant QSP 022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Bigeleisen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Chemistry of isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Science 147, 463–471, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='3657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='463  (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=" Zel'dovich, Y." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Frankevich, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Magnetic-spin effects in chemistry and  molecular physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Uspekhi 31, 385–408, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='1070/pu1988v031n05abeh003544  (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  2/5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Wolfsberg, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Hook, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Paneth, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Rebelo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Faure, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Principles of isotope geology (John Wiley and Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', New York, NY, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Hoefs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Hoefs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Stable isotope geochemistry, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' 285 (Springer, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Fry, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Stable isotope ecology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' 521 (Springer, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Van Hook, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Isotope effects in chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Nukleonika 56, 217–240 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Magnetic isotope effect: nuclear spin control of chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' The J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1021/jp011261d (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Cook, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Enzyme mechanism from isotope effects (Crc Press, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Grissom, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Magnetic field effects in biology: A survey of possible mechanisms with emphasis  on radical-pair recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Kohen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Isotope effects in chemistry and biology (cRc Press, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Magnetic isotope effect in chemistry and biochemistry (Nova Science Publishers  New York, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Koltover, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Magnetic field-dependent molecular and chemical processes in biochemistry, ge-  netics and medicine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Magnetic control of enzymatic phosphorylation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Bukhvostov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Napolov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Kuznetsov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' A new platform for anti-cancer  experimental pharmacology: the dna repair enzyme affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Brit J Pharmacol Toxicol 5, 35–41  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Bukhvostov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Ermakov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Kuznetsov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Nuclear spin selectivity in en-  zymatic catalysis: A caution for applied biophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Biochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='abb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Arkhangelskaya, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Vorobyeva, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Leonov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=', Osipov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Mag-  netic isotope effect on the repair of radiation-induced DNA damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=', Ermakov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Kuznetsov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' A specific role of mag-  netic isotopes in biological and ecological systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' physics and biophysics beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Biophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Magnesium magnetic isotope effects in microbiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Microbiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1007/s00203-021-02219-4 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Sechzer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=', Lieberman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1016/0006-3223(86)90308-2 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Ettenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Pharmacol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Biochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Magnetic control of the  DNA synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='cplett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Buchachenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Breslavskaya, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' & Simon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Entangled radicals may explain lithium effects on hyperactivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Reports 11, DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=', Salahub, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' & Simon, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Radical pairs may explain reactive  oxygen species-mediated effects of hypomagnetic field on neurogenesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' PLOS Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='031405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='125026 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='  Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Role of nuclear volume in driving equilibrium stable isotope fractionation of mer-  cury, thallium, and other very heavy elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Geochimica et Cosmochimica Acta 71, 2170–2189,  DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='gca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='004 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Shchepinov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='  Reactive oxygen species, isotope effect, essential nutrients, and enhanced  longevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content=' Rejuvenation Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='1089/rej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
+page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
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+page_content='  5/5' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNFAT4oBgHgl3EQfbB0a/content/2301.08554v1.pdf'}
diff --git a/kdAyT4oBgHgl3EQf_fou/content/tmp_files/2301.00908v1.pdf.txt b/kdAyT4oBgHgl3EQf_fou/content/tmp_files/2301.00908v1.pdf.txt
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+Draft version January 4, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+The Spitzer-HETDEX Exploratory Large Area Survey. IV. Model-Based Multi-wavelength
+Photometric Catalog
+Gene C. K. Leung
+,1 Steven L. Finkelstein
+,1 John R. Weaver
+,2 Casey Papovich
+,3, 4
+Rebecca L. Larson
+,1 Katherine Chworowsky
+,1 Robin Ciardullo
+,5, 6 Eric Gawiser
+,7
+Caryl Gronwall
+,5, 6 Shardha Jogee
+,1 Lalitwadee Kawinwanichakij
+,8 Rachel S. Somerville
+,9
+Isak G. B. Wold
+,10 and L. Y. Aaron Yung
+10
+1Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA
+2Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA
+3Department of Physics and Astronomy, Texas A&M University, College Station, TX, USA
+4George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX,
+USA
+5Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA
+6Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
+7Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Rd., Piscataway, NJ 08854,
+USA
+8Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa 277-8583, Japan (Kavli IPMU,
+WPI)
+9Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA
+10Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA
+ABSTRACT
+We present a 0.3–4.5 µm 16-band photometric catalog for the Spitzer/HETDEX Exploratory Large-
+Area (SHELA) survey. SHELA covers a ∼ 27 deg2 field within the footprint of the Hobby-Eberly
+Telescope Dark Energy Experiment (HETDEX). Here we present new DECam imaging and a rizKs-
+band-selected catalog of four million sources extracted using a fully model-based approach. We validate
+our photometry by comparing with the model-based DECam Legacy Survey. We analyze the differences
+between model-based and aperture photometry by comparing with the previous SHELA catalog, finding
+that our model-based photometry can measure point sources to fainter fluxes and better capture the
+full emission of resolved sources. The catalog is 80% (50%) complete at riz ∼ 24.7 (25.1) AB mag, and
+the optical photometry reaches a 5σ depth of ∼ 25.5 AB mag. We measure photometric redshifts and
+achieve 1σ scatter of ∆z/(1 + z) of 0.04 with available spectroscopic redshifts at 0 ≤ z ≤ 1. This large
+area, multi-wavelength photometric catalog, combined with spectroscopic information from HETDEX,
+will enable a wide range of extragalactic science investigations.
+1. INTRODUCTION
+Hobby-Eberly Telescope Dark Energy Experiment
+(HETDEX) (Gebhardt et al. 2021) is an unbiased
+integral-field spectroscopic survey aiming to measure
+Lyα emission from one million galaxies at z = 1.9 − 3.5
+in a 540 deg2 region.
+The SHELA survey (Papovich
+et al. 2016) is a wide-field multi-wavelength photomet-
+ric survey targeting a ∼ 27 deg2 field within the foot-
+print of HETDEX. It is designed to study the physi-
+cal processes, intrinsic and environmental, driving the
+growth of galaxies from z = 1.9 − 3.5. The SHELA sur-
+vey has obtained optical imaging using DECam, near-
+infrared (NIR) imaging using NEWFIRM, and mid-
+infrared imaging using Spitzer/IRAC, and is supple-
+mented by a large amount of publicly available ground-
+based imaging data.
+Early SHELA observations have resulted in three pub-
+lished catalogs.
+Papovich et al. (2016) presented a
+3.6 and 4.5 µm catalog of the SHELA Spitzer/IRAC
+dataset.
+Wold et al. (2019) used data from the first
+phase of SHELA DECam observations and presented
+an riz-selected catalog with DECam ugriz photome-
+try along with the existing IRAC data and JKs data
+from the VICS82 survey (Geach et al. 2017).
+Ste-
+vans et al. (2021) presented a Ks-selected catalog com-
+bining the previous dataset with Ks-band observations
+from the NEWFIRM HETDEX Survey. Since then, we
+have completed the second and final phase of our DE-
+Cam imaging campaign in SHELA. In this paper, we
+arXiv:2301.00908v1  [astro-ph.GA]  3 Jan 2023
+
+ID2
+Leung et al.
+present a new photometric catalog by combining our
+complete 27-deg2 DECam imaging dataset with the ex-
+isting SHELA NEWFIRM and Spitzer/IRAC observa-
+tions. Apart from the new DECam imaging data that
+increase both the width and depth of our survey, we
+discuss below several key differences of this new catalog
+from the previous ones.
+In this catalog, we benefit from a number of publicly
+available imaging datasets in addition to our own imag-
+ing campaign. A crucial addition from the previous cat-
+alogs is the Hyper Suprime-Cam Subaru Strategic Pro-
+gram (HSC-SSP, Aihara et al. 2021), which has observed
+the SHELA field in grizy as a part of its autumn field.
+The publicly available HSC-SSP images in the SHELA
+field are ∼ 0.5 − 1 mag deeper than the DECam images
+in Wold et al. (2019). Combining these deep HSC-SSP
+images with our DECam dataset allows precise measure-
+ment of the optical spectral energy distribution (SED)
+and constrains the physical properties of galaxies. For
+example, deeper g-band imaging is important in reduc-
+ing the contamination rate in the selection of z ≳ 3 − 5
+galaxies.
+The full coverage of the SHELA field with both DE-
+Cam and NEWFIRM supplemented by HSC-SSP now
+allows us to select galaxies using the optical and NIR
+bands simultaneously. As opposed to riz only in Wold
+et al. (2019), we will be able to detect, using the Ks
+band, very red galaxies that peak in the NIR, such as
+massive quiescent galaxies, dusty star-forming galaxies
+and z ≥ 7 Lyman break galaxies. On the other hand,
+the Ks-band selected catalog in Stevans et al. (2021) is
+limited by its depth of ∼ 22.4 mag, and the inclusion of
+deep optical imaging in this catalog that is ∼ 2 − 3 mag
+deeper will significantly increase the completeness, such
+as for z ∼ 2 − 5 star-forming galaxies whose rest-frame
+UV emission falls in the optical wavelengths.
+The previous SHELA catalogs were created entirely
+or partially based on aperture photometry using tools
+such as SourceExtractor (Bertin & Arnouts 1996).
+In such an approach, the images in each band will be
+homogenized to a common PSF, leading to the degra-
+dation of spatial resolution in the images. An alterna-
+tive approach is to model the light profile of sources
+convolved with the measured PSF. Recently, the image
+modeling code the Tractor (Lang et al. 2016) has
+been developed to to perform such model-based pho-
+tometry. It has since been implemented by a number of
+imaging surveys, including the DECam Legacy Survey
+(DECaLS, Dey et al. 2019) and COSMOS2020 catalog
+(Weaver et al. 2022). The model-based approach is cru-
+cial in extracting IRAC photometry, which has a sub-
+stantially larger PSF than the optical bands. For this
+catalog, we extract fully model-based photometry using
+the software package The Farmer (Weaver et al. 2022),
+which generates photometry using the Tractor.
+The large area, multi-wavelength photometric catalog
+of SHELA will enable a variety of extragalactic studies.
+In particular, studies that focus on galaxies spanning a
+wide range of physical parameters, galaxies residing in
+a variety of environments or the search for rare objects
+will benefit the most from the large area of SHELA.
+This paper is organized as follows. In Section 2, we
+describe our observations and the composition of our
+imaging dataset. Section 3 describes the data reduction
+procedures for the images. The photometry extraction
+using The Farmer and the photometry validation pro-
+cess are described in Section 4. Section 5 presents the
+photometric redshift measurements. We give a descrip-
+tion of the catalog published with this paper in Sec-
+tion 6. In this paper we assume a Planck Collaboration
+et al. (2020) cosmology of H0 = 67.4 km s−1 Mpc−1,
+Ωm = 0.315 and ΩΛ = 0.685.
+2. OBSERVATIONS AND DATA
+We observed the SHELA field using the DECam im-
+ager on the Blanco 4m telescope at CTIO in the u-, g-,
+r-, i-, z- filters. New data were obtained over 3 nights in
+the 2019B semester (PI: Papovich; NOAO PID: 2019B-
+0080). Our program also includes data obtained over 5
+nights in the 2013B semester presented in Wold et al.
+(2019). The complete program consists of eight slightly
+overlapping pointings covering a total area of 23 deg2.
+The 2013B data covered six of the eight pointings, while
+the new 2019B data completed the last two pointings
+and obtained additional exposures in the previous posi-
+tions.
+The SHELA field has been observed with DECam by
+other programs, most notably the Dark Energy Survey
+(DES, Dark Energy Survey Collaboration et al. 2016)
+and DECaLS (Dey et al. 2019). We supplement our data
+with archival DECam imaging data within the footprint
+of the SHELA field in the ugriz bands. We also include
+archival data in Y band, which was not observed by our
+proprietary program. All such archival data available
+through the NOIRLab Astro Data Archive1 released
+prior to March 3 2021 were included for this catalog.
+This results in a total of [200, 719, 735, 688, 720, 468]
+exposures in the [u, g, r, i, z, Y ] bands. We show the
+DECam ugrizY weight maps in the SHELA field in Fig-
+ure 1 to demonstrate the footprint and relative depths
+of the DECam data.
+1 https://astroarchive.noirlab.edu/
+
+SHELA catalog
+3
+1
+0
+1
+u
+T1
+T2
+T3
+T4
+T5
+T6
+T7
+1
+0
+1
+g
+1
+0
+1
+r
+1
+0
+1
+                         Declination (J2000)
+i
+1
+0
+1
+z
+16
+18
+20
+22
+24
+26
+Right ascention (J2000)
+1
+0
+1
+Y
+Figure 1. SHELA DECam ugrizY weight maps demonstrating the survey’s footprint and relative depths. The field is covered
+by eight DECam pointings, and is divided into seven tiles of 1.◦98 × 2.◦145, labelled T1 to T7, so that the tiles are roughly
+square. Our survey combines ugriz images from our observations with grizY archival images released prior to March 3 2021.
+We note that T1 has substantially deeper Y -band imaging due to a large number of repeated exposures in the archive.
+
+4
+Leung et al.
+We incorporate multi-wavelength imaging data from
+surveys overlapping with the SHELA field.
+Previous
+SHELA programs have obtained data in the near- and
+mid-infrared wavelengths.
+This includes NEWFIRM
+Ks-band images from the NEWFIRM HETDEX Sur-
+vey (Stevans et al. 2021) and mid-infrared mosaics in
+3.6 µm and 4.5 µm with Spitzer/IRAC presented in Pa-
+povich et al. (2016). We also make use of publicly avail-
+able data from PDR3 of HSC-SSP (Aihara et al. 2021)
+within the SHELA field. Finally we include J and (shal-
+lower) Ks images from the VICS82 survey (Geach et al.
+2017) obtained with VISTA and CFHT. We show the
+coverage of all the surveys included in this catalog in
+Figure 2. Figure 3 shows the filter transmission curves
+of all the photometric bands used in this catalog.
+3. DATA REDUCTION
+To
+perform
+model-based
+photometry
+with
+The
+Farmer, images in each photometric band need to be
+resampled and stacked to the same pixel grids. In this
+section, we describe the procedures to obtain the stacked
+images and zero-points in each photometric band.
+3.1. DECam Images
+We begin the stacking of DECam images with the
+NOAO DECam Community Pipeline (CP) resampled
+images of each individual exposure.
+The NOAO DE-
+Cam CP resampled images have been photometrically
+and astrometrically calibrated to a common pixel scale
+of 0.′′27 per pixel. The DECam exposures in our data
+set were observed through a seven-year period, and the
+images were reduced using different versions of the CP.
+Therefore, we re-calibrate the astrometry and flux scal-
+ing of each image uniformly prior to stacking.
+3.1.1. Astrometric and Flux-scaling Calibration
+We tie the astrometry of each pre-stack image to the
+Gaia EDR3 catalog (Gaia Collaboration et al. 2021).
+For each image, we first generate an initial source cat-
+alog using SEP (Barbary 2016), a Python library that
+implements the core algorithm of SourceExtractor
+(Bertin & Arnouts 1996). We match sources in this cat-
+alog to the reported coordinates of astrometry stars in
+the Gaia EDR3 catalog. We first identify stars using
+their Gaia catalog colors (Bailer-Jones et al. 2019). We
+then select astrometry stars as those with a G band S/N
+greater than 10 and proper motion less than 10′′yr−1
+in the Gaia EDR3 catalog, and no extraction flags in
+our SEP catalog. We then calculate the median x- and
+y-offset required to match the good stars to the Gaia
+coordinates in each image.
+Before stacking, it is common to first scale the images
+to a common count level to account for differences in
+exposure time and extinction.
+To do so, we measure
+a flux scaling factor for each image from a set of PSF
+stars. PSF stars are defined as astrometry stars from
+above with no neighbours within 5′′ in our initial cat-
+alog. We measure the flux of these PSF stars in a 2.′′5
+radius aperture, as well as an aperture that, on median,
+contains 70% of the 2.′′5 aperture flux. The 2.′′5 aper-
+ture is a proxy for the total flux except for very bright
+stars, but it includes substantial noise for fainter stars.
+The 70% aperture provides a more robust measurement
+of flux for stars of a range of brightness. We divide the
+70% aperture flux by a factor of 0.7 to obtain the total
+flux measurements of the PSF stars.
+We then calculate the flux scaling factor required to
+scale each image to the count level of a reference image.
+For each band, the reference image is selected in the T4
+tile at the center of the SHELA field, and is required
+to have seeing in the lower half of all images and at
+least 1000 PSF stars identified. The count level of the
+reference image is then propagated to the rest of the
+SHELA field from a list of overlapping images satisfying
+the seeing and PSF star criteria, resulting in a catalog
+of PSF stars at the reference count level for the entire
+SHELA field. The flux scaling factor is calculated by the
+median ratio between the catalog counts and measured
+counts of the PSF stars in the image.
+3.1.2. Image Stacking
+We stack the exposures to seven tiles of 1.◦98 × 2.◦145
+with a pixel scale of 0.′′27 per pixel using SWarp (Bertin
+2010). Each tile overlaps with its neighbouring tile by
+approximately 3′. We apply the astrometric correction
+and flux scaling factors obtained above using an exter-
+nal header file. We remove any CCDs within a DECam
+exposure if the number of bad pixels in the data quality
+mask (DQM) exceeds 20% of the total number of pixels.
+We also exclude the S7 CCD from all exposures as it
+is known to have a defective amplifier. This leaves us
+with 61 working CCDs for most DECam pointings. For
+a small number of exposures, some other CCDs display
+a strong variable background and discontinuity between
+the two amplifiers, and are removed from our stacking.
+Bleed trails from bright stars that are not removed by
+the CP are identified and masked. We then stack the ex-
+posures using the surface-brightness-optimized weights
+following Gawiser et al. (2006):
+wSB
+i
+=
+�
+1
+pirmsi
+�2
+,
+(1)
+where pi is the flux scaling factor and rmsi is the pixel-
+by-pixel rms of the background in the science image.
+Pixels with a nonzero DQM value are assigned a weight
+
+SHELA catalog
+5
+14
+16
+18
+20
+22
+24
+26
+Right ascention (J2000)
+1
+0
+1
+Declination (J2000)
+DECam ugrizY,
+HSC grz,
+VICS82
+NEWFIRM
+HSC i
+HSC Y
+Figure 2. Exposure map of the IRAC imaging overlaid with the coverage of data from all the surveys used in this catalog.
+The area included this catalog is defined by the coverage of our own DECam program, which is completely filled by the HSC
+grz-bands and VICS82 imaging.
+0.5
+1
+2
+5
+Wavelength ( m)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Transmission
+u
+g
+r
+i
+z
+Y
+J
+K
+[3.6]
+[4.5]
+DECam
+HSC
+NEWFIRM
+VISTA
+CFHT
+IRAC
+Figure 3. Ralative transmission curves for the photometric bands used. The effects of the filter, atmosphere, detector and
+optics are included.
+of zero. We utilize the recommended Lanczos3 inter-
+polation function to resample and co-add the images.
+3.2. Non-DECam Images
+We construct non-DECam images in the same tiles
+and tangent points. The source images in these bands
+have been previously calibrated, and stacked to different
+tile schemes than our SHELA tiles. We resample and
+moasic the source images to match with the DECam
+stacks. We first update the astrometry of each image to
+match Gaia EDR3 using the same astrometric correc-
+tion procedures described in Section 3.1.1. We do not
+calculate a flux-scaling factor or surface-brightness op-
+timized weight, as these are calibrated stacked images.
+We use the available DQM and weight maps for masking
+and weighting. The exceptions are NHS Ks and IRAC
+images, for which only exposure maps are available in-
+stead of weight maps. For these bands, we scale the ex-
+posure maps with the background rms to create inverse-
+variance weight maps. The publicly available HSC-SSP
+images are previously stacked to “patches” with over-
+lapping edges with neighbouring ones, so we trim each
+image accordingly to avoid double-counting weights in
+those regions. The images are then resampled and mo-
+saiced using the same SWarp configurations as the DE-
+Cam images.
+3.3. Detection Images
+For each tile, we construct a detection image by co-
+adding the DECam r, i, z and NHS Ks bands using the
+CHI-MEAN co-addition in SWarp. The CHI-MEAN co-
+addition method is the normalized quadrature sum of
+the signal-to-noise in each band offset to be centered at
+zero (see Drlica-Wagner et al. 2018). We select the r,
+i, z and Ks bands to optimize the detection of higher
+redshift galaxies, as many will drop out in the u and
+g bands.
+3.4. Zero-point Determination
+We calibrate the zero-point of the DECam images
+to the Pan-STARRS1 (PS1) DR2 catalog for the
+grizY bands, and to the SDSS DR16 for the u band,
+which is not covered by PS1. We first identify F0 stars
+with S/N > 10 in the reference catalogs using a color
+cut on their catalog magnitudes.
+We select F0 stars
+because of their relatively flat SED and large number
+density. We determine the expected colors of an F0 star
+using the PS1 and SDSS filter transmission curves and
+the Kurucz (1993) F0 SED template.
+We adopt the
+
+6
+Leung et al.
+following color cuts for the PS1 and SDSS catalogs to
+obtain > 2000 F0 stars in the SHELA field:
+(gPS1 − rPS1 − 0.11)2 + (rPS1 − iPS1 + 0.04)2
++(iPS1 − zPS1 + 0.08)2 + (zPS1 − yPS1 + 0.04)2
+<0.272
+(2)
+(uSDSS − gSDSS − 0.96)2 + (gSDSS − rSDSS − 0.14)2
++(rSDSS − iSDSS + 0.03)2 + (iSDSS − zSDSS + 0.09)2
+<0.22.
+(3)
+We calculate the zero-points of the g, r, i, z and
+Y bands using the PS1 catalog and that of the u band
+using the SDSS catalog following the procedures in Wold
+et al. (2019).
+Since the DECam filter transmission
+curves are not identical to those of PS1 or SDSS, we
+calculate the expected DECam magnitude using a lin-
+ear color relation with the reference catalog magnitude.
+For example, in the g band, we assume
+gAB
+PS1 = gDECam + ZPT + α(gAB
+PS1 − rAB
+PS1),
+(4)
+where gDECam is the g band instrumental DECam mag-
+nitude, ZPT is the zero-point, and α is the color slope.
+We solve for the zero-point and the color slope in each
+band and tile by performing a least square fit to the lin-
+ear relation. The DECam magnitude is measured using
+the 70% flux aperture described in Section 3.1.1.
+For the u band, a sharp 4000˚A break in stellar spectra
+makes the color a poor predictor of the DECam flux (see
+Wold et al. 2019). We instead compute the zero point
+by computing the expected magnitude offset between
+the DECam and SDSS filters (∆ufilter) from the filter
+transmission curves and F0 SED template. The zero-
+point is given by
+ZPT = median(uAB
+SDSS − uDECam − ∆ufilter).
+(5)
+We subtract 0.04 from the zero-point to bring the SDSS
+magnitudes in alignment with AB magnitudes2.
+The HSC-SSP, NHS, VICS82 and IRAC stacks are
+constructed from calibrated images, and their zero-
+points are preserved throughout the stacking procedure.
+We verify the final zero-points using a similar procedure
+as above, and find that they are consistent within un-
+certainty. We thus adopt the documented zero-points
+for these bands.
+4. SOURCE DETECTION AND PHOTOMETRY
+2 https://www.sdss.org/dr16/algorithms/fluxcal/
+4.1. The Farmer
+Source detection and photometry are performed us-
+ing The Farmer, a software package that utilizes The
+Tractor to perform source modeling and photometry
+on multi-wavelength data. A detailed description of The
+Farmer software package can be found in Weaver et al.
+(2022). To facilitate parallel computation, each tile is
+divided into 156 “bricks” with dimensions of approxi-
+mately 10′ × 10′. Source detection, modeling and forced
+photometry are performed in each brick independently.
+In this section, we describe each step in The Farmer.
+4.1.1. Source Detection
+Before source detection, we mask extremely extended
+and resolved sources in the brick images since they can
+result in non-convergence in the modeling process and
+lead to inaccurate photometry of sources in their vicin-
+ity. We perform an initial source detection procedure
+on the detection image in each brick using SEP with a
+minimum detection area of 8000 pixels to create a seg-
+mentation map of very extended sources. The object re-
+gions in the segmentation map are dilated on each side
+by 15 pixels to create an extended object mask. In the
+HSC-SSP images, the extended object mask is dilated by
+95 pixels because of the larger bright star halos for the
+instrument. The extended object masks are combined
+with the masks for the detection image and modeling
+images of each band.
+Source detection in The Farmer utilizes SEP. We use
+the weight maps generated in Section 3.1.2 and masks
+generated above. The values of the source detection pa-
+rameters used are shown in Table 1. After source detec-
+tion, The Farmer uses the segmentation map to iden-
+tify crowded regions with multiple nearby sources, and
+groups them into “blobs”. The blobs are the smallest
+units for source modeling, meaning that all the sources
+in a blob are modelled simultaneously.
+4.1.2. PSF Creation
+To model the source profiles across multiple bands
+with inhomogeneous PSFs, we need to first model the
+PSF in each band and in each tile.
+This is achieved
+using PSFEx, which is integrated into The Farmer.
+A catalog of bright sources is first constructed using
+SourceExtractor in each tile.
+Unsaturated point
+sources are then selected visually using the half-light ra-
+dius versus apparent magnitude diagram. The selected
+point sources are then used for PSF modeling, where
+we allow spatial variations in each tile using a third de-
+gree polynomial. This allows a spatially dependent PSF
+model which accounts for variations in the image quality
+across each the tile.
+
+SHELA catalog
+7
+Table 1. Source Detection Parameters
+Parameter
+Value
+THRESH
+1.5
+MINAREA
+3
+DEBLEND NTHRESH
+32
+DEBLEND CONT
+0.0001
+FILTER KERNEL
+gauss 1.5 3×3.conv
+FILTER TYPE
+matched
+BW
+128
+BH
+128
+FW
+3
+FH
+3
+Table 2. Farmer Decision Tree Pa-
+rameters
+Parameter
+Value
+PS SG THRESH1
+0.8
+EXP DEV THRESH
+0.5
+CHISQ FORCE EXP DEV
+1.05
+CHISQ FORCE COMP
+1.5
+4.1.3. Source Modeling
+After creating the PSF models, source modeling is per-
+formed on a user-defined set of “modeling bands”, which
+is usually a subset of all the available bands. In this cat-
+alog, our modeling bands include the DECam r, i, z and
+NEWFIRM Ks bands. This is because all the detection
+bands have to be used in modeling so that every de-
+tected source can be modeled. Additionally, we include
+the HSC r and z bands, which have improved depth and
+spatial resolution over the same DECam bands.
+The
+HSC i band is not included since it does not fully cover
+the SHELA field. The modeling bands are then simulta-
+neously fitted to determine the best-fit source profiles for
+each source. The Farmer allows five different models
+to describe the source profiles:
+1. PointSource models are used to describe unre-
+solved sources. They are simply the PSF of each
+individual band, parametrized by the flux and
+source centroid.
+2. SimpleGalaxy
+models
+are
+used
+to
+describe
+barely resolved sources. They are a special case
+of an exponential light profile, where the profile
+is circularly symmetric and has a fixed effective
+radius of 0.′′45.
+3. ExpGalaxy models are exponential light profiles
+parametrized by the flux, effective radius, axis ra-
+tio, position angle and source centroid.
+4. DevGalaxy models are de Vaucouleurs light pro-
+files parametrized by the flux, effective radius, axis
+ratio, position angle and source centroid.
+5. CompositeGalaxy models are the concentric su-
+perposition of the ExpGalaxy and DevGalaxy
+models. They are parametrized by the source cen-
+troid, total flux and flux ratio between the two
+models, as well as the effective radius, axis ratio,
+position angle of each of the two models.
+The best fit model is chosen using a decision tree in The
+Farmer. Briefly, it progresses from a simpler model to
+a more complex model if either the improvement in χ2
+is greater than a user-defined threshold or the simpler
+model χ2 exceeds a certain value. We select the decision
+tree parameters by testing different parameters in steps
+on a small number of bricks. A set of parameters that
+balances between sufficient modeling and overfitting is
+chosen by visually inspecting the resulting source counts
+as a function of magnitude. The decision tree parame-
+ters are shown in Table 2. Figure 4 shows the distribu-
+tion of sources within a certain model as a function of
+HSC r band magnitude. Faint sources with r ≳ 24 are
+mostly modelled by the PointSource and SimpleGalaxy
+models, as they are unresolved or barely resolved due
+to their low surface brightness.
+The ExpGalaxy and
+DevGalaxy models dominate an intermediate range of
+20 ≲ r ≲ 24, while the most complex CompositeGalaxy
+model are mostly found at r ≲ 22, as the sources become
+better resolved at brighter magnitudes.
+4.1.4. Forced photometry
+After a source profile model is chosen using the model-
+ing bands, forced photometry is performed on all of the
+remaining bands that are not used in the modeling pro-
+cess. This is achieved by fitting each force photometric
+band individually by fixing the source profiles, convolved
+with the PSF of each band, and only allowing the flux
+normalizations to vary. Additionally, a small positional
+offset is allowed in each band to account for any po-
+tential imperfect astrometry. A Gaussian prior with a
+standard deviation of 0.3 pixels is applied to the posi-
+tion to prevent excessive offsets. In Figure 5, we show
+the image, model map and residual map of an example
+blob containing a pair of sources separated by ∼ 1′′ for
+
+8
+Leung et al.
+14
+16
+18
+20
+22
+24
+26
+r Mag (AB)
+102
+103
+104
+105
+Number
+PointSource
+SimpleGalaxy
+DevGalaxy
+ExpGalaxy
+CompositeGalaxy
+Total
+Figure 4. Distribution of source models as a function of
+HSC r magnitude in T1.
+Sources are more resolved at
+brighter magnitudes. The unresolved PointSource model and
+barely resolved SimpleGalaxy model dominate fainter mag-
+nitudes of r ≳ 24. The resolved ExpGalaxy and DevGalaxy
+models dominate brighter magnitudes of 20 ≲ r ≲ 24. The
+most complex CompositeGalaxy model is mostly found at
+r ≲ 22.
+the HSC r, NEWFIRM Ks, and IRAC [3.6] bands. The
+forced photometry results in the complete photometry
+in all the photometric bands in this catalog.
+4.1.5. Duplicates
+Since there is a small overlap between tiles, we iden-
+tify and remove duplicate sources in these regions. Any
+source within 0.′′5 of a source in another tile is considered
+a duplicate, and the source that has a larger χ2 in the
+DECam r-band is removed from the combined catalog.
+We note that the raw data covering these overlapping re-
+gions are identical. The only difference between the tile
+images is the different resampling schemes. We verified
+that the photometry of duplicate pairs are consistent
+within their uncertainties.
+4.2. Completeness
+We perform simulations to estimate the complete-
+ness of the catalog. We insert fake point sources with
+no color, i.e., equal AB magnitudes between bands, to
+the science images, and attempt to recover them using
+the same detection and photometry procedures as the
+catalog sources.
+For each tile, we randomly generate
+15,000 locations and fluxes for source insertion.
+The
+fluxes are drawn from a power-law-plus-constant distri-
+bution between 20 and 27 mag. Point source profiles
+are constructed by multiplying the normalized position-
+dependent PSF stamps in each band with the desired
+HSC r
+2"
+Image
+Model
+Residual
+NEWFIRM K
+2"
+IRAC [3.6]
+2"
+Figure 5. Demonstration of model fitting and forced pho-
+tometry by The Farmer. A blob containing a pair of neigh-
+boring sources separated by ∼ 1′′ is shown.
+The image,
+model and residual maps are shown from left to right. The
+HSC r, NEWFIRM Ks and IRAC [3.6] bands are shown
+from top to bottom. 2′′-diamater apertures centered at the
+source positions are shown in red for reference. The sources
+are modeled simultaneously using the rizKs bands. Forced
+photometry is performed to all the bands by fixing the source
+profiles, convolved with the PSF of each band, and only al-
+lowing the flux normalizations to vary.
+flux. These fake point sources are then inserted into the
+science images of each band at the same set of locations.
+We then construct the CHI-MEAN detection image, gen-
+erate bricks and detect sources in the same procedures.
+Figure 6 shows for fraction of fake sources recovered as
+a function of magnitude.
+Our simulations show that
+our catalog is 80% complete at ≈ 24.7 mag. After this
+threshold, the completeness declines to 50% at ≈ 25.1
+mag.
+4.3. Photometric Errors
+We estimate point-source photometric errors using the
+same set of simulations and compare them with the flux
+errors reported by The Farmer. After source detec-
+tion, we proceed to measure the fluxes of the recovered
+sources in each band by performing our modeling and
+forced photometry on the blobs containing the recovered
+sources.
+We then measure the photometric errors by
+comparing the recovered and input fluxes in each tile and
+photometric band. As an example, Figure 7 shows the
+median and 68th-percentile of the fractional difference
+
+SHELA catalog
+9
+20
+21
+22
+23
+24
+25
+26
+27
+Input magnitude
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Recovery fraction
+T1
+T2
+T3
+T4
+T5
+T6
+T7
+Figure 6. The fraction of simulated sources recovered as
+a function of input magnitude. The simulated sources have
+zero color, i.e. equal AB magnitudes between bands. The
+simulations show an 80% (50%) completeness threshold of
+∼ 24.7 (∼ 25.1) mag.
+between the input and recovered fluxes as a function
+of the input magnitude for the DECam r, NEWFIRM
+Ks and IRAC [3.6] bands in T1. We also show the me-
+dian error reported by The Farmer in the same plot.
+The recovered error is generally larger than but follows
+a similar trend as the error reported by The Farmer,
+showing that the errors reported by The Farmer are
+underestimated. We therefore estimate the true photo-
+metric error by applying a correction to the error re-
+ported by The Farmer.
+We parametrize the fractional photometric error as
+�σtotal
+f
+�2
+= σ2
+sys + C2
+�σFarmer
+f
+�2
+,
+(6)
+where σtot is the total photometric error, f is the flux
+density, σsys is the dimensionless systematic error, C is
+a dimensionless correction factor, and σFarmer is the er-
+ror reported by The Farmer in flux density. In each
+magnitude bin, we take half of the central 68-percentiles
+of the recovered fractional flux error as σtotal/f, the me-
+dian Farmer error as σFarmer, and the flux corresponding
+to the magnitude bin center as f. We then perform a
+least squares fit to Equation 6 to find the values of σsys
+and C. As the recovered fluxes fluctuate strongly at very
+faint magnitudes, we only make use of the data points
+brighter than than m = 25 in our fitting for the optical
+bands and m = 23 for the infrared bands.
+The systematic error term is typically ∼ 5% of the
+flux. A major source of systematic uncertainty is from
+the PSF model. PSFEx does not return uncertainties
+for the PSF stamps and the Tractor engine assumes
+the PSF models to be the truth. Therefore, the error
+reported by The Farmer does not account for the effect
+of the uncertainty in the PSF model.
+The correction factor is typically ∼ 1.2 − 1.5 for the
+optical and VICS82 bands. Larger correction factors of
+2 and 5 are required for the NHS Ks and IRAC bands,
+respectively.
+The need for a correction factor for the
+statistical error is likely due to correlated pixels. The
+Farmer error, which is calculated by a weighted quadra-
+ture sum of pixel errors, gives the total error under the
+assumption of independent data points, i.e., uncorre-
+lated pixel values, and therefore have zero covariance
+terms. The science images have been interpolated and
+resampled from their native pixel scales, which leads to
+correlation between neighbouring pixels. Furthermore,
+when the PSF is significantly wider than the pixel size,
+a considerable number of pixels within the PSF can be-
+come correlated. These factors invalidate the assump-
+tion of uncorrelated pixels, and lead to non-zero covari-
+ance terms and thus underestimation of the true statis-
+tical error. In fact, the bands that require the largest
+correction factors are the IRAC bands, followed by the
+NHS Ks band, which have the largest PSF and native
+pixel size.
+For each source, the total photometric error from the
+simulations is given by
+σtotal,i =
+�
+(σsysfi)2 + (CσFarmer,i)2,
+(7)
+where fi and σFarmer,i are the measured flux and error
+from The Farmer for the source, and σsys and C are
+the values measured by the simulations for the tile and
+band. In the catalog, we report the total error derived
+from Equation 7 as well as the error returned by The
+Farmer. The total error is likely an conservative upper
+limit of the actual error, since a small fraction of recov-
+ered sources could be mismatched to a different source,
+which can lead to overestimation of the recovered flux
+error. We also note that we have only simulated point
+sources in our analysis, since there is an infinite number
+of possible resolved source profiles. As a result, these
+errors are only formally applicable to point sources.
+4.4. Detection Limits
+We estimate our point-source detection limits using
+the photometric errors obtained in the previous section.
+For each tile and band, we calculate the S/N by dividing
+the flux by the photometric error.
+We then estimate
+the 5σ detection limit by finding the median magnitude
+for point sources at 4.8 < S/N < 5.2. We calculated
+
+10
+Leung et al.
+20
+22
+24
+26
+DECam r Mag
+0.4
+0.2
+0.0
+0.2
+0.4
+(Fout
+Fin) / Fin
+Farmer error / F_in
+Fit
+Systematic
+Statistical
+Recovered flux
+20
+22
+24
+26
+NEWFIRM K Mag
+0.4
+0.2
+0.0
+0.2
+0.4
+20
+22
+24
+26
+[3.6] Mag
+0.4
+0.2
+0.0
+0.2
+0.4
+Figure 7. Demonstration of our measurement of flux error by simulation of injected sources in the DECam r, NEWFIRM
+Ks and IRAC [3.6] bands in T1. The data points and error bars show the median and central 68-percentiles of the fractional
+flux difference between the injected and recovered sources, respectively, in each magnitude bin. The shaded region shows the
+median fractional error reported by The Farmer (σFarmer/Fin). The dotted, dashed and solid black lines show the best-fit
+systematic, statistical and total errors, respectively, as defined in Equation 6. The recovered error is substantially larger than
+the error reported by The Farmer in the NEWFIRM and IRAC bands. This is likely due to a combination of larger PSF and
+native pixel sizes, resulting in more correlated pixels.
+Table 3. Point-source Detection Limits (5σ, AB)
+DECam
+HSC
+VISTA
+CFHT
+NEWFIRM
+IRAC
+Tile
+u
+g
+r
+i
+z
+Y
+g
+r
+i
+z
+Y
+J
+Ks
+J
+Ks
+Ks
+[3.6]
+[4.5]
+Simulation
+1
+25.1
+25.5
+25.1
+24.8
+24.3
+22.9
+26.1
+25.6
+25.1
+24.8
+23.4
+22.7
+21.9
+22.4
+21.8
+22.6
+21.9
+21.9
+2
+25.3
+25.4
+25.0
+24.7
+23.9
+22.5
+26.0
+25.7
+25.4
+24.5
+24.2
+22.3
+21.7
+-
+-
+22.4
+21.9
+22.0
+3
+25.4
+25.4
+25.1
+24.7
+24.1
+22.3
+26.1
+25.8
+25.7
+24.5
+24.3
+22.3
+21.8
+-
+-
+22.2
+21.9
+22.0
+4
+25.5
+25.3
+25.0
+24.5
+24.1
+22.3
+25.9
+25.6
+25.6
+24.3
+24.3
+22.4
+21.8
+22.0
+21.6
+21.5
+22.0
+22.0
+5
+25.4
+25.4
+25.0
+24.5
+24.1
+22.1
+25.7
+25.2
+25.0
+24.6
+24.1
+21.7
+21.7
+22.0
+21.7
+21.9
+22.0
+22.0
+6
+25.4
+25.6
+25.3
+24.7
+24.0
+22.1
+25.1
+25.1
+24.7
+24.7
+23.5
+22.2
+21.7
+-
+-
+22.3
+21.9
+22.0
+7
+25.3
+25.1
+24.8
+24.4
+24.1
+22.3
+25.4
+24.9
+24.8
+24.6
+-
+22.1
+21.0
+-
+-
+22.5
+21.9
+22.0
+The Farmer
+1
+25.7
+25.9
+25.6
+25.3
+24.7
+23.6
+26.4
+26.1
+25.3
+25.0
+23.9
+23.1
+22.4
+23.0
+22.4
+23.4
+23.7
+23.7
+2
+25.7
+25.8
+25.4
+25.0
+24.5
+22.5
+26.3
+26.2
+25.7
+24.8
+24.5
+22.7
+22.1
+-
+-
+23.2
+23.7
+23.7
+3
+25.8
+25.7
+25.4
+25.0
+24.4
+22.5
+26.3
+26.2
+25.8
+24.8
+24.5
+22.9
+22.3
+-
+-
+23.2
+23.7
+23.7
+4
+25.7
+25.7
+25.2
+24.8
+24.4
+22.4
+26.3
+26.2
+25.8
+24.7
+24.5
+22.9
+22.3
+22.6
+22.1
+23.2
+23.7
+23.7
+5
+25.8
+25.8
+25.3
+25.0
+24.3
+22.4
+26.2
+25.9
+25.2
+24.9
+24.4
+22.6
+22.2
+22.6
+22.2
+23.1
+23.7
+23.7
+6
+25.8
+26.2
+25.6
+25.0
+24.5
+22.4
+26.2
+25.6
+25.1
+25.0
+23.8
+22.8
+22.2
+-
+-
+23.1
+23.7
+23.7
+7
+25.8
+25.4
+25.3
+24.7
+24.3
+22.5
+26.1
+25.6
+25.0
+24.9
+-
+22.6
+21.8
+-
+-
+23.3
+23.7
+23.7
+Note—Simulation detection limits are calculated using flux errors obtained by injecting and recovering simulated point sources in
+the science images. The Farmer detection limits are calculated using the flux errors reported by the Tractor. See Section 4.3
+for a detailed discussion of the differences between the two flux error measurements.
+
+SHELA catalog
+11
+detection limits both using the simulation-based errors
+and the values from The Farmer, and the results are
+listed in Table 3. The simulation-based detection limits
+are generally shallower than The Farmer-based ones
+by ∼ 0.3−0.5 AB mag in the ugrizY -bands and VICS82
+bands, ∼ 0.8 AB mag in the NEWFIRM Ks-band, and
+∼ 1.7 AB mag in both IRAC bands.
+Here we compare our simulation-based 5σ detection
+limits with previous studies. Comparing with the previ-
+ous SHELA DECam catalog by Wold et al. (2019), our
+DECam u-band depths of ∼ 24.9 − 25.2 AB mag are
+in agreement with their results, valued between their
+deeper “sky aperture” and shallower “simulation” de-
+tection limits. Our DECam griz-bands depths are gen-
+erally ∼ 0.5 AB mag deeper than the deeper “sky aper-
+ture” in Wold et al. (2019) thanks to our additional ex-
+posures. Our NEWFIRM Ks-band reaches a depth of
+∼ 22.3 AB mag, similar to the 22.4 AB mag in Ste-
+vans et al. (2021) based on 2′′-diameter apertures on
+the same Ks dataset. Our IRAC 3.6 µm- and 4.5 µm-
+bands reach very uniform depths of ∼ 22.0 AB mag, and
+are in excellent agreement with Papovich et al. (2016)
+using the same IRAC dataset. In the VICS82 bands,
+our detection limits of ∼ 22.3 and ∼ 21.7 AB mag in
+J and Ks are ∼ 0.8 AB mag deeper than those re-
+ported in Geach et al. (2017) based on 2′′-diameter aper-
+tures. This difference could be due to the better seeing
+in VICS82. Our model-based photometry and errors are
+based on fitting the light profile convolved with the PSF
+model, and are effectively an optimal extraction based
+on the actual seeing of the images. We speculate that
+the 2′′-diameter apertures in Geach et al. (2017) could
+be too large for their images, which have a typical seeing
+FWHM of 0.8−0.9′′ at VISTA, resulting in the inclusion
+of noise and thus a shallower measured detection limit.
+In comparison, the NEWFIRM Ks images in Stevans
+et al. (2021), which are also measured by 2′′-diameter
+apertures and are in agreement with our results, have a
+typical seeing FWHM of ∼ 1.2′′.
+4.5. Number Counts
+An important test of the detection and photometry
+of the catalog is to compare the galaxy number counts
+to the literature. In Figure 8, we plot the differential
+galaxy number counts against the measured DECam r-
+band magnitude. We also plot in the same figure the
+best-fit line log(N) = −3.52 + 0.34R to 20 < R < 24
+galaxies presented in Gawiser et al. (2006). Known stars
+in our catalog are removed by cross-matching with the
+SDSS DR17 star catalog. Our measured number counts
+agree with the reported relation up to r ≈ 25, beyond
+which the number counts fall below the relation. The
+18
+20
+22
+24
+26
+r Mag (AB)
+102
+103
+104
+105
+N/mag/deg2
+Gawiser+06
+T1
+T2
+T3
+T4
+T5
+T6
+T7
+Figure 8. Differential number counts of galaxies versus DE-
+Cam r-band magnitude. Poisson error bars are smaller than
+the data point markers. The black line shows the best-fit
+line in Gawiser et al. (2006). Our measured number counts
+agree with the reported relation up to our 50% completeness
+limit of r ≈ 25, beyond which the number counts fall below
+the relation.
+agreement at r < 25 shows that a smooth transition
+from the resolved to unresolved models is achieved dur-
+ing the modeling process. The decline beyond r ≈ 25
+is consistent with our completeness simulation results,
+which show an 80% completeness threshold of r ≈ 24.7.
+We use the DECam r band here instead of the deeper
+HSC r band because the HSC r filter is bluer than the
+MOSAIC II R filter in Gawiser et al. (2006) by an ef-
+fective wavelength of 330 ˚A. The DECam filter, bluer
+by only 130 ˚A, provides a more direct comparison with
+Gawiser et al. (2006). The HSC number count follows
+a similar trend, but is slightly shifted towards fainter
+magnitudes.
+4.6. Comparison to DECaLS
+The DECaLS Survey (Dey et al. 2019) DR9 presents
+photometry using DECam imaging in the grz-bands ob-
+served through March 2019. The DECaLS data encom-
+pass most of the observing dates of those used in this
+catalog, and the footprint covers the SHELA field. Thus,
+it provides an independent reference catalog to verify
+the photometry in our catalog. Moreover, the DECaLS
+photometry is also based on source modeling by the
+Tractor, allowing an “apples-to-apples” comparison
+with this catalog. In Figure 9, we plot the difference
+in measured magnitude between this catalog and DE-
+CaLS as a function of the magnitude in the catalog for
+the DECam grz-bands. The photometry is in excellent
+
+12
+Leung et al.
+17.5
+20.0
+22.5
+25.0
+0.5
+0.0
+0.5
+ Mag [L22-LS] (AB)
+g_DECam
+ = -0.01
+17.5
+20.0
+22.5
+25.0
+Mag (AB)
+r_DECam
+ = 0.03
+17.5
+20.0
+22.5
+25.0
+z_DECam
+ = 0.03
+Figure 9. Difference between the measured magnitudes in this catalog (L22) and DECaLS DR9 (LS) as a function of magnitude
+in this catalog. The median magnitude difference is shown in each panel. DECaLS covers the grz-bands in DECam. Both
+catalogs employ model-based photometry. The photometry is in excellent agreement between the catalogs with a median offset
+of ≤ 0.03 mag.
+agreement between the two catalogs. The median mag-
+nitude offset is −0.01 mag in g, and 0.03 mag in r and
+z.
+4.7. Comparison to Previous SHELA Catalog
+We also compare our photometry with the previous
+aperture photometry catalog presented in the DECam
+catalog of Wold et al. (2019). In Figure 10, we show
+the magnitude difference as a function of magnitude for
+the DECam ugriz and HSC griz bands.
+We use the
+“AUTO” magnitudes based on Kron apertures in Wold
+et al. (2019) for this comparison. Since this catalog in-
+cludes additional photometric bands than Wold et al.
+(2019), the HSC bands here are compared against the
+corresponding DECam bands. The magnitudes in this
+catalog are generally within 0.1 mag of those in Wold
+et al. (2019), with a absolute median offset of < 0.07
+mag.
+We further examine the relation between the magni-
+tude offset and the light profile models of the sources in
+the IRAC bands, where source blending due to its lower
+resolution is expected to lead to the largest difference
+between model-based and aperture photometry.
+We
+compare the difference between the model-based mag-
+nitudes in this catalog and the “AUTO” magnitudes
+based on Kron apertures in the IRAC [3.6] band from
+the IRAC catalog of Papovich et al. (2016). In Figure 11,
+we show the magnitude difference for sources with the
+PointSource, SimpleGalaxy, DevGalaxy and ExpGalaxy
+models. The PointSource model, which represents un-
+resolved sources and dominates at > 24 AB mag, have
+generally similar magnitudes as those in Papovich et al.
+(2016), showing a median offset of -0.01 mag.
+This
+suggests that point source fluxes are accurately mea-
+sured in both model-based and aperture photometry. By
+contrast, the SimpleGalaxy, DevGalaxy and ExpGalaxy
+models, which represent resolved sources, have gener-
+ally brighter magnitudes compared with Papovich et al.
+(2016), showing a median offset of ∼ −0.1 mag. This
+suggests that a substantial fraction of the flux can be
+missed by the use of a fixed aperture on these resolved
+sources, and the modeling of the light profile and PSF
+is needed to capture all the flux from these sources.
+5. PHOTOMETRIC REDSHIFTS
+We measure photometric redshifts using the photo-
+metric redshift code EAZY Brammer et al. (2008). We
+include fluxes in all the photometric bands that have
+χ2 < 10 and we use the photometric uncertainties in-
+clusive of the systematic error terms and scaling factors
+derived from our simulations. We require the objects
+to have five valid flux measurements to ensure a well-
+cosntrained SED. We use a redshift grid from 0.01 to 15
+with a step size of 0.01. We utilize the included EAZY
+template set, “tweak fsps QSF v12 v3,” which uses a
+Chabrier (2003) initial mass function, a Kriek & Con-
+roy (2013) dust attenuation law and solar metallicity.
+We also include an additional set of six templates in
+Larson et al. (2022) covering bluer colors than the in-
+cluded templates, which improves photometric redshifts
+for bluer galaxies.
+We compare our photometric redshifts with spectro-
+scopic redshifts in SDSS DR17. The vast majority of the
+sources with available spectroscopic redshifts are at z ≤
+1. In Figure 12, we compare the results for the 13,895
+sources at z ≤ 1. The median (zphot−zSDSS)/(1+zSDSS)
+is 0.002, and the normalized median absolute deviation
+(see Brammer et al. 2008), defined as
+σNMAD = 1.48 × median
+�����
+∆z − median(∆z)
+1 + zSDSS
+����
+�
+, (8)
+is 0.037. We find that 8% of the sources are 5σNMAD
+outliers. The outliers are mostly attributed to imper-
+fect source modeling for nearby galaxies, where bright,
+
+SHELA catalog
+13
+0.5
+0.0
+0.5
+u_DECam
+ = 0.02
+g_DECam
+ = -0.06
+g_hsc
+ = -0.07
+0.5
+0.0
+0.5
+ Mag [L22-W19] (AB)
+r_DECam
+ = 0.00
+r_hsc
+ = 0.06
+i_DECam
+ = 0.05
+17.5
+20.0
+22.5
+25.0
+0.5
+0.0
+0.5
+i_hsc
+ = 0.05
+17.5
+20.0
+22.5
+25.0
+Mag (AB)
+z_DECam
+ = 0.02
+17.5
+20.0
+22.5
+25.0
+z_hsc
+ = 0.02
+Figure 10.
+Comparison between the measured magnitudes in this catalog (L22) and the AUTO magnitudes using Kron
+apertures in Wold et al. (2019, W19). The photometry in this catalog are generally within 0.1 mag of that in Wold et al. (2019),
+with a absolute median offset of < 0.07 mag.
+0.5
+0.0
+0.5
+PointSource
+ = -0.01
+SimpleGalaxy
+ = -0.11
+20
+25
+              ch1 Mag (AB)
+0.5
+0.0
+0.5
+                           Mag [L22-P16] (AB)
+DevGalaxy
+ = -0.14
+20
+25
+ExpGalaxy
+ = -0.13
+Figure 11. Comparison between IRAC [3.6] magnitudes in
+this catalog and the aperture-based SHELA IRAC catalog
+(Papovich et al. 2016, P16) for sources with different models.
+The magnitudes in P16 are measured in Kron apertures. The
+unresolved PointSource model results in fainter magnitudes,
+while the resolved SimpleGalaxy, DevGalaxy and ExpGalaxy
+models result in brighter magnitudes in this catalog. This
+suggests that model-based photometry is more capable at
+capturing all the flux from resolved source.
+resolved structures are not adequately described by any
+of the light profiles used. An indicator of an inadequate
+fit is the χ2 of the source modeling process. We find that
+the median χ2 in the DECam r-band for the 5σNMAD
+outliers is 3.0 compared with 1.9 for all the sources, sug-
+gesting a generally worse fit in the outliers. This shows
+that the photometry of these outliers are less accurate
+than the rest of the sample. The number of outliers can
+be significantly reduced by applying a χ2 threshold dur-
+ing sample selection. We note that this effect is the most
+prominent for the bright, z ≤ 1 sources that are being
+compared here, since these sources can be sufficiently
+resolved that none of the model light profiles used can
+adequately describe the observed light distribution. In
+general, the χ2 values are close to unity at magnitudes
+≳ 22 mag, indicating good photometry. Furthermore,
+the comparison here includes any available photometric
+redshift measurements in the catalog, while in actual
+applications, it is common to apply additional phys-
+ically motivated selection criteria such as flux and/or
+S/N thresholds in specific photometric bands, which is
+expected to further reduce the outlier fraction. None of
+the zSDSS ≤ 1 galaxies compared here have a photomet-
+ric redshift of zphot ≥ 10.
+6. THE SHELA PHOTOMETRIC CATALOG
+We publish with this paper the full SHELA photo-
+metric catalog.
+In Table 3, we show a sample of the
+SHELA photometric catalog.
+The catalog contains a
+
+14
+Leung et al.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+zSDSS
+0
+1
+2
+3
+4
+5
+zphot
+100
+101
+102
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+zSDSS
+0.4
+0.2
+0.0
+0.2
+0.4
+(zphot
+zSDSS) / (1+zSDSS)
+Median = 0.002
+NMAD = 0.037
+100
+101
+102
+Figure 12. Comparison between photometric redshifts in this catalog and SDSS spectroscopic redshifts for 13,895 sources. In
+the left panel, the dashed black line shows where the photometric redshift and spectroscopic redshift are equal. In the right
+panel, the black points and error bars show the median and 68%-tiles, respectively. Outliers are mainly due to bright and highly
+resolved sources at z ≤ 1 that are not adequately modelled by the available light profiles and tend to get set to specific values
+of zphot.
+unique source ID and J2000 coordinates for each source.
+We also report the best-fit light profile model selected by
+The Farmer. For each band, we report the measured
+flux, the flux errors from both The Farmer and our
+simulations, and the χ2 of the light profile fit. Fluxes
+and flux errors are in µJy. We recommend using the
+simulation-based flux errors for most purposes. We ad-
+vise against using fluxes with a corresponding χ2 ≫ 10
+for science. Finally, we list the photometric redshift in-
+formation from EAZY, inculding the best-fit redshift,
+68- and 95-percentiles, number of filters used in the fit,
+and the χ2 of the fit. Entries with no valid data are
+shown as −99.
+7. SUMMARY
+In this paper, we have presented the ugrizY JKs plus
+3.6 and 4.5 µm photometric catalog that reaches a 5σ
+depth ∼ 25.5 AB mag for over four millions sources
+in the ∼ 27 deg2 SHELA field.
+We performed fully
+model-based photometry using The Farmer, and val-
+idated our photometry with the model-based DECaLS
+DR9 catalog. We compared the model-based photome-
+try with the previous aperture photometry catalogs. We
+find that model-based photometry can accurately mea-
+sure point source fluxes and capture the full extended
+emission of resolved sources. We also presented photo-
+metric redshifts for the catalog, which show good agree-
+ment with available spectrosopic redshifts.
+The large
+area, multi-wavelength photometric catalog of SHELA
+will enable a wide range extragalactic studies.
+We acknowledge that the location where most of this
+work took place, the University of Texas at Austin, sits
+on the Indigenous lands of Turtle Island, the ancestral
+name for what now is called North America. Moreover,
+we would like to acknowledge the Alabama-Coushatta,
+Caddo, Carrizo/Comecrudo, Coahuiltecan, Comanche,
+Kickapoo, Lipan Apache, Tonkawa and Ysleta Del Sur
+Pueblo, and all the American Indian and Indigenous
+Peoples and communities who have been or have become
+a part of these lands and territories in Texas. GCKL and
+SLF acknowledge support from the NSF through AAG
+grant awards 1908817 and 2009905. The Institute for
+Gravitation and the Cosmos is supported by the Eberly
+College of Science and the Office of the Senior Vice Pres-
+ident for Research at the Pennsylvania State University.
+This research draws upon DECam data as distributed
+by the Astro Data Archive at NSF’s NOIRLab. NOIR-
+Lab is managed by the Association of Universities for
+Research in Astronomy (AURA) under a cooperative
+agreement with the National Science Foundation.
+This project used data obtained with the Dark Energy
+Camera (DECam), which was constructed by the Dark
+Energy Survey (DES) collaboration.
+Funding for the
+DES Projects has been provided by the US Department
+of Energy, the US National Science Foundation, the
+Ministry of Science and Education of Spain, the Science
+and Technology Facilities Council of the United King-
+dom, the Higher Education Funding Council for Eng-
+land, the National Center for Supercomputing Applica-
+tions at the University of Illinois at Urbana-Champaign,
+the Kavli Institute for Cosmological Physics at the Uni-
+
+SHELA catalog
+15
+versity of Chicago, Center for Cosmology and Astro-
+Particle Physics at the Ohio State University, the
+Mitchell Institute for Fundamental Physics and Astron-
+omy at Texas A&M University, Financiadora de Estudos
+e Projetos, Funda¸c˜ao Carlos Chagas Filho de Amparo
+`a Pesquisa do Estado do Rio de Janeiro, Conselho Na-
+cional de Desenvolvimento Cient´ıfico e Tecnol´ogico and
+the Minist´erio da Ciˆencia, Tecnologia e Inova¸c˜ao, the
+Deutsche Forschungsgemeinschaft and the Collaborat-
+ing Institutions in the Dark Energy Survey.
+The Collaborating Institutions are Argonne National
+Laboratory, the University of California at Santa Cruz,
+the University of Cambridge, Centro de Investigaciones
+En´ergeticas, Medioambientales y Tecnol´ogicas–Madrid,
+the University of Chicago, University College Lon-
+don, the DES-Brazil Consortium, the University of
+Edinburgh, the Eidgen¨ossische Technische Hochschule
+(ETH) Z¨urich, Fermi National Accelerator Laboratory,
+the University of Illinois at Urbana-Champaign, the In-
+stitut de Ci`encies de l’Espai (IEEC/CSIC), the Insti-
+tut de F´ısica d’Altes Energies, Lawrence Berkeley Na-
+tional Laboratory, the Ludwig-Maximilians Universit¨at
+M¨unchen and the associated Excellence Cluster Uni-
+verse, the University of Michigan, NSF’s NOIRLab, the
+University of Nottingham, the Ohio State University,
+the OzDES Membership Consortium, the University of
+Pennsylvania, the University of Portsmouth, SLAC Na-
+tional Accelerator Laboratory, Stanford University, the
+University of Sussex, and Texas A&M University.
+Based on observations at Cerro Tololo Inter-American
+Observatory, NSF’s NOIRLab (NOIRLab Prop.
+ID
+2019B-0080; PI: C. Papovich), which is managed by the
+Association of Universities for Research in Astronomy
+(AURA) under a cooperative agreement with the Na-
+tional Science Foundation.
+The Hyper Suprime-Cam (HSC) collaboration in-
+cludes the astronomical communities of Japan and Tai-
+wan, and Princeton University. The HSC instrumenta-
+tion and software were developed by the National As-
+tronomical Observatory of Japan (NAOJ), the Kavli
+Institute for the Physics and Mathematics of the Uni-
+verse (Kavli IPMU), the University of Tokyo, the High
+Energy Accelerator Research Organization (KEK), the
+Academia Sinica Institute for Astronomy and Astro-
+physics in Taiwan (ASIAA), and Princeton University.
+Funding was contributed by the FIRST program from
+the Japanese Cabinet Office, the Ministry of Educa-
+tion, Culture, Sports, Science and Technology (MEXT),
+the Japan Society for the Promotion of Science (JSPS),
+Japan Science and Technology Agency (JST), the Toray
+Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA,
+and Princeton University.
+This paper makes use of software developed for Vera
+C. Rubin Observatory. We thank the Rubin Observa-
+tory for making their code available as free software at
+http://pipelines.lsst.io/.
+This paper is based on data collected at the Subaru
+Telescope and retrieved from the HSC data archive sys-
+tem, which is operated by the Subaru Telescope and
+Astronomy Data Center (ADC) at NAOJ. Data analy-
+sis was in part carried out with the cooperation of Cen-
+ter for Computational Astrophysics (CfCA), NAOJ. We
+are honored and grateful for the opportunity of observ-
+ing the Universe from Maunakea, which has the cultural,
+historical and natural significance in Hawaii.
+This work has made use of data from the Euro-
+pean Space Agency (ESA) mission Gaia (https://www.
+cosmos.esa.int/gaia), processed by the Gaia Data Pro-
+cessing and Analysis Consortium (DPAC, https://www.
+cosmos.esa.int/web/gaia/dpac/consortium).
+Funding
+for the DPAC has been provided by national institu-
+tions, in particular the institutions participating in the
+Gaia Multilateral Agreement.
+The Pan-STARRS1 Surveys (PS1) and the PS1 public
+science archive have been made possible through contri-
+butions by the Institute for Astronomy, the University
+of Hawaii, the Pan-STARRS Project Office, the Max-
+Planck Society and its participating institutes, the Max
+Planck Institute for Astronomy, Heidelberg and the Max
+Planck Institute for Extraterrestrial Physics, Garching,
+The Johns Hopkins University, Durham University, the
+University of Edinburgh, the Queen’s University Belfast,
+the Harvard-Smithsonian Center for Astrophysics, the
+Las Cumbres Observatory Global Telescope Network
+Incorporated, the National Central University of Tai-
+wan, the Space Telescope Science Institute, the National
+Aeronautics and Space Administration under Grant No.
+NNX08AR22G issued through the Planetary Science Di-
+vision of the NASA Science Mission Directorate, the
+National Science Foundation Grant No. AST-1238877,
+the University of Maryland, Eotvos Lorand University
+(ELTE), the Los Alamos National Laboratory, and the
+Gordon and Betty Moore Foundation.
+
+16
+Leung et al.
+Table 3. SHELA Catalog Sample
+ID
+R.A.
+Decl.
+Model
+fu,DECam
+efarmer
+u,DECam
+esim
+u,DECam
+χ2
+u,DECam
+fg,DECam
+efarmer
+g,DECam
+esim
+g,DECam
+χ2
+g,DECam
+fr,DECam
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+(7)
+(8)
+(9)
+(10)
+(11)
+(12)
+(13)
+100401830
+15.933
+-0.526
+PointSource
+0.009
+0.045
+0.078
+0.737
+-0.052
+0.031
+0.05
+0.744
+0.015
+100401831
+15.882
+-0.526
+DevGalaxy
+0.038
+0.044
+0.076
+1.117
+0.119
+0.03
+0.048
+0.896
+0.816
+100401832
+15.871
+-0.527
+PointSource
+0.321
+0.038
+0.07
+0.961
+0.288
+0.028
+0.045
+1.009
+0.414
+100401833
+15.871
+-0.526
+ExpGalaxy
+0.84
+0.082
+0.156
+1.135
+0.863
+0.054
+0.09
+0.996
+1.371
+100401834
+15.851
+-0.526
+PointSource
+-0.081
+0.041
+0.071
+1.065
+0.148
+0.025
+0.041
+1.002
+0.524
+100401835
+15.941
+-0.526
+PointSource
+0.138
+0.034
+0.061
+1.148
+0.311
+0.024
+0.04
+1.45
+0.86
+100401836
+15.906
+-0.526
+ExpGalaxy
+1.321
+0.044
+0.127
+0.829
+1.736
+0.026
+0.059
+0.965
+3.904
+100401837
+15.84
+-0.528
+SimpleGalaxy
+0.392
+0.066
+0.119
+0.663
+0.489
+0.032
+0.052
+1.067
+1.07
+100401838
+15.84
+-0.528
+DevGalaxy
+3.561
+0.059
+0.292
+0.968
+4.179
+0.027
+0.108
+0.906
+6.082
+100401839
+15.841
+-0.529
+PointSource
+0.368
+0.05
+0.092
+0.814
+0.674
+0.025
+0.043
+0.943
+1.515
+100401840
+15.839
+-0.528
+SimpleGalaxy
+0.128
+0.062
+0.108
+0.816
+0.309
+0.031
+0.051
+0.922
+0.61
+100401841
+15.838
+-0.527
+DevGalaxy
+2.725
+0.064
+0.238
+0.831
+10.376
+0.038
+0.253
+1.023
+44.144
+100401844
+15.946
+-0.526
+PointSource
+-0.056
+0.037
+0.065
+0.753
+0.1
+0.028
+0.045
+0.259
+0.219
+100401845
+15.899
+-0.526
+PointSource
+0.257
+0.039
+0.071
+1.025
+0.282
+0.025
+0.042
+0.497
+0.415
+100401846
+15.842
+-0.528
+SimpleGalaxy
+0.092
+0.064
+0.112
+0.396
+0.249
+0.033
+0.053
+0.622
+0.384
+100401847
+15.842
+-0.527
+ExpGalaxy
+10.869
+0.055
+0.841
+1.219
+24.075
+0.036
+0.572
+4.057
+36.264
+100401848
+15.841
+-0.526
+PointSource
+0.349
+0.042
+0.078
+0.744
+0.393
+0.027
+0.045
+1.06
+0.792
+100401850
+15.919
+-0.526
+PointSource
+0.172
+0.044
+0.078
+0.677
+0.118
+0.027
+0.043
+1.112
+0.322
+100401851
+15.891
+-0.526
+PointSource
+0.322
+0.043
+0.079
+0.805
+0.245
+0.028
+0.045
+0.568
+0.455
+100401852
+15.846
+-0.526
+PointSource
+0.102
+0.042
+0.074
+0.532
+0.111
+0.027
+0.044
+0.85
+0.374
+Note—(1) Unique object ID number. (2) Object R.A. (J2000) in decimal degrees. (3) Object decl. (J2000) in decimal degrees.
+(4) Best-fit light profile model.
+(5 - 76) Measured flux in µJy, flux error from The Farmer in µJy, flux error from our
+simulations in µJy, and χ2 in the profile fitting for each band. (77) EAZY mimimum χ2 photometric redshift. (78 - 81) EAZY
+lower and upper 68- and 95-percentiles of photometric redshift. (82) Number of filters used in EAZY fit. (83) EAZY minimum
+χ2 value.
+
+SHELA catalog
+17
+Table 3. (Continued)
+efarmer
+r,DECam
+esim
+r,DECam
+χ2
+r,DECam
+fi,DECam
+efarmer
+i,DECam
+esim
+i,DECam
+χ2
+i,DECam
+fz,DECam
+efarmer
+z,DECam
+esim
+z,DECam
+χ2
+z,DECam
+fy,DECam
+efarmer
+y,DECam
+(14)
+(15)
+(16)
+(17)
+(18)
+(19)
+(20)
+(21)
+(22)
+(23)
+(24)
+(25)
+(26)
+0.048
+0.075
+1.71
+0.011
+0.072
+0.113
+1.466
+-0.292
+0.116
+0.172
+2.005
+0.663
+0.993
+0.048
+0.08
+1.183
+2.26
+0.081
+0.142
+0.889
+5.478
+0.13
+0.378
+1.016
+6.553
+0.827
+0.044
+0.069
+1.337
+0.571
+0.063
+0.099
+0.809
+0.585
+0.108
+0.163
+0.759
+-0.106
+0.799
+0.094
+0.154
+1.209
+2.46
+0.136
+0.223
+0.962
+3.154
+0.24
+0.401
+0.925
+5.08
+1.65
+0.046
+0.073
+1.271
+0.754
+0.06
+0.096
+0.926
+0.818
+0.112
+0.172
+0.904
+1.266
+0.75
+0.039
+0.067
+1.077
+2.241
+0.064
+0.119
+0.931
+4.924
+0.099
+0.327
+1.123
+6.765
+0.75
+0.045
+0.146
+0.984
+4.908
+0.065
+0.173
+0.966
+5.475
+0.116
+0.368
+0.824
+6.947
+0.779
+0.058
+0.097
+0.781
+1.805
+0.082
+0.137
+0.941
+3.212
+0.149
+0.291
+0.748
+4.264
+0.961
+0.047
+0.212
+1.17
+7.301
+0.067
+0.233
+0.894
+10.286
+0.123
+0.638
+0.926
+11.028
+0.796
+0.047
+0.089
+0.835
+1.667
+0.06
+0.105
+1.1
+1.569
+0.111
+0.189
+0.983
+3.031
+0.782
+0.057
+0.091
+0.779
+0.917
+0.082
+0.13
+0.839
+1.306
+0.149
+0.233
+0.974
+2.235
+0.967
+0.065
+1.447
+1.13
+74.321
+0.094
+2.133
+1.183
+99.153
+0.175
+5.897
+1.056
+112.973
+1.101
+0.044
+0.069
+1.248
+0.309
+0.068
+0.106
+1.937
+0.534
+0.101
+0.153
+1.034
+-0.333
+0.85
+0.042
+0.067
+0.463
+0.685
+0.063
+0.1
+0.587
+1.227
+0.11
+0.178
+1.19
+1.358
+0.748
+0.058
+0.092
+1.365
+0.507
+0.082
+0.128
+0.748
+0.622
+0.149
+0.224
+1.229
+2.376
+0.958
+0.059
+1.189
+5.45
+46.151
+0.083
+1.327
+4.818
+48.856
+0.151
+2.911
+2.691
+53.52
+0.946
+0.043
+0.072
+1.177
+1.061
+0.061
+0.099
+0.67
+1.159
+0.104
+0.169
+1.128
+2.616
+0.738
+0.042
+0.066
+1.281
+0.283
+0.062
+0.097
+1.379
+0.301
+0.1
+0.148
+1.02
+-0.526
+0.795
+0.04
+0.065
+1.176
+0.529
+0.06
+0.095
+1.312
+1.042
+0.105
+0.166
+1.058
+1.326
+0.753
+0.044
+0.071
+1.333
+0.359
+0.064
+0.101
+0.843
+0.427
+0.108
+0.162
+0.742
+-0.214
+0.793
+
+18
+Leung et al.
+Table 3. (Continued)
+esim
+y,DECam
+χ2
+y,DECam
+fg,hsc
+efarmer
+g,hsc
+esim
+g,hsc
+χ2
+g,hsc
+fr,hsc
+efarmer
+r,hsc
+esim
+r,hsc
+χ2
+r,hsc
+fi,hsc
+efarmer
+i,hsc
+esim
+i,hsc
+χ2
+i,hsc
+fz,hsc
+efarmer
+z,hsc
+(27)
+(28)
+(29)
+(30)
+(31)
+(32)
+(33)
+(34)
+(35)
+(36)
+(37)
+(38)
+(39)
+(40)
+(41)
+(42)
+1.809
+2.87
+-0.019
+0.032
+0.043
+2.764
+-0.048
+0.035
+0.056
+1.342
+-0.002
+0.047
+0.058
+0.543
+0.018
+0.081
+1.553
+1.123
+0.118
+0.027
+0.035
+0.925
+0.62
+0.033
+0.061
+1.093
+2.261
+0.056
+0.15
+1.041
+5.224
+0.094
+1.455
+0.88
+0.267
+0.022
+0.03
+1.39
+0.297
+0.027
+0.045
+1.346
+0.273
+0.041
+0.053
+1.692
+0.469
+0.071
+3.02
+1.033
+0.758
+0.056
+0.078
+0.767
+1.323
+0.073
+0.132
+0.929
+2.324
+0.123
+0.205
+1.115
+4.208
+0.193
+1.368
+1.003
+0.138
+0.021
+0.029
+1.388
+0.429
+0.027
+0.048
+1.254
+0.495
+0.042
+0.06
+0.846
+0.697
+0.076
+1.42
+1.129
+0.263
+0.03
+0.04
+1.363
+0.661
+0.035
+0.064
+1.275
+1.872
+0.075
+0.144
+1.109
+4.532
+0.088
+1.475
+0.967
+1.617
+0.027
+0.061
+1.029
+3.664
+0.036
+0.179
+0.943
+4.623
+0.054
+0.28
+1.018
+5.309
+0.086
+1.768
+0.877
+0.538
+0.035
+0.049
+1.5
+1.049
+0.044
+0.085
+2.056
+1.78
+0.06
+0.128
+0.907
+3.177
+0.111
+1.582
+0.974
+4.039
+0.031
+0.13
+0.999
+5.682
+0.037
+0.27
+1.489
+6.726
+0.046
+0.4
+1.435
+9.674
+0.088
+1.435
+1.074
+0.577
+0.023
+0.035
+1.388
+1.188
+0.028
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+1.527
+1.355
+0.035
+0.091
+3.018
+1.576
+0.07
+1.765
+1.157
+0.303
+0.034
+0.046
+1.169
+0.667
+0.043
+0.076
+1.02
+0.947
+0.058
+0.091
+1.051
+0.898
+0.114
+6.807
+0.985
+9.681
+0.048
+0.302
+1.044
+39.347
+0.073
+1.828
+1.591
+70.062
+0.096
+4.129
+2.137
+97.593
+0.162
+1.548
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+0.026
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+0.189
+0.073
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+1.669
+0.464
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+1.364
+0.581
+0.278
+0.022
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+0.026
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+0.821
+0.719
+0.04
+0.066
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+1.023
+0.072
+1.751
+0.739
+0.149
+0.035
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+0.341
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+0.939
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+1.476
+0.589
+0.117
+3.53
+1.112
+23.398
+0.055
+0.718
+4.947
+35.523
+0.069
+1.65
+9.816
+44.173
+0.079
+2.604
+10.12
+48.97
+0.138
+1.352
+0.946
+0.302
+0.022
+0.031
+1.292
+0.567
+0.028
+0.052
+0.971
+0.841
+0.036
+0.066
+1.094
+0.954
+0.072
+1.448
+0.733
+0.153
+0.028
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+0.728
+0.223
+0.031
+0.051
+0.469
+0.233
+0.05
+0.064
+1.403
+0.401
+0.08
+1.373
+1.086
+0.273
+0.023
+0.031
+1.095
+0.368
+0.027
+0.046
+0.72
+0.647
+0.041
+0.063
+1.543
+0.747
+0.07
+1.444
+1.832
+0.094
+0.022
+0.03
+0.953
+0.283
+0.027
+0.045
+2.751
+0.277
+0.034
+0.046
+1.307
+0.33
+0.074
+
+SHELA catalog
+19
+Table 3. (Continued)
+esim
+z,hsc
+χ2
+z,hsc
+fy,hsc
+efarmer
+y,hsc
+esim
+y,hsc
+χ2
+y,hsc
+fK,NHS
+efarmer
+K,NHS
+esim
+K,NHS
+χ2
+K,NHS
+fJ,CFHT
+efarmer
+J,CFHT
+esim
+J,CFHT
+χ2
+J,CFHT
+(43)
+(44)
+(45)
+(46)
+(47)
+(48)
+(49)
+(50)
+(51)
+(52)
+(53)
+(54)
+(55)
+(56)
+0.099
+1.358
+0.293
+0.164
+0.189
+1.443
+-1.659
+0.322
+0.627
+1.103
+-99.0
+-99.0
+-99.0
+-99.0
+0.3
+1.118
+6.788
+0.259
+1.046
+0.995
+25.276
+0.365
+1.946
+0.993
+-99.0
+-99.0
+-99.0
+-99.0
+0.09
+1.106
+0.169
+0.202
+0.228
+0.907
+-0.327
+0.283
+0.541
+1.04
+-99.0
+-99.0
+-99.0
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+
+22
+Leung et al.
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+
diff --git a/kdAyT4oBgHgl3EQf_fou/content/tmp_files/load_file.txt b/kdAyT4oBgHgl3EQf_fou/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9b2a9102e292c63a4e2e1fac997bb18c814ca753
--- /dev/null
+++ b/kdAyT4oBgHgl3EQf_fou/content/tmp_files/load_file.txt
@@ -0,0 +1,2827 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf,len=2826
+page_content='Draft version January 4, 2023  Typeset using LATEX twocolumn style in AASTeX631  The Spitzer-HETDEX Exploratory Large Area Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Model-Based Multi-wavelength  Photometric Catalog  Gene C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Leung  ,1 Steven L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Finkelstein  ,1 John R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Weaver  ,2 Casey Papovich  ,3, 4  Rebecca L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Larson  ,1 Katherine Chworowsky  ,1 Robin Ciardullo  ,5, 6 Eric Gawiser  ,7  Caryl Gronwall  ,5, 6 Shardha Jogee  ,1 Lalitwadee Kawinwanichakij  ,8 Rachel S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Somerville  ,9  Isak G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Wold  ,10 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Aaron Yung  10  1Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA  2Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA  3Department of Physics and Astronomy, Texas A&M University, College Station, TX, USA  4George P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX,  USA  5Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA  6Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA  7Department of Physics and Astronomy, Rutgers, The State University of New Jersey, 136 Frelinghuysen Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', Piscataway, NJ 08854,  USA  8Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Kashiwa 277-8583, Japan (Kavli IPMU,  WPI)  9Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY 10010, USA  10Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA  ABSTRACT  We present a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 µm 16-band photometric catalog for the Spitzer/HETDEX Exploratory Large-  Area (SHELA) survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' SHELA covers a ∼ 27 deg2 field within the footprint of the Hobby-Eberly  Telescope Dark Energy Experiment (HETDEX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Here we present new DECam imaging and a rizKs-  band-selected catalog of four million sources extracted using a fully model-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We validate  our photometry by comparing with the model-based DECam Legacy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We analyze the differences  between model-based and aperture photometry by comparing with the previous SHELA catalog, finding  that our model-based photometry can measure point sources to fainter fluxes and better capture the  full emission of resolved sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The catalog is 80% (50%) complete at riz ∼ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7 (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1) AB mag, and  the optical photometry reaches a 5σ depth of ∼ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 AB mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We measure photometric redshifts and  achieve 1σ scatter of ∆z/(1 + z) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='04 with available spectroscopic redshifts at 0 ≤ z ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This large  area, multi-wavelength photometric catalog, combined with spectroscopic information from HETDEX,  will enable a wide range of extragalactic science investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' INTRODUCTION  Hobby-Eberly Telescope Dark Energy Experiment  (HETDEX) (Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2021) is an unbiased  integral-field spectroscopic survey aiming to measure  Lyα emission from one million galaxies at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='9 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  in a 540 deg2 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The SHELA survey (Papovich  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2016) is a wide-field multi-wavelength photomet-  ric survey targeting a ∼ 27 deg2 field within the foot-  print of HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' It is designed to study the physi-  cal processes, intrinsic and environmental, driving the  growth of galaxies from z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='9 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The SHELA sur-  vey has obtained optical imaging using DECam, near-  infrared (NIR) imaging using NEWFIRM, and mid-  infrared imaging using Spitzer/IRAC, and is supple-  mented by a large amount of publicly available ground-  based imaging data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Early SHELA observations have resulted in three pub-  lished catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2016) presented a  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 µm catalog of the SHELA Spitzer/IRAC  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019) used data from the first  phase of SHELA DECam observations and presented  an riz-selected catalog with DECam ugriz photome-  try along with the existing IRAC data and JKs data  from the VICS82 survey (Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Ste-  vans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2021) presented a Ks-selected catalog com-  bining the previous dataset with Ks-band observations  from the NEWFIRM HETDEX Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Since then, we  have completed the second and final phase of our DE-  Cam imaging campaign in SHELA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In this paper, we  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='00908v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='GA]  3 Jan 2023  ID2  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  present a new photometric catalog by combining our  complete 27-deg2 DECam imaging dataset with the ex-  isting SHELA NEWFIRM and Spitzer/IRAC observa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Apart from the new DECam imaging data that  increase both the width and depth of our survey, we  discuss below several key differences of this new catalog  from the previous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  In this catalog, we benefit from a number of publicly  available imaging datasets in addition to our own imag-  ing campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' A crucial addition from the previous cat-  alogs is the Hyper Suprime-Cam Subaru Strategic Pro-  gram (HSC-SSP, Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2021), which has observed  the SHELA field in grizy as a part of its autumn field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The publicly available HSC-SSP images in the SHELA  field are ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 − 1 mag deeper than the DECam images  in Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Combining these deep HSC-SSP  images with our DECam dataset allows precise measure-  ment of the optical spectral energy distribution (SED)  and constrains the physical properties of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' For  example, deeper g-band imaging is important in reduc-  ing the contamination rate in the selection of z ≳ 3 − 5  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The full coverage of the SHELA field with both DE-  Cam and NEWFIRM supplemented by HSC-SSP now  allows us to select galaxies using the optical and NIR  bands simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' As opposed to riz only in Wold  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019), we will be able to detect, using the Ks  band, very red galaxies that peak in the NIR, such as  massive quiescent galaxies, dusty star-forming galaxies  and z ≥ 7 Lyman break galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' On the other hand,  the Ks-band selected catalog in Stevans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2021) is  limited by its depth of ∼ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4 mag, and the inclusion of  deep optical imaging in this catalog that is ∼ 2 − 3 mag  deeper will significantly increase the completeness, such  as for z ∼ 2 − 5 star-forming galaxies whose rest-frame  UV emission falls in the optical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The previous SHELA catalogs were created entirely  or partially based on aperture photometry using tools  such as SourceExtractor (Bertin & Arnouts 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  In such an approach, the images in each band will be  homogenized to a common PSF, leading to the degra-  dation of spatial resolution in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' An alterna-  tive approach is to model the light profile of sources  convolved with the measured PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Recently, the image  modeling code the Tractor (Lang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2016) has  been developed to to perform such model-based pho-  tometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' It has since been implemented by a number of  imaging surveys, including the DECam Legacy Survey  (DECaLS, Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2019) and COSMOS2020 catalog  (Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The model-based approach is cru-  cial in extracting IRAC photometry, which has a sub-  stantially larger PSF than the optical bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' For this  catalog, we extract fully model-based photometry using  the software package The Farmer (Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2022),  which generates photometry using the Tractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The large area, multi-wavelength photometric catalog  of SHELA will enable a variety of extragalactic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  In particular, studies that focus on galaxies spanning a  wide range of physical parameters, galaxies residing in  a variety of environments or the search for rare objects  will benefit the most from the large area of SHELA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In Section 2, we  describe our observations and the composition of our  imaging dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Section 3 describes the data reduction  procedures for the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The photometry extraction  using The Farmer and the photometry validation pro-  cess are described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Section 5 presents the  photometric redshift measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We give a descrip-  tion of the catalog published with this paper in Sec-  tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In this paper we assume a Planck Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2020) cosmology of H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4 km s−1 Mpc−1,  Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='315 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' OBSERVATIONS AND DATA  We observed the SHELA field using the DECam im-  ager on the Blanco 4m telescope at CTIO in the u-, g-,  r-, i-, z- filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' New data were obtained over 3 nights in  the 2019B semester (PI: Papovich;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' NOAO PID: 2019B-  0080).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our program also includes data obtained over 5  nights in the 2013B semester presented in Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The complete program consists of eight slightly  overlapping pointings covering a total area of 23 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The 2013B data covered six of the eight pointings, while  the new 2019B data completed the last two pointings  and obtained additional exposures in the previous posi-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The SHELA field has been observed with DECam by  other programs, most notably the Dark Energy Survey  (DES, Dark Energy Survey Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2016)  and DECaLS (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We supplement our data  with archival DECam imaging data within the footprint  of the SHELA field in the ugriz bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We also include  archival data in Y band, which was not observed by our  proprietary program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' All such archival data available  through the NOIRLab Astro Data Archive1 released  prior to March 3 2021 were included for this catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This results in a total of [200, 719, 735, 688, 720, 468]  exposures in the [u, g, r, i, z, Y ] bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We show the  DECam ugrizY weight maps in the SHELA field in Fig-  ure 1 to demonstrate the footprint and relative depths  of the DECam data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  1 https://astroarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='noirlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='edu/  SHELA catalog  3  1  0  1  u  T1  T2  T3  T4  T5  T6  T7  1  0  1  g  1  0  1  r  1  0  1  Declination (J2000)  i  1  0  1  z  16  18  20  22  24  26  Right ascention (J2000)  1  0  1  Y  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' SHELA DECam ugrizY weight maps demonstrating the survey’s footprint and relative depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The field is covered  by eight DECam pointings, and is divided into seven tiles of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='◦98 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='◦145, labelled T1 to T7, so that the tiles are roughly  square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our survey combines ugriz images from our observations with grizY archival images released prior to March 3 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We note that T1 has substantially deeper Y -band imaging due to a large number of repeated exposures in the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We incorporate multi-wavelength imaging data from  surveys overlapping with the SHELA field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Previous  SHELA programs have obtained data in the near- and  mid-infrared wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This includes NEWFIRM  Ks-band images from the NEWFIRM HETDEX Sur-  vey (Stevans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2021) and mid-infrared mosaics in  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6 µm and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 µm with Spitzer/IRAC presented in Pa-  povich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We also make use of publicly avail-  able data from PDR3 of HSC-SSP (Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2021)  within the SHELA field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Finally we include J and (shal-  lower) Ks images from the VICS82 survey (Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  2017) obtained with VISTA and CFHT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We show the  coverage of all the surveys included in this catalog in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Figure 3 shows the filter transmission curves  of all the photometric bands used in this catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' DATA REDUCTION  To  perform  model-based  photometry  with  The  Farmer, images in each photometric band need to be  resampled and stacked to the same pixel grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In this  section, we describe the procedures to obtain the stacked  images and zero-points in each photometric band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' DECam Images  We begin the stacking of DECam images with the  NOAO DECam Community Pipeline (CP) resampled  images of each individual exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The NOAO DE-  Cam CP resampled images have been photometrically  and astrometrically calibrated to a common pixel scale  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='′′27 per pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The DECam exposures in our data  set were observed through a seven-year period, and the  images were reduced using different versions of the CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Therefore, we re-calibrate the astrometry and flux scal-  ing of each image uniformly prior to stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Astrometric and Flux-scaling Calibration  We tie the astrometry of each pre-stack image to the  Gaia EDR3 catalog (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  For each image, we first generate an initial source cat-  alog using SEP (Barbary 2016), a Python library that  implements the core algorithm of SourceExtractor  (Bertin & Arnouts 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We match sources in this cat-  alog to the reported coordinates of astrometry stars in  the Gaia EDR3 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We first identify stars using  their Gaia catalog colors (Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We  then select astrometry stars as those with a G band S/N  greater than 10 and proper motion less than 10′′yr−1  in the Gaia EDR3 catalog, and no extraction flags in  our SEP catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We then calculate the median x- and  y-offset required to match the good stars to the Gaia  coordinates in each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Before stacking, it is common to first scale the images  to a common count level to account for differences in  exposure time and extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  To do so, we measure  a flux scaling factor for each image from a set of PSF  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' PSF stars are defined as astrometry stars from  above with no neighbours within 5′′ in our initial cat-  alog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We measure the flux of these PSF stars in a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='′′5  radius aperture, as well as an aperture that, on median,  contains 70% of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='′′5 aperture flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='′′5 aper-  ture is a proxy for the total flux except for very bright  stars, but it includes substantial noise for fainter stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The 70% aperture provides a more robust measurement  of flux for stars of a range of brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We divide the  70% aperture flux by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7 to obtain the total  flux measurements of the PSF stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We then calculate the flux scaling factor required to  scale each image to the count level of a reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  For each band, the reference image is selected in the T4  tile at the center of the SHELA field, and is required  to have seeing in the lower half of all images and at  least 1000 PSF stars identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The count level of the  reference image is then propagated to the rest of the  SHELA field from a list of overlapping images satisfying  the seeing and PSF star criteria, resulting in a catalog  of PSF stars at the reference count level for the entire  SHELA field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The flux scaling factor is calculated by the  median ratio between the catalog counts and measured  counts of the PSF stars in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Image Stacking  We stack the exposures to seven tiles of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='◦98 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='◦145  with a pixel scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='′′27 per pixel using SWarp (Bertin  2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Each tile overlaps with its neighbouring tile by  approximately 3′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We apply the astrometric correction  and flux scaling factors obtained above using an exter-  nal header file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We remove any CCDs within a DECam  exposure if the number of bad pixels in the data quality  mask (DQM) exceeds 20% of the total number of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We also exclude the S7 CCD from all exposures as it  is known to have a defective amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This leaves us  with 61 working CCDs for most DECam pointings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' For  a small number of exposures, some other CCDs display  a strong variable background and discontinuity between  the two amplifiers, and are removed from our stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Bleed trails from bright stars that are not removed by  the CP are identified and masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We then stack the ex-  posures using the surface-brightness-optimized weights  following Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2006):  wSB  i  =  �  1  pirmsi  �2  ,  (1)  where pi is the flux scaling factor and rmsi is the pixel-  by-pixel rms of the background in the science image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Pixels with a nonzero DQM value are assigned a weight  SHELA catalog  5  14  16  18  20  22  24  26  Right ascention (J2000)  1  0  1  Declination (J2000)  DECam ugrizY,  HSC grz,  VICS82  NEWFIRM  HSC i  HSC Y  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Exposure map of the IRAC imaging overlaid with the coverage of data from all the surveys used in this catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The area included this catalog is defined by the coverage of our own DECam program, which is completely filled by the HSC  grz-bands and VICS82 imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  1  2  5  Wavelength ( m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  Relative Transmission  u  g  r  i  z  Y  J  K  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6]  [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5]  DECam  HSC  NEWFIRM  VISTA  CFHT  IRAC  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Ralative transmission curves for the photometric bands used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The effects of the filter, atmosphere, detector and  optics are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We utilize the recommended Lanczos3 inter-  polation function to resample and co-add the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Non-DECam Images  We construct non-DECam images in the same tiles  and tangent points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The source images in these bands  have been previously calibrated, and stacked to different  tile schemes than our SHELA tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We resample and  moasic the source images to match with the DECam  stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We first update the astrometry of each image to  match Gaia EDR3 using the same astrometric correc-  tion procedures described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We do not  calculate a flux-scaling factor or surface-brightness op-  timized weight, as these are calibrated stacked images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We use the available DQM and weight maps for masking  and weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The exceptions are NHS Ks and IRAC  images, for which only exposure maps are available in-  stead of weight maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' For these bands, we scale the ex-  posure maps with the background rms to create inverse-  variance weight maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The publicly available HSC-SSP  images are previously stacked to “patches” with over-  lapping edges with neighbouring ones, so we trim each  image accordingly to avoid double-counting weights in  those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The images are then resampled and mo-  saiced using the same SWarp configurations as the DE-  Cam images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Detection Images  For each tile, we construct a detection image by co-  adding the DECam r, i, z and NHS Ks bands using the  CHI-MEAN co-addition in SWarp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The CHI-MEAN co-  addition method is the normalized quadrature sum of  the signal-to-noise in each band offset to be centered at  zero (see Drlica-Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We select the r,  i, z and Ks bands to optimize the detection of higher  redshift galaxies, as many will drop out in the u and  g bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Zero-point Determination  We calibrate the zero-point of the DECam images  to the Pan-STARRS1 (PS1) DR2 catalog for the  grizY bands, and to the SDSS DR16 for the u band,  which is not covered by PS1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We first identify F0 stars  with S/N > 10 in the reference catalogs using a color  cut on their catalog magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We select F0 stars  because of their relatively flat SED and large number  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We determine the expected colors of an F0 star  using the PS1 and SDSS filter transmission curves and  the Kurucz (1993) F0 SED template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We adopt the  6  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  following color cuts for the PS1 and SDSS catalogs to  obtain > 2000 F0 stars in the SHELA field:  (gPS1 − rPS1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='11)2 + (rPS1 − iPS1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='04)2  +(iPS1 − zPS1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='08)2 + (zPS1 − yPS1 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='04)2  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='272  (2)  (uSDSS − gSDSS − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='96)2 + (gSDSS − rSDSS − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='14)2  +(rSDSS − iSDSS + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='03)2 + (iSDSS − zSDSS + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='09)2  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  (3)  We calculate the zero-points of the g, r, i, z and  Y bands using the PS1 catalog and that of the u band  using the SDSS catalog following the procedures in Wold  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Since the DECam filter transmission  curves are not identical to those of PS1 or SDSS, we  calculate the expected DECam magnitude using a lin-  ear color relation with the reference catalog magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  For example, in the g band, we assume  gAB  PS1 = gDECam + ZPT + α(gAB  PS1 − rAB  PS1),  (4)  where gDECam is the g band instrumental DECam mag-  nitude, ZPT is the zero-point, and α is the color slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We solve for the zero-point and the color slope in each  band and tile by performing a least square fit to the lin-  ear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The DECam magnitude is measured using  the 70% flux aperture described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  For the u band, a sharp 4000˚A break in stellar spectra  makes the color a poor predictor of the DECam flux (see  Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We instead compute the zero point  by computing the expected magnitude offset between  the DECam and SDSS filters (∆ufilter) from the filter  transmission curves and F0 SED template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The zero-  point is given by  ZPT = median(uAB  SDSS − uDECam − ∆ufilter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  (5)  We subtract 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='04 from the zero-point to bring the SDSS  magnitudes in alignment with AB magnitudes2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The HSC-SSP, NHS, VICS82 and IRAC stacks are  constructed from calibrated images, and their zero-  points are preserved throughout the stacking procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We verify the final zero-points using a similar procedure  as above, and find that they are consistent within un-  certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We thus adopt the documented zero-points  for these bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' SOURCE DETECTION AND PHOTOMETRY  2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='org/dr16/algorithms/fluxcal/  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The Farmer  Source detection and photometry are performed us-  ing The Farmer, a software package that utilizes The  Tractor to perform source modeling and photometry  on multi-wavelength data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' A detailed description of The  Farmer software package can be found in Weaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' To facilitate parallel computation, each tile is  divided into 156 “bricks” with dimensions of approxi-  mately 10′ × 10′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Source detection, modeling and forced  photometry are performed in each brick independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  In this section, we describe each step in The Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Source Detection  Before source detection, we mask extremely extended  and resolved sources in the brick images since they can  result in non-convergence in the modeling process and  lead to inaccurate photometry of sources in their vicin-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We perform an initial source detection procedure  on the detection image in each brick using SEP with a  minimum detection area of 8000 pixels to create a seg-  mentation map of very extended sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The object re-  gions in the segmentation map are dilated on each side  by 15 pixels to create an extended object mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In the  HSC-SSP images, the extended object mask is dilated by  95 pixels because of the larger bright star halos for the  instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The extended object masks are combined  with the masks for the detection image and modeling  images of each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Source detection in The Farmer utilizes SEP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We use  the weight maps generated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2 and masks  generated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The values of the source detection pa-  rameters used are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' After source detec-  tion, The Farmer uses the segmentation map to iden-  tify crowded regions with multiple nearby sources, and  groups them into “blobs”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The blobs are the smallest  units for source modeling, meaning that all the sources  in a blob are modelled simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' PSF Creation  To model the source profiles across multiple bands  with inhomogeneous PSFs, we need to first model the  PSF in each band and in each tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This is achieved  using PSFEx, which is integrated into The Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  A catalog of bright sources is first constructed using  SourceExtractor in each tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Unsaturated point  sources are then selected visually using the half-light ra-  dius versus apparent magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The selected  point sources are then used for PSF modeling, where  we allow spatial variations in each tile using a third de-  gree polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This allows a spatially dependent PSF  model which accounts for variations in the image quality  across each the tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  SHELA catalog  7  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Source Detection Parameters  Parameter  Value  THRESH  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  MINAREA  3  DEBLEND NTHRESH  32  DEBLEND CONT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0001  FILTER KERNEL  gauss 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='conv  FILTER TYPE  matched  BW  128  BH  128  FW  3  FH  3  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Farmer Decision Tree Pa-  rameters  Parameter  Value  PS SG THRESH1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8  EXP DEV THRESH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  CHISQ FORCE EXP DEV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='05  CHISQ FORCE COMP  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Source Modeling  After creating the PSF models, source modeling is per-  formed on a user-defined set of “modeling bands”, which  is usually a subset of all the available bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In this cat-  alog, our modeling bands include the DECam r, i, z and  NEWFIRM Ks bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This is because all the detection  bands have to be used in modeling so that every de-  tected source can be modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Additionally, we include  the HSC r and z bands, which have improved depth and  spatial resolution over the same DECam bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The  HSC i band is not included since it does not fully cover  the SHELA field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The modeling bands are then simulta-  neously fitted to determine the best-fit source profiles for  each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The Farmer allows five different models  to describe the source profiles:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' PointSource models are used to describe unre-  solved sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' They are simply the PSF of each  individual band, parametrized by the flux and  source centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' SimpleGalaxy  models  are  used  to  describe  barely resolved sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' They are a special case  of an exponential light profile, where the profile  is circularly symmetric and has a fixed effective  radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='′′45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' ExpGalaxy models are exponential light profiles  parametrized by the flux, effective radius, axis ra-  tio, position angle and source centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' DevGalaxy models are de Vaucouleurs light pro-  files parametrized by the flux, effective radius, axis  ratio, position angle and source centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' CompositeGalaxy models are the concentric su-  perposition of the ExpGalaxy and DevGalaxy  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' They are parametrized by the source cen-  troid, total flux and flux ratio between the two  models, as well as the effective radius, axis ratio,  position angle of each of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The best fit model is chosen using a decision tree in The  Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Briefly, it progresses from a simpler model to  a more complex model if either the improvement in χ2  is greater than a user-defined threshold or the simpler  model χ2 exceeds a certain value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We select the decision  tree parameters by testing different parameters in steps  on a small number of bricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' A set of parameters that  balances between sufficient modeling and overfitting is  chosen by visually inspecting the resulting source counts  as a function of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The decision tree parame-  ters are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Figure 4 shows the distribu-  tion of sources within a certain model as a function of  HSC r band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Faint sources with r ≳ 24 are  mostly modelled by the PointSource and SimpleGalaxy  models, as they are unresolved or barely resolved due  to their low surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The ExpGalaxy and  DevGalaxy models dominate an intermediate range of  20 ≲ r ≲ 24, while the most complex CompositeGalaxy  model are mostly found at r ≲ 22, as the sources become  better resolved at brighter magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Forced photometry  After a source profile model is chosen using the model-  ing bands, forced photometry is performed on all of the  remaining bands that are not used in the modeling pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This is achieved by fitting each force photometric  band individually by fixing the source profiles, convolved  with the PSF of each band, and only allowing the flux  normalizations to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Additionally, a small positional  offset is allowed in each band to account for any po-  tential imperfect astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' A Gaussian prior with a  standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3 pixels is applied to the posi-  tion to prevent excessive offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In Figure 5, we show  the image, model map and residual map of an example  blob containing a pair of sources separated by ∼ 1′′ for  8  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  14  16  18  20  22  24  26  r Mag (AB)  102  103  104  105  Number  PointSource  SimpleGalaxy  DevGalaxy  ExpGalaxy  CompositeGalaxy  Total  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Distribution of source models as a function of  HSC r magnitude in T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Sources are more resolved at  brighter magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The unresolved PointSource model and  barely resolved SimpleGalaxy model dominate fainter mag-  nitudes of r ≳ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The resolved ExpGalaxy and DevGalaxy  models dominate brighter magnitudes of 20 ≲ r ≲ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The  most complex CompositeGalaxy model is mostly found at  r ≲ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  the HSC r, NEWFIRM Ks, and IRAC [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6] bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The  forced photometry results in the complete photometry  in all the photometric bands in this catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Duplicates  Since there is a small overlap between tiles, we iden-  tify and remove duplicate sources in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Any  source within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='′′5 of a source in another tile is considered  a duplicate, and the source that has a larger χ2 in the  DECam r-band is removed from the combined catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We note that the raw data covering these overlapping re-  gions are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The only difference between the tile  images is the different resampling schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We verified  that the photometry of duplicate pairs are consistent  within their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Completeness  We perform simulations to estimate the complete-  ness of the catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We insert fake point sources with  no color, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', equal AB magnitudes between bands, to  the science images, and attempt to recover them using  the same detection and photometry procedures as the  catalog sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  For each tile, we randomly generate  15,000 locations and fluxes for source insertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The  fluxes are drawn from a power-law-plus-constant distri-  bution between 20 and 27 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Point source profiles  are constructed by multiplying the normalized position-  dependent PSF stamps in each band with the desired  HSC r  2"  Image  Model  Residual  NEWFIRM K  2"  IRAC [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6]  2"  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Demonstration of model fitting and forced pho-  tometry by The Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' A blob containing a pair of neigh-  boring sources separated by ∼ 1′′ is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The image,  model and residual maps are shown from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The  HSC r, NEWFIRM Ks and IRAC [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6] bands are shown  from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2′′-diamater apertures centered at the  source positions are shown in red for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The sources  are modeled simultaneously using the rizKs bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Forced  photometry is performed to all the bands by fixing the source  profiles, convolved with the PSF of each band, and only al-  lowing the flux normalizations to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' These fake point sources are then inserted into the  science images of each band at the same set of locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We then construct the CHI-MEAN detection image, gen-  erate bricks and detect sources in the same procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Figure 6 shows for fraction of fake sources recovered as  a function of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Our simulations show that  our catalog is 80% complete at ≈ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' After this  threshold, the completeness declines to 50% at ≈ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1  mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Photometric Errors  We estimate point-source photometric errors using the  same set of simulations and compare them with the flux  errors reported by The Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' After source detec-  tion, we proceed to measure the fluxes of the recovered  sources in each band by performing our modeling and  forced photometry on the blobs containing the recovered  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We then measure the photometric errors by  comparing the recovered and input fluxes in each tile and  photometric band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' As an example, Figure 7 shows the  median and 68th-percentile of the fractional difference  SHELA catalog  9  20  21  22  23  24  25  26  27  Input magnitude  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  Recovery fraction  T1  T2  T3  T4  T5  T6  T7  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The fraction of simulated sources recovered as  a function of input magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The simulated sources have  zero color, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' equal AB magnitudes between bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The  simulations show an 80% (50%) completeness threshold of  ∼ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7 (∼ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1) mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  between the input and recovered fluxes as a function  of the input magnitude for the DECam r, NEWFIRM  Ks and IRAC [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6] bands in T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We also show the me-  dian error reported by The Farmer in the same plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The recovered error is generally larger than but follows  a similar trend as the error reported by The Farmer,  showing that the errors reported by The Farmer are  underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We therefore estimate the true photo-  metric error by applying a correction to the error re-  ported by The Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We parametrize the fractional photometric error as  �σtotal  f  �2  = σ2  sys + C2  �σFarmer  f  �2  ,  (6)  where σtot is the total photometric error, f is the flux  density, σsys is the dimensionless systematic error, C is  a dimensionless correction factor, and σFarmer is the er-  ror reported by The Farmer in flux density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In each  magnitude bin, we take half of the central 68-percentiles  of the recovered fractional flux error as σtotal/f, the me-  dian Farmer error as σFarmer, and the flux corresponding  to the magnitude bin center as f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We then perform a  least squares fit to Equation 6 to find the values of σsys  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' As the recovered fluxes fluctuate strongly at very  faint magnitudes, we only make use of the data points  brighter than than m = 25 in our fitting for the optical  bands and m = 23 for the infrared bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The systematic error term is typically ∼ 5% of the  flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' A major source of systematic uncertainty is from  the PSF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' PSFEx does not return uncertainties  for the PSF stamps and the Tractor engine assumes  the PSF models to be the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Therefore, the error  reported by The Farmer does not account for the effect  of the uncertainty in the PSF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The correction factor is typically ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 for the  optical and VICS82 bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Larger correction factors of  2 and 5 are required for the NHS Ks and IRAC bands,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The need for a correction factor for the  statistical error is likely due to correlated pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The  Farmer error, which is calculated by a weighted quadra-  ture sum of pixel errors, gives the total error under the  assumption of independent data points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', uncorre-  lated pixel values, and therefore have zero covariance  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The science images have been interpolated and  resampled from their native pixel scales, which leads to  correlation between neighbouring pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Furthermore,  when the PSF is significantly wider than the pixel size,  a considerable number of pixels within the PSF can be-  come correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' These factors invalidate the assump-  tion of uncorrelated pixels, and lead to non-zero covari-  ance terms and thus underestimation of the true statis-  tical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In fact, the bands that require the largest  correction factors are the IRAC bands, followed by the  NHS Ks band, which have the largest PSF and native  pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  For each source, the total photometric error from the  simulations is given by  σtotal,i =  �  (σsysfi)2 + (CσFarmer,i)2,  (7)  where fi and σFarmer,i are the measured flux and error  from The Farmer for the source, and σsys and C are  the values measured by the simulations for the tile and  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In the catalog, we report the total error derived  from Equation 7 as well as the error returned by The  Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The total error is likely an conservative upper  limit of the actual error, since a small fraction of recov-  ered sources could be mismatched to a different source,  which can lead to overestimation of the recovered flux  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We also note that we have only simulated point  sources in our analysis, since there is an infinite number  of possible resolved source profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' As a result, these  errors are only formally applicable to point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Detection Limits  We estimate our point-source detection limits using  the photometric errors obtained in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  For each tile and band, we calculate the S/N by dividing  the flux by the photometric error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We then estimate  the 5σ detection limit by finding the median magnitude  for point sources at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8 < S/N < 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We calculated  10  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  20  22  24  26  DECam r Mag  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  (Fout  Fin) / Fin  Farmer error / F_in  Fit  Systematic  Statistical  Recovered flux  20  22  24  26  NEWFIRM K Mag  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  20  22  24  26  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6] Mag  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Demonstration of our measurement of flux error by simulation of injected sources in the DECam r, NEWFIRM  Ks and IRAC [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6] bands in T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The data points and error bars show the median and central 68-percentiles of the fractional  flux difference between the injected and recovered sources, respectively, in each magnitude bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The shaded region shows the  median fractional error reported by The Farmer (σFarmer/Fin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The dotted, dashed and solid black lines show the best-fit  systematic, statistical and total errors, respectively, as defined in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The recovered error is substantially larger than  the error reported by The Farmer in the NEWFIRM and IRAC bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This is likely due to a combination of larger PSF and  native pixel sizes, resulting in more correlated pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Point-source Detection Limits (5σ, AB)  DECam  HSC  VISTA  CFHT  NEWFIRM  IRAC  Tile  u  g  r  i  z  Y  g  r  i  z  Y  J  Ks  J  Ks  Ks  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6]  [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5]  Simulation  1  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
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+page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
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+page_content='7  Note—Simulation detection limits are calculated using flux errors obtained by injecting and recovering simulated point sources in  the science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The Farmer detection limits are calculated using the flux errors reported by the Tractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3  for a detailed discussion of the differences between the two flux error measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  SHELA catalog  11  detection limits both using the simulation-based errors  and the values from The Farmer, and the results are  listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The simulation-based detection limits  are generally shallower than The Farmer-based ones  by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 AB mag in the ugrizY -bands and VICS82  bands, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8 AB mag in the NEWFIRM Ks-band, and  ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7 AB mag in both IRAC bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Here we compare our simulation-based 5σ detection  limits with previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Comparing with the previ-  ous SHELA DECam catalog by Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019), our  DECam u-band depths of ∼ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='9 − 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2 AB mag are  in agreement with their results, valued between their  deeper “sky aperture” and shallower “simulation” de-  tection limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our DECam griz-bands depths are gen-  erally ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 AB mag deeper than the deeper “sky aper-  ture” in Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019) thanks to our additional ex-  posures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our NEWFIRM Ks-band reaches a depth of  ∼ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3 AB mag, similar to the 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4 AB mag in Ste-  vans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2021) based on 2′′-diameter apertures on  the same Ks dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our IRAC 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6 µm- and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 µm-  bands reach very uniform depths of ∼ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0 AB mag, and  are in excellent agreement with Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2016)  using the same IRAC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In the VICS82 bands,  our detection limits of ∼ 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3 and ∼ 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7 AB mag in  J and Ks are ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8 AB mag deeper than those re-  ported in Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2017) based on 2′′-diameter aper-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This difference could be due to the better seeing  in VICS82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our model-based photometry and errors are  based on fitting the light profile convolved with the PSF  model, and are effectively an optimal extraction based  on the actual seeing of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We speculate that  the 2′′-diameter apertures in Geach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2017) could  be too large for their images, which have a typical seeing  FWHM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='9′′ at VISTA, resulting in the inclusion  of noise and thus a shallower measured detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  In comparison, the NEWFIRM Ks images in Stevans  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2021), which are also measured by 2′′-diameter  apertures and are in agreement with our results, have a  typical seeing FWHM of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Number Counts  An important test of the detection and photometry  of the catalog is to compare the galaxy number counts  to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In Figure 8, we plot the differential  galaxy number counts against the measured DECam r-  band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We also plot in the same figure the  best-fit line log(N) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='52 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='34R to 20 < R < 24  galaxies presented in Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Known stars  in our catalog are removed by cross-matching with the  SDSS DR17 star catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our measured number counts  agree with the reported relation up to r ≈ 25, beyond  which the number counts fall below the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The  18  20  22  24  26  r Mag (AB)  102  103  104  105  N/mag/deg2  Gawiser+06  T1  T2  T3  T4  T5  T6  T7  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Differential number counts of galaxies versus DE-  Cam r-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Poisson error bars are smaller than  the data point markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The black line shows the best-fit  line in Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Our measured number counts  agree with the reported relation up to our 50% completeness  limit of r ≈ 25, beyond which the number counts fall below  the relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  agreement at r < 25 shows that a smooth transition  from the resolved to unresolved models is achieved dur-  ing the modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The decline beyond r ≈ 25  is consistent with our completeness simulation results,  which show an 80% completeness threshold of r ≈ 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We use the DECam r band here instead of the deeper  HSC r band because the HSC r filter is bluer than the  MOSAIC II R filter in Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2006) by an ef-  fective wavelength of 330 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The DECam filter, bluer  by only 130 ˚A, provides a more direct comparison with  Gawiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The HSC number count follows  a similar trend, but is slightly shifted towards fainter  magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Comparison to DECaLS  The DECaLS Survey (Dey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2019) DR9 presents  photometry using DECam imaging in the grz-bands ob-  served through March 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The DECaLS data encom-  pass most of the observing dates of those used in this  catalog, and the footprint covers the SHELA field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Thus,  it provides an independent reference catalog to verify  the photometry in our catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Moreover, the DECaLS  photometry is also based on source modeling by the  Tractor, allowing an “apples-to-apples” comparison  with this catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In Figure 9, we plot the difference  in measured magnitude between this catalog and DE-  CaLS as a function of the magnitude in the catalog for  the DECam grz-bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The photometry is in excellent  12  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  Mag [L22-LS] (AB)  g_DECam  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='01  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  Mag (AB)  r_DECam  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='03  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  z_DECam  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='03  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Difference between the measured magnitudes in this catalog (L22) and DECaLS DR9 (LS) as a function of magnitude  in this catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The median magnitude difference is shown in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' DECaLS covers the grz-bands in DECam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Both  catalogs employ model-based photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The photometry is in excellent agreement between the catalogs with a median offset  of ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='03 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  agreement between the two catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The median mag-  nitude offset is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='01 mag in g, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='03 mag in r and  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Comparison to Previous SHELA Catalog  We also compare our photometry with the previous  aperture photometry catalog presented in the DECam  catalog of Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In Figure 10, we show  the magnitude difference as a function of magnitude for  the DECam ugriz and HSC griz bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We use the  “AUTO” magnitudes based on Kron apertures in Wold  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019) for this comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Since this catalog in-  cludes additional photometric bands than Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  (2019), the HSC bands here are compared against the  corresponding DECam bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The magnitudes in this  catalog are generally within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1 mag of those in Wold  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019), with a absolute median offset of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='07  mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We further examine the relation between the magni-  tude offset and the light profile models of the sources in  the IRAC bands, where source blending due to its lower  resolution is expected to lead to the largest difference  between model-based and aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We  compare the difference between the model-based mag-  nitudes in this catalog and the “AUTO” magnitudes  based on Kron apertures in the IRAC [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6] band from  the IRAC catalog of Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In Figure 11,  we show the magnitude difference for sources with the  PointSource, SimpleGalaxy, DevGalaxy and ExpGalaxy  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The PointSource model, which represents un-  resolved sources and dominates at > 24 AB mag, have  generally similar magnitudes as those in Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  (2016), showing a median offset of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='01 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This  suggests that point source fluxes are accurately mea-  sured in both model-based and aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' By  contrast, the SimpleGalaxy, DevGalaxy and ExpGalaxy  models, which represent resolved sources, have gener-  ally brighter magnitudes compared with Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  (2016), showing a median offset of ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This  suggests that a substantial fraction of the flux can be  missed by the use of a fixed aperture on these resolved  sources, and the modeling of the light profile and PSF  is needed to capture all the flux from these sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' PHOTOMETRIC REDSHIFTS  We measure photometric redshifts using the photo-  metric redshift code EAZY Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We  include fluxes in all the photometric bands that have  χ2 < 10 and we use the photometric uncertainties in-  clusive of the systematic error terms and scaling factors  derived from our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We require the objects  to have five valid flux measurements to ensure a well-  cosntrained SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We use a redshift grid from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='01 to 15  with a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We utilize the included EAZY  template set, “tweak fsps QSF v12 v3,” which uses a  Chabrier (2003) initial mass function, a Kriek & Con-  roy (2013) dust attenuation law and solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We also include an additional set of six templates in  Larson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2022) covering bluer colors than the in-  cluded templates, which improves photometric redshifts  for bluer galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We compare our photometric redshifts with spectro-  scopic redshifts in SDSS DR17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The vast majority of the  sources with available spectroscopic redshifts are at z ≤  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In Figure 12, we compare the results for the 13,895  sources at z ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The median (zphot−zSDSS)/(1+zSDSS)  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='002, and the normalized median absolute deviation  (see Brammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2008), defined as  σNMAD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='48 × median  �����  ∆z − median(∆z)  1 + zSDSS  ����  �  , (8)  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='037.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We find that 8% of the sources are 5σNMAD  outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The outliers are mostly attributed to imper-  fect source modeling for nearby galaxies, where bright,  SHELA catalog  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  u_DECam  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='02  g_DECam  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='06  g_hsc  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  Mag [L22-W19] (AB)  r_DECam  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='00  r_hsc  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='06  i_DECam  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='05  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  i_hsc  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='05  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  Mag (AB)  z_DECam  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='02  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  z_hsc  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='02  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Comparison between the measured magnitudes in this catalog (L22) and the AUTO magnitudes using Kron  apertures in Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019, W19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The photometry in this catalog are generally within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='1 mag of that in Wold et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' (2019),  with a absolute median offset of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='07 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  PointSource  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='01  SimpleGalaxy  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='11  20  25  ch1 Mag (AB)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5  Mag [L22-P16] (AB)  DevGalaxy  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='14  20  25  ExpGalaxy  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='13  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Comparison between IRAC [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6] magnitudes in  this catalog and the aperture-based SHELA IRAC catalog  (Papovich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2016, P16) for sources with different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The magnitudes in P16 are measured in Kron apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The  unresolved PointSource model results in fainter magnitudes,  while the resolved SimpleGalaxy, DevGalaxy and ExpGalaxy  models result in brighter magnitudes in this catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This  suggests that model-based photometry is more capable at  capturing all the flux from resolved source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  resolved structures are not adequately described by any  of the light profiles used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' An indicator of an inadequate  fit is the χ2 of the source modeling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We find that  the median χ2 in the DECam r-band for the 5σNMAD  outliers is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0 compared with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='9 for all the sources, sug-  gesting a generally worse fit in the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' This shows  that the photometry of these outliers are less accurate  than the rest of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The number of outliers can  be significantly reduced by applying a χ2 threshold dur-  ing sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We note that this effect is the most  prominent for the bright, z ≤ 1 sources that are being  compared here, since these sources can be sufficiently  resolved that none of the model light profiles used can  adequately describe the observed light distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In  general, the χ2 values are close to unity at magnitudes  ≳ 22 mag, indicating good photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Furthermore,  the comparison here includes any available photometric  redshift measurements in the catalog, while in actual  applications, it is common to apply additional phys-  ically motivated selection criteria such as flux and/or  S/N thresholds in specific photometric bands, which is  expected to further reduce the outlier fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' None of  the zSDSS ≤ 1 galaxies compared here have a photomet-  ric redshift of zphot ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' THE SHELA PHOTOMETRIC CATALOG  We publish with this paper the full SHELA photo-  metric catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  In Table 3, we show a sample of the  SHELA photometric catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The catalog contains a  14  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  zSDSS  0  1  2  3  4  5  zphot  100  101  102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  zSDSS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='4  (zphot  zSDSS) / (1+zSDSS)  Median = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='002  NMAD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='037  100  101  102  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Comparison between photometric redshifts in this catalog and SDSS spectroscopic redshifts for 13,895 sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In  the left panel, the dashed black line shows where the photometric redshift and spectroscopic redshift are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' In the right  panel, the black points and error bars show the median and 68%-tiles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Outliers are mainly due to bright and highly  resolved sources at z ≤ 1 that are not adequately modelled by the available light profiles and tend to get set to specific values  of zphot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  unique source ID and J2000 coordinates for each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We also report the best-fit light profile model selected by  The Farmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' For each band, we report the measured  flux, the flux errors from both The Farmer and our  simulations, and the χ2 of the light profile fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Fluxes  and flux errors are in µJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We recommend using the  simulation-based flux errors for most purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We ad-  vise against using fluxes with a corresponding χ2 ≫ 10  for science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Finally, we list the photometric redshift in-  formation from EAZY, inculding the best-fit redshift,  68- and 95-percentiles, number of filters used in the fit,  and the χ2 of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Entries with no valid data are  shown as −99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' SUMMARY  In this paper, we have presented the ugrizY JKs plus  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 µm photometric catalog that reaches a 5σ  depth ∼ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='5 AB mag for over four millions sources  in the ∼ 27 deg2 SHELA field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We performed fully  model-based photometry using The Farmer, and val-  idated our photometry with the model-based DECaLS  DR9 catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We compared the model-based photome-  try with the previous aperture photometry catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We  find that model-based photometry can accurately mea-  sure point source fluxes and capture the full extended  emission of resolved sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We also presented photo-  metric redshifts for the catalog, which show good agree-  ment with available spectrosopic redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The large  area, multi-wavelength photometric catalog of SHELA  will enable a wide range extragalactic studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  We acknowledge that the location where most of this  work took place, the University of Texas at Austin, sits  on the Indigenous lands of Turtle Island, the ancestral  name for what now is called North America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Moreover,  we would like to acknowledge the Alabama-Coushatta,  Caddo, Carrizo/Comecrudo, Coahuiltecan, Comanche,  Kickapoo, Lipan Apache, Tonkawa and Ysleta Del Sur  Pueblo, and all the American Indian and Indigenous  Peoples and communities who have been or have become  a part of these lands and territories in Texas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' GCKL and  SLF acknowledge support from the NSF through AAG  grant awards 1908817 and 2009905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The Institute for  Gravitation and the Cosmos is supported by the Eberly  College of Science and the Office of the Senior Vice Pres-  ident for Research at the Pennsylvania State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This research draws upon DECam data as distributed  by the Astro Data Archive at NSF’s NOIRLab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' NOIR-  Lab is managed by the Association of Universities for  Research in Astronomy (AURA) under a cooperative  agreement with the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This project used data obtained with the Dark Energy  Camera (DECam), which was constructed by the Dark  Energy Survey (DES) collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Funding for the  DES Projects has been provided by the US Department  of Energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the US National Science Foundation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the  Ministry of Science and Education of Spain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Science  and Technology Facilities Council of the United King-  dom,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Higher Education Funding Council for Eng-  land,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the National Center for Supercomputing Applica-  tions at the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  the Kavli Institute for Cosmological Physics at the Uni-  SHELA catalog  15  versity of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Center for Cosmology and Astro-  Particle Physics at the Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the  Mitchell Institute for Fundamental Physics and Astron-  omy at Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Financiadora de Estudos  e Projetos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Funda¸c˜ao Carlos Chagas Filho de Amparo  `a Pesquisa do Estado do Rio de Janeiro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Conselho Na-  cional de Desenvolvimento Cient´ıfico e Tecnol´ogico and  the Minist´erio da Ciˆencia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Tecnologia e Inova¸c˜ao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the  Deutsche Forschungsgemeinschaft and the Collaborat-  ing Institutions in the Dark Energy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The Collaborating Institutions are Argonne National  Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the University of California at Santa Cruz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  the University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Centro de Investigaciones  En´ergeticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Medioambientales y Tecnol´ogicas–Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  the University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' University College Lon-  don,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the DES-Brazil Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the University of  Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Eidgen¨ossische Technische Hochschule  (ETH) Z¨urich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Fermi National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  the University of Illinois at Urbana-Champaign,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the In-  stitut de Ci`encies de l’Espai (IEEC/CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Insti-  tut de F´ısica d’Altes Energies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Lawrence Berkeley Na-  tional Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Ludwig-Maximilians Universit¨at  M¨unchen and the associated Excellence Cluster Uni-  verse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' NSF’s NOIRLab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the  University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  the OzDES Membership Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the University of  Pennsylvania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' SLAC Na-  tional Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Stanford University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the  University of Sussex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' and Texas A&M University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Based on observations at Cerro Tololo Inter-American  Observatory, NSF’s NOIRLab (NOIRLab Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  ID  2019B-0080;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' PI: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Papovich), which is managed by the  Association of Universities for Research in Astronomy  (AURA) under a cooperative agreement with the Na-  tional Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The Hyper Suprime-Cam (HSC) collaboration in-  cludes the astronomical communities of Japan and Tai-  wan, and Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' The HSC instrumenta-  tion and software were developed by the National As-  tronomical Observatory of Japan (NAOJ), the Kavli  Institute for the Physics and Mathematics of the Uni-  verse (Kavli IPMU), the University of Tokyo, the High  Energy Accelerator Research Organization (KEK), the  Academia Sinica Institute for Astronomy and Astro-  physics in Taiwan (ASIAA), and Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Funding was contributed by the FIRST program from  the Japanese Cabinet Office, the Ministry of Educa-  tion, Culture, Sports, Science and Technology (MEXT),  the Japan Society for the Promotion of Science (JSPS),  Japan Science and Technology Agency (JST), the Toray  Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA,  and Princeton University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This paper makes use of software developed for Vera  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Rubin Observatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We thank the Rubin Observa-  tory for making their code available as free software at  http://pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='lsst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This paper is based on data collected at the Subaru  Telescope and retrieved from the HSC data archive sys-  tem, which is operated by the Subaru Telescope and  Astronomy Data Center (ADC) at NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Data analy-  sis was in part carried out with the cooperation of Cen-  ter for Computational Astrophysics (CfCA), NAOJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' We  are honored and grateful for the opportunity of observ-  ing the Universe from Maunakea, which has the cultural,  historical and natural significance in Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  This work has made use of data from the Euro-  pean Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='int/gaia), processed by the Gaia Data Pro-  cessing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  Funding  for the DPAC has been provided by national institu-  tions, in particular the institutions participating in the  Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The Pan-STARRS1 Surveys (PS1) and the PS1 public  science archive have been made possible through contri-  butions by the Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the University  of Hawaii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Pan-STARRS Project Office,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Max-  Planck Society and its participating institutes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the Max  Planck Institute for Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Heidelberg and the Max  Planck Institute for Extraterrestrial Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  The Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' Durham University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the  University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
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+page_content='  the Harvard-Smithsonian Center for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the  Las Cumbres Observatory Global Telescope Network  Incorporated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' the National Central University of Tai-  wan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
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+page_content=' AST-1238877,  the University of Maryland, Eotvos Lorand University  (ELTE), the Los Alamos National Laboratory, and the  Gordon and Betty Moore Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='  16  Leung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
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+page_content='3847/1538-4357/ac0cf6  Weaver, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', Kauffmann, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', Ilbert, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' 2022,  ApJS, 258, 11, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content='3847/1538-4365/ac3078  Wold, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', Kawinwanichakij, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', Stevans, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
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+page_content='3847/1538-4365/aaee85' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdAyT4oBgHgl3EQf_fou/content/2301.00908v1.pdf'}
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+PRECISION DOSE-FINDING CANCER CLINICAL TRIALS IN THE
+SETTING OF BROADENED ELIGIBILITY
+BY REBECCA B. SILVA1,a, BIN CHENG1,b, RICHARD D. CARVAJAL2,d AND SHING
+M. LEE1,c
+1Department of Biostatistics, Columbia University Mailman School of Public Health,
+ars4025@cumc.columbia.edu; bbc2159@cumc.columbia.edu; csml2114@cumc.columbia.edu
+2Herbert Irving Comprehensive Cancer Center, Columbia University, drdc2150@cumc.columbia.edu
+Broadening eligibility criteria in cancer trials has been advocated to rep-
+resent the true patient population more accurately. While the advantages are
+clear in terms of generalizability and recruitment, novel dose-finding designs
+are needed to ensure patient safety. These designs should be able to recom-
+mend precise doses for subpopulations if such subpopulations with different
+toxicity profiles exist. While dose-finding designs accounting for patient het-
+erogeneity have been proposed, all existing methods assume the source of
+heterogeneity is known and thus pre-specify the subpopulations or only allow
+inclusion of a few patient characteristics. We propose a precision dose-finding
+design to address the setting of unknown patient heterogeneity in phase I can-
+cer clinical trials where eligibility is expanded, and multiple eligibility criteria
+could potentially lead to different optimal doses for patient subgroups. The
+design offers a two-in-one approach to dose-finding by simultaneously select-
+ing patient criteria that differentiate the maximum tolerated dose (MTD) and
+recommending the subpopulation-specific MTD if needed, using marginal
+models to sequentially incorporate patient covariates. Our simulation study
+compares the proposed design to the naive approach of assuming patient ho-
+mogeneity and our design recommends multiple doses when heterogeneity
+exists and a single dose when no heterogeneity exists. The proposed dose-
+finding design addresses the challenges of broadening eligibility criteria in
+cancer trials and the desire for a more precise dose in the context of early
+phase clinical trials.
+1. Introduction.
+The goal of a dose-finding phase I cancer clinical trial is to select a
+safe dose of a new therapy for subsequent phase II and III trials. The recommended dose is
+often known as the maximum tolerated dose (MTD), assuming toxicity and efficacy of a treat-
+ment increase with dosage. The majority of phase I trials recommend one MTD, assuming
+all eligible patients have similar tolerability to the new treatment. Over the past decade, the
+expansion of eligibility criteria for cancer trials which were initially designed for cytotoxic
+chemotherapies has been advocated. Eligibility criteria have been found to be too restrictive
+and not reflective of the actual patient population that could benefit from a therapy [Fehren-
+backer et al. (2009); Gerber et al. (2016); Malik and Lu (2019); Liu et al. (2021); Harvey
+et al. (2021)], thus limiting the generalizability of study results and significantly slowing
+down patient accrual [Malik and Lu (2019)]. One study examining the ineligibility rates of
+326 patients consecutively diagnosed with non-small cell lung cancer (NSCLC) found that
+about 80% of patients are ineligible based on traditional eligibility criteria [Fehrenbacker et
+al. (2009)].
+Several authors have found that the inclusion of patients with brain metastases, HIV, or
+prior cancers, does not affect trial outcomes and better reflects the actual patient population
+Keywords and phrases: subpopulation-specific dose-finding, subgroup dose-finding, eligibility criteria, patient
+heterogeneity, sequential design, phase I cancer.
+1
+arXiv:2301.04578v1  [stat.AP]  11 Jan 2023
+
+2
+[George (1996); Kim et al. (2015, 2017); Beaver et al. (2017); Malik and Lu (2019)]. In
+2017, the American Society of Clinical Oncology (ASCO), Friends of Cancer Research, and
+the US Food and Drug Administration published new recommendations for exclusion criteria
+that should be reconsidered including brain metastases, HIV and AIDS, organ dysfunction,
+prior or concurrent cancers, and a minimum age for enrollment [Kim et al. (2017); Food and
+Drug Administration (2020)]. Moreover, eligibility criteria such as concomitant medications,
+prior therapies, performance status, liver or renal dysfunction, and laboratory values such as
+levels of bilirubin, platelets, hemoglobin and alkaline phosphatase, continue to be questioned
+and evaluated by ASCO and Friends of Cancer Research [Harvey et al. (2021); Liu et al.
+(2021); Magnuson et al. (2021); Osarogiagbon et al. (2021)].
+While the broadening of eligibility criteria has clear advantages, it also has to be done
+cautiously to ensure patient safety. Specifically, dose-finding methods should account for the
+potential of heterogeneous populations where patient subpopulations could have different
+tolerances and take a precision medicine approach to dose estimation by recommending a
+subpopulation-specific MTD if needed. Methods have been proposed in the phase I setting
+to account for a pre-specified patient covariate or subpopulations. All these methods either
+assume that the covariates of interest or subpopulations with different toxicity profiles are
+known beforehand, or adaptively combine subgroups based on similar risk without the iden-
+tification of covariates. Therefore, these methods cannot be applied to address the challenges
+in the setting of broadened eligibility criteria where it is not known beforehand whether
+different MTDs are needed. Extensions of the Continual Reassessment Method (CRM) [Pi-
+antadosi and Liu (1996)] and the Escalation with Overdose Control allow for the addition of a
+patient-specific covariate to the dose-toxicity model [Babb and Rogatko (2001); Cheng et al.
+(2004)]. Moreover, there are methods that allow for the pre-specification of subpopulations
+including the two-sample CRM [O’Quigley, Shen, and Gamst (1999)], the two-sample CRM
+for ordered groups [O’Quigley and Paoletti (2003)], and isotonic designs for multiple ordered
+risk groups [Yuan and Chappell (2004)] and partially ordered risk groups [Conaway (2017)].
+More recently, Morita, Thall, and Takeda (2017) proposed a hierarchical Bayesian approach
+that uses subgroup-specific intercepts and Chapple and Thall (2018) proposed Spike and Slab
+priors on subgroup parameters to account for multiple groups.
+While precision medicine has been generally advocated for later stages in drug develop-
+ment and with larger data sets, in our early stage of drug development, it would be ideal to
+have a method that can screen for the expanded eligibility criteria, assess whether it is neces-
+sary to account for them, and be able to recommend more than one MTD if needed. Beaver et
+al. (2017) propose that “a logical approach to defining eligibility could allow for detection of
+safety signals in early clinical trials that use broad eligibility criteria”. As eligibility criteria
+in oncology trials expand, having a method that can learn and assess for signals of patient
+heterogeneity in toxicity will be key to ensuring the safety of patients who have traditionally
+been thought to be at increased risk of adverse outcomes.
+As a motivating example, we use the phase I trial evaluating intermittent dosing of a
+mitogen-activated protein kinase kinase (MEK) inhibitor, Selumetinib, to treat patients with
+uveal melanoma [Khan et al. (2022)]. The study evaluated six doses of Selumetinib (100, 125,
+150, 175, 200, 225 mg) and estimated the MTD using the Time-to-Event CRM (TITE-CRM)
+with a target toxicity level of 25% in 28 patients. Given the rarity of uveal melanoma, accrual
+took over three years with three large sites. The principal investigator of the trial identified
+three criteria (creatinine levels, organ and marrow function, and the presence of known or
+suspected brain metastases) that could have been loosened to allow more patients to benefit
+from the trial. For example, creatinine level captures renal function and between 12% and
+53% percent of cancer patients have chronic kidney disease [Kitchlu et al. (2018)]. If eligi-
+bility criteria were to expand, the design of this phase I trial would require accounting for
+
+PRECISION DOSE-FINDING DESIGN
+3
+these three criteria, since knowledge about their association with toxicity is limited. Some of
+these criteria may affect the probability of toxicity of Selumetinib while others may not. For
+example, renal impairment has been associated with increased risk of toxicity, leading inves-
+tigators to account for degree of renal dysfunction when determining dose [Lameire (2014);
+Leal et al. (2011); LoRusso et al. (2012)]. However, dose reductions have been found to not
+always be necessary [Leal et al. (2011)]. If the loosened criteria are not accounted for and at
+least one criterion is associated with increased risk of toxicity, the design could overestimate
+the MTD for these patients.
+Our proposed design, the Precision CRM (P-CRM), addresses the case where we want to
+consider and learn about multiple patient characteristics, which contribute to a broader patient
+population. For example, if the eligibility criteria were expanded in the Selumetinib trial, we
+would want to consider evaluating three patient criteria since the MTD could differ based on
+one or more of these criteria. Our method selects patient criterion to add to the dose-toxicity
+model sequentially, based on their marginal correlation with toxicity after accounting for
+dose. With each new enrollment cohort, given the additional information on toxicity and pa-
+tient characteristics, the criteria are tested for inclusion or removal. Dose assignment is based
+on the selected patient criteria given the current data and the final subpopulation-specific
+MTD is determined using the selected covariates in the final model after the total sample size
+is reached. This method provides a two-in-one approach to dose-finding; the method learns
+about the criteria and the need to differentiate the MTD and recommends a dose for each
+patient subpopulation based on these criteria.
+The rest of the paper is outlined as follows. The precision dose-finding method is described
+in Section 2. In Section 3, we present a simulation study based on the redesigned Selumetinib
+trial and compare the performance of the P-CRM to the naive approach of not accounting for
+any patient heterogeneity. We also discuss sample size considerations for the design. Finally,
+the significance and limitations of our proposed design are discussed in Section 4.
+2. Methods.
+Drug safety is most commonly measured by the occurrence of a severe
+adverse event, or toxicity, in the first cycle of treatment, known as a dose-limiting toxicity
+(DLT). In model-based designs, a total of J doses, d1,...,dJ, are evaluated, and the proba-
+bility of toxicity, or DLT, is estimated using a dose-toxicity model. The MTD, defined as the
+dose whose toxicity is closest to pT , the highest rate of toxicity that clinicians deem accept-
+able, or target toxicity level, is estimated sequentially. For all future notation, let Yi be the
+binary toxicity outcome for the ith patient and Xi be the dose assigned to this patient.
+2.1. Dose-toxicity Model.
+Traditionally, it is assumed that the entire patient population
+has the same probability of toxicity at each dose, and therefore has the same MTD. Assuming
+a logistic model and complete patient homogeneity, π(dj,β0,β1) = P(Yi = 1|Xi = dj) is
+modeled as
+(1)
+logit{π(dj,β0,β1)} = β0 + β1dj,j = 1,...,J.
+In the late 1990s, the patient homogeneity assumption in dose-finding models started being
+questioned. A dose-toxicity model incorporating a patient covariate, z, was proposed, where
+π(dj,zi,θ) = P(Yi = 1|Xi = dj,zi) is modeled as,
+(2)
+logit{π(dj,zi,θ)} = β0 + β1dj + γzi,
+where θ = (β0,β1,γ)T [Piantadosi and Liu (1996)]. Equation (2) represents an overly restric-
+tive model; it assumes that we know beforehand which covariate contributes to the hetero-
+geneity and that heterogeneity is completely due to z, both of which are strong assumptions.
+
+4
+In this paper, we consider a potentially large collection of patient covariates, z =
+(z1,...,zM)T , which could be responsible for dose heterogeneity. We model π(dj,zi,θ) =
+P(Yi = 1|Xi = dj,zi) as
+(3)
+logit{π(dj,zi,θ)} = β0 + β1dj +
+M
+�
+m=1
+γmzmi,
+where θ = (β0,β1,γ1,...,γM)T . In addition, we make the following sparsity assumption
+M
+�
+m=1
+1γm̸=0 = K ≪ M.
+(4)
+In the setting of expanded eligibility criteria, Equation (3) represents a more realistic model
+which is potentially high dimensional, but we assume that only a small subset of these criteria
+are responsible for the heterogeneity. Assumption (4) is consistent with clinicians’ belief that
+there are only a few expanded criteria/covariates or none at all for which we need to actually
+differentiate the MTD.
+Therefore, assuming there are M total patient covariates suspected to be associated with
+toxicity of which a small subset actually differentiate the MTDs, our objective is two-fold.
+One is to identify the subset of K covariates that are responsible for the heterogeneity in
+MTDs, and the other is to implement a procedure that determines the subpopulation-specific
+MTDs.
+2.2. Precision Dose-finding Procedure.
+Let Γ = {γi,i = 1,...,M}, and Γ0 = {γi : γi ̸=
+0}. By Equation (3), Γ0 is an unknown small subset of Γ that must be estimated because it is
+responsible for the heterogeneity in MTDs. The estimation of Γ imposes a large challenge.
+Since K is unknown, and thus must be estimated, we need to search among all 2M subsets
+of Γ for Γ0. Furthermore, in dose-finding studies, patients are assigned to doses in sequential
+cohorts, not simultaneously to all doses as done in a parallel group design. This means that
+information concerning parameters in model (3) is to be accumulated gradually as more and
+more toxicity data become available, suggesting that it would be too ambitious to estimate
+Γ0 in a single effort.
+Consistent with the sequential nature of dose finding studies, we propose to estimate Γ0
+sequentially. The proposed design, which we call the Precision CRM (P-CRM), comprises
+of two stages. In the first stage, patient characteristic information is collected, but dose as-
+signment is not patient-specific. In the second stage, once enough information is obtained, the
+method sequentially selects patient covariates to add to the dose-toxicity model based on their
+marginal correlation with toxicity after accounting for dose level. With each new enrollment
+cohort, given the additional information on toxicity and patient characteristics, the selected
+covariate or covariates are tested for removal. Please note these are enrollment cohorts and
+not dose cohorts, and thus patients within a cohort are assigned to different dose levels de-
+pending on their patient characteristics. Dose assignment is then based on the selected patient
+covariates given the current data. The final covariate-specific MTD is determined using the
+selected covariates in the final model after the total sample size is reached. The dose-finding
+stages of the P-CRM, in more detail, are as follows.
+Stage I: For the first N1 patients, the regular one-sample CRM method based on working
+model (1) is used [O’Quigley, Pepe, and Fisher (1990)]. Covariate information is collected
+but not used until Stage II. The skeleton can be elicited from physicians or calibrated through
+the approach by Lee and Cheung (2009) after specifying a target toxicity level, pT , cohort
+size, and initial guess of MTD. The dose label, or standardized dose, dj, for the jth dose is
+determined through equation logit(p0j) = ˜β0 + ˜β1dj, where ˜β0 and ˜β1 are the prior guesses
+
+PRECISION DOSE-FINDING DESIGN
+5
+of parameters β0 and β1 in Model (1), respectively, and p0j is the skeleton, j = 1,...,J. The
+dose labels are updated using a new skeleton based on the posterior probabilities of toxicity
+from N1 patients.
+Stage II: After N1 patients have been assigned and dose and toxicity are observed, patient
+covariates are sequentially assessed for inclusion after each patient enrollment cohort.
+1. In this stage, patients are enrolled in cohorts of size L.
+2. Once the first cohort is enrolled, we start the selection of covariates. For each covariate
+zm,m = 1,...,M, we fit the following working model
+(5)
+logit{π(dj,zm,i,θm)} = β0m + β1mdj + γmzm,i
+and denote the p-value associated with covariate zm as p1(zm). Let m1 = arg minm p1(zm).
+If p1(zm1) ≥ α∗
+1 for some prespecified α∗
+1 > 0, then no covariate is added and dose as-
+signment is conducted as in Stage I. If p1(zm1) < α∗
+1, then covariate zm1 is selected, and
+the (n + 1)-th cohort of patients is assigned to the dose closest to the target toxicity level
+pT :
+(6)
+x[n+1] = argminx∈{d1,...,dJ}|π(x;zm1,n+1, ˆθm1,n) − pT |.
+3. Once the data from a new enrollment cohort are collected, we proceed to select the next
+covariate and consider removal of the selected covariate. For each of the remaining co-
+variates zm, m ̸= m1, we fit working model (5). Let p2(zm) be the p-value associated with
+covariate zm,m ̸= m1, and let m2 = arg minm̸=m1 p2(zm). Covariate zm2 is selected if
+and only if p2(zm2) < α∗
+2 for some prespecified α∗
+2.
+Similarly, given that q covariates zm1,...,zmq are selected, to select the (q + 1)-th
+covariate, the remaining M − q covariates are tested one by one under Model (5) where
+m /∈ {m1,...,mq}. Let pq+1(zm) be the p-value associated with covariate zm in Model
+(5), and let mq+1 = arg minm/∈{m1,...,mq} pq+1(zm). Then, covariate zmq+1 is selected if
+and only if pq+1(zmq+1) < α∗
+q+1 for some α∗
+q+1 > 0.
+4. To assess for removal of the selected covariates, supposing q covariates zm1,...,zmq have
+been selected, we fit a working model
+(7)
+logit{π(dj,zm,i,θm)} = β0m + β1mdj +
+q
+�
+l=1
+γmlzml,i.
+Let m∗
+q = arg maxm{pq(zm1),...,pq(zmq)}. Covariate zm∗
+q is removed if pq(zm∗
+q) >
+α∗∗
+q .
+5. Given q remaining covariates that have not been removed, dose assignment follows from
+Equation (6) based on working model (7). If after removal, no covariates remain, dose
+assignment is conducted as in Stage I.
+6. Repeat Steps 3 to 5 until Nmax patients are assigned a recommended dose in this manner.
+Once Nmax is reached, estimate the MTD as follows:
+If no covariates remain selected, the MTD is estimated using the one-sample CRM and
+a single MTD is recommended for all patients. If k > 0 covariates zm1,...,zmk remain in
+the model, let z∗
+k = (zm1,...,zmk)T , then the MTD is
+(8)
+MTDz∗
+k = argminx∈{d1,...,dJ}|π(x;z∗
+k, ˆθm,Nmax) − pT |,
+where π(x;z∗
+k, ˆθm,Nmax) comes from the final model
+(9)
+logit{π(dj,z∗
+k,i,θm)} = β0m + β1mdj +
+k
+�
+l=1
+γmlzml,i.
+
+6
+Regarding the choice of cohort size, L, we recommend using three to five patients de-
+pending on the total sample size and number of criteria being considered, M. The inclusion
+thresholds α∗
+q and exclusion thresholds α∗∗
+q ,q = 1,...,M, should depend on the number of
+marginal models that are fit for a new cohort. If M is the total number of covariates being
+tested in the trial and q is the number of covariates already selected in the model, we recom-
+mend using α∗
+q = α(M − q)/M for inclusion and α∗∗
+q = α/q for exclusion, with α = 0.20.
+Our suggested choice of the inclusion threshold α∗
+q is a decreasing function in q, indicat-
+ing that we put more penalty for including more covariates, in consistency with the sparsity
+assumption (4).
+3. Numerical Studies.
+3.1. Simulation Setting: Redesigning the Selumetinib study.
+We performed a simulation
+study to examine the operating characteristics of the P-CRM dose-finding design. We re-
+designed the Selumetinib trial to consider expanding three patient criteria and evaluated the
+need for a subpopulation-specific MTD. The design was compared to the one-sample CRM
+to demonstrate how ignoring patient heterogeneity if it exists affects MTD recommendation.
+We tested three binary patient criteria (M = 3), and examined five scenarios of dose-
+toxicity relationships with six dose levels (J = 6). For each scenario, we considered each
+criterion having a prevalence of 50% and 25%, P(zm = 1) = 0.50,0.25, for m = 1,2,3.
+The dose-toxicity relationships for each scenario are illustrated in Figure 1 and the exact
+probability of DLT at each dose is provided in the Appendix. None of the scenarios were
+generated from the dose-toxicity models specified previously.
+The scenarios include cases when zero or one of the three eligibility criteria are strongly
+associated with toxicity, leading to one or two subgroups with different dose-toxicity rela-
+tionships, respectively. In Scenarios 1 through 4, z2 is associated with toxicity and the MTD
+for newly eligible patients satisfying the loosened criterion, z2 = 1, differs from the MTD for
+patients who meet the originally published criterion, z2 = 0. In both Scenarios 1 and 2, the
+MTDs of the subpopulations differ by one dose but in Scenario 1, the lower MTD is dose
+level one while in Scenario 2 the lower MTD is dose level two. These scenarios explore how
+the method performs when the MTDs exist on the boundary of the range of dose levels versus
+inside the range of dose levels. In Scenarios 3 and 4, the MTDs of the subpopulations dif-
+fer by two and three dose levels, respectively, representing increasingly more heterogeneous
+subpopulations and a greater need to differentiate the MTDs. Lastly, in Scenario 5, no crite-
+ria are associated with toxicity so no heterogeneity exists and the MTD is the same for all
+patients.
+In Stage I of the P-CRM and in the one-sample CRM, we started the CRM at dose level
+two and a target toxicity level, pT , of 0.25, as in the original trial. We used the logistic
+dose-toxicity model with a fixed intercept of 3 and assumed a normal prior of mean 0 and
+variance 1.34 on the dose covariate, β. The dose-toxicity model was calibrated to select a
+dose that yields between a 17% and 33% DLT rate [Lee and Cheung (2009)]. The cohort
+size was three. In the P-CRM, we set N1 = 15 and after N1 patients were assigned a dose
+using the CRM, we obtained the updated scaled dose as dj = log(
+p∗
+0j
+1−p∗
+0j ) − 3;j = 1,...,J,
+where p∗
+0j is the posterior probability of toxicity at dose j, assuming a dose effect coefficient
+of 1 and intercept of 3. In Stage II, we used a fixed-intercept of 3 in each marginal logistic
+regression model and a cohort size of three. Although three patients were added at a time,
+dose-assignment depended on their respective covariate pattern and the covariates chosen
+in the dose-finding algorithm. Additionally, we did not allow skipping a dose that had not
+yet been assigned to a patient. We ran 2,000 simulations of the Selumetinib trial with four
+different total sample sizes of Nmax = 30,45,60,72 to evaluate the impact of sample size.
+
+PRECISION DOSE-FINDING DESIGN
+7
+FIG 1. Dose-toxicity relationships for Scenarios 1-5. Scenarios 1, 2, 3, and 4 have one criterion associated with
+toxicity, differentiating the MTD by one, two, or three doses levels. In Scenario 5, no criteria are associated with
+toxicity so there is one MTD for the population. Horizontal dotted line represents the target toxicity level of 0.25.
+Star-shaped point represents the covariate-specific MTD.
+For each scenario and criteria prevalence, we were interested in three performance mea-
+sures: the probability of correct covariate selection, the probability of correct MTD selection
+(PCS), and the weighted probability of MTD selection (WPS) [Morita et al. (2017)]. The
+probability of correct covariate selection was defined as the frequency in which the k co-
+variates remaining in the final model from Equation 9 were the true criteria associated with
+toxicity. For Scenarios 1 through 4, the correct final model includes one additional covariate
+besides dose, z2, and for Scenario 5 it includes no additional covariates. The PCS was ob-
+tained by estimating the MTD for the observed patients based on the covariates chosen in the
+final model at Nmax = 30,45,60,72. For each observed patient covariate pattern, we used
+the final model obtained from the dose-finding procedure to estimate the covariate-specific
+probability of toxicity at each dose and estimated the covariate-specific MTD as discussed in
+Section 2. If no covariates were chosen, the one-sample CRM was used to estimate the MTD.
+To obtain a weighted probability of selection, we used the measure defined by Morita et al.
+(2017), which accounts for dose selection close to the MTD using a weight characterized by
+the distance in probability of toxicity at the selected dose and the target toxicity level. Let
+rj,k be the true probability of DLT at dose level j, j = 1,...,6 for the kth subgroup, where
+subgroups are defined by the true MTD as shown in Figure 1 such that k = 1,2 for Scenarios
+1 through 4, and k = 1 for Scenario 5, ordered by increasing true MTD, and pT is the target
+toxicity level. Then the WPS for subgroup k, WPSk is defined as
+WPSk =
+J
+�
+j=1
+wj,kP(dj is selected as the MTD for subgroup k),
+where
+wj,k =
+(maxj′=1,...,J |rj′,k − pT |) − |rj,k − pT |
+(maxj′=1,...,J |rj′,k − pT |) − (minj′=1,...,J |rj′,k − pT |).
+The WPS gives no weight to the least desirable dose and relatively higher weight for doses
+yielding true toxicity rates closer to the target toxicity level, while the PCS gives no weight to
+all doses except the true MTD. The subgroup PCS and WPS from the P-CRM were compared
+to the subgroup PCS and WPS from the one-sample CRM.
+
+Scenario 1
+Scenario 2
+Scenario 3
+Scenario 4
+Scenario 5
+1.00
+.00
+1.00
+.00
+.00
+0.75 -
+0.75 -
+0.75 
+0.75.
+0.75 -
+0.50 
+0.50 -
+0.50 
+0.50
+0.50 -
+0
+α
+0.25
+0.25 +
+0.25 +
+0.25
+0.25 +
+101
+101
+0.0/0
+0.0 
+D2
+D3
+D5
+D6
+D2
+D3
+D4
+D5
+D6
+D1
+D3
+D4
+D6
+D3
+D6
+D1
+D2 
+D3 
+D4
+D5
+01
+D4
+D3
+D2
+D5
+01
+D
+Dose level
+Dose level
+Dose level
+Dose level
+Dose level
+Z2 = 0
+Z2= 0
+Z²= 0
+Z2 =
+Z2= 0
+Entire Population
+Z2 = 18
+TABLE 1
+Probability of Criteria Selection with three total covariates, M = 3,
+each with prevalence of 50% and 25% for Nmax = 45
+Prevalence = 50%
+Scenario
+No criteria
+Correct criterion
+Correct criterion with others
+Incorrect criteria
+1
+30
+48
+6
+16
+2
+30
+44
+6
+19
+3
+11
+68
+10
+11
+4
+6
+73
+14
+7
+5
+56
+NA
+NA
+44
+Prevalence = 25%
+Scenario
+No criteria
+Correct criterion
+Correct criterion with others
+Incorrect criteria
+1
+38
+43
+4
+16
+2
+39
+41
+3
+16
+3
+15
+67
+8
+10
+4
+7
+79
+10
+4
+5
+63
+NA
+NA
+37
+* Selection probabilities may not equal 100 due to rounding.
+3.2. Simulation Results.
+The probability of criteria selection for the P-CRM when the
+prevalence of each criterion is 50% and 25% is given in Table 1 for Nmax = 45. The PCS and
+WPS for the P-CRM and one-sample CRM across total sample size for prevalence of each
+criterion of 50% and 25% are shown in Figures 2 and 3, respectively. Additional simulation
+results for criteria selection for Nmax = 60,72 and probability of selection of each dose under
+the P-CRM and one-sample CRM for a prevalence of 50% can be found in the Appendix.
+Table 1 gives the probability of criteria selection when P(zm = 1) = 0.50,0.25,m =
+1,2,3 and Nmax = 45. The probability of correct criteria selection is in bold; in Scenarios
+1 through 4, only one criterion, z2, should be selected, and in Scenario 5, no criteria should
+be selected. The P-CRM demonstrates higher criteria selection when the dose-toxicity rela-
+tionships between subpopulations are more heterogeneous. For example, for a prevalence of
+50%, when the MTDs are at least two dose levels apart, the design has a high selection rate
+of the true criterion, z2, of around 70%, but in the more challenging case of Scenarios 1 and
+2, where the MTDs are only one dose apart, the method selects z2 between 40 to 50% of the
+time. In these cases, when the method does not correctly select the true criterion, it will most
+often select no criteria, defaulting to the naive method of not including any covariates in the
+dose-toxicity model and assuming a homogeneous population. In Scenario 5, when no crite-
+ria are associated with toxicity, the method selects none of them about 56% of the time. When
+the prevalence of each criterion is 25%, the percentage of correct criterion selection remains
+similar, slightly decreasing for Scenarios 1 through 3 by an average of 3%. For larger sample
+sizes of Nmax = 60 and Nmax = 72, the probability of correct criterion selection improves,
+particularly for the cases of only slight heterogeneity, shown in the Appendix.
+Given that the design has no prior assumptions about which criterion are associated with
+toxicity, the design selects the correct criterion at a desirable rate for heterogeneous popu-
+lations where the optimal doses differ by more than one dose. The design has a controlled
+probability of incorrect criteria selection or false positive rate (FPR), except for Scenario 5
+when the population is homogeneous. If the investigator wishes to lower the FPR, they could
+consider a more conservative adjustment of α, but the design will be less likely to screen
+for patient heterogeneity. When the prevalence decreases by half, the method performs al-
+most as well in selecting criteria as when the criteria prevalence is 50%, implying the correct
+selection rate will only fall substantially when the criteria are very rare.
+Figure 2 (A and B) compare the subpopulation-specific PCS and WPS, respectively, be-
+tween the P-CRM and one-sample CRM when prevalence of each criterion is 50% across
+Nmax = 30,45,60,72. As heterogeneity of the subgroups grows and as total sample size
+
+PRECISION DOSE-FINDING DESIGN
+9
+FIG 2. A) Proportion of Correct Selection of MTD (PCS) and B) Weighted Probability of Selection (WPS) for
+each subgroup under the P-CRM and one-sample CRM across Nmax = 30,45,60,72 for criteria prevalence of
+50%.
+grows, the P-CRM demonstrates substantially improved PCS and WPS over the one-sample
+CRM. Performance of the one-sample CRM indicates that when there is more than one true
+MTD, ignoring patient criteria will lead to incorrectly dosing at least one subgroup, and often
+both. When the subgroups differ in true MTD by at least two doses, the one-sample CRM
+most often assigns a potentially sub-therapeutic dose to one subgroup and unsafe dose to the
+other, misidentifying each true respective MTD, whereas the P-CRM most often assigns the
+true subgroup MTD. Under the P-CRM, the PCS increases more over Nmax for more het-
+erogeneous populations. For scenarios where heterogeneity exists, the average PCS exceeds
+60% by Nmax = 60. For a homogeneous population, the PCS is close to 60% by Nmax = 30.
+The high WPS for all scenarios indicates that the P-CRM rarely assigns a dose that is more
+than one dose away from the true MTD. Detail of selection rates across each dose for the
+P-CRM and one-sample CRM are given in Appendix Tables 3 and 4 for criteria prevalence
+of 50%, showing that the subgroup with a lower MTD is more likely to be recommended a
+dose lower than the subgroup without the expanded eligibility criterion when the exact MTD
+assignment in incorrect.
+Figure 3 (A and B) compare the PCS and WPS, respectively, between the P-CRM and the
+one-sample CRM when the prevalence of each criterion is 25% across Nmax = 30,45,60,72.
+Compared to when the prevalence is 50%, the PCS for the subgroups that satisfy the broad-
+ened eligibility criteria (zi = 1,i = 1,2) is lower on average by 14%. However, even though
+estimating the exact subpopulation-specific MTD for this subgroup is more challenging, the
+design is more likely to assign a lower dose to the less tolerant subgroup and therefore rec-
+ommend a dose closer to the true MTD, as evidenced by the still favorable WPS compared
+
+A
+Scenario 1
+Scenario 2
+Scenario 3
+Scenario 4
+Scenario 5
+100+
+100-
+100-
+100-
+100-
+80-
+80.
+80
+80.
+80.
+101
+60
+60
+60
+60
+60
+PCS
+PCS
+PCS
+10
+10
+40.
+40
+40.
+40.
+40-
+20-
+20-
+20-
+20
+20.
+0-
+0-
+0+
+0-
+0
+45 
+60
+72
+30
+45 
+60
+72
+30
+45
+60
+72
+30
+45
+60
+72
+45
+60
+72
+30
+30
+Nmax
+Nmax
+Nmax
+Z²=１
+Z2 = 0
+Z²=１
+Z² = O
+Z2 = 0
+Z²=１
+Z2 = 0
+Entire Population
+P-CRM
+ One-sample CRM
+B
+Scenario 1
+Scenario 2
+Scenario 3
+Scenario 4
+Scenario 5
+100-
+100-
+100+
+100+
+100-
+0
+口
+O
+Q
+80
+80
+80.
+口
+80
+80
+O:
+0
+口
+09
+09
+60
+60
+09
+WPS
+10
+WPS
+WPS
+WPS
+WPS
+40-
+40.
+40-
+40-
+40-
+20-
+20-
+20-
+20-
+0+
+0-
+0-
+0-
+0-
+30
+45 
+60
+72
+30
+45
+60
+72
+30
+45 
+60
+72
+30
+45 
+60
+72
+30
+45
+60
+72
+Nmax
+Nmax
+Nmax
+Nmax
+Z²=１
+Z2 = 0
+Z²=１
+Z2 = 0
+2=１
+Z2 = 0
+Z²=１
+Z2 = 0
+Entire Population
+P-CRM
+One-sample CRM10
+FIG 3. A) Proportion of Correct Selection of MTD (PCS) and B) Weighted Probability of Selection (WPS) for
+each subgroup under the P-CRM and one-sample CRM across Nmax = 30,45,60,72 for criteria prevalence of
+25%.
+to the one-sample CRM. Under the one-sample CRM when the criteria prevalence is 25%,
+the subgroup satisfying z2 = 1 is completely ignored and for any heterogeneity where the
+subgroups have MTDs more than one dose apart, the PCS is close to 0% with a decreasing
+WPS as heterogeneity increases.
+Overall, the P-CRM performs well when there is one criterion differentiating the optimal
+dose for the patient population when the probability of toxicity at each subgroup’s MTD
+differs by at least 20%. The design identifies the true criterion more often for more heteroge-
+neous populations. Regarding dose assignment, we find that even when the design does not
+correctly identify the true MTD, particularly when the prevalence of the criteria is low, the
+design tends to recommend a dose that is close to the true MTD rather than assign the same
+dose as the one recommended for the more prevalent subgroup. Finally, the poor performance
+of the one-sample CRM highlights how assuming patient homogeneity and ignoring patient
+criteria leads to overly toxic or sub-therapeutic dose recommendations.
+3.3. Sample Size Considerations.
+The total sample size required for the design depends
+on the number of eligibility criteria that are being considered, M, the number of criteria for
+which we need to differentiate the MTD, K, the number of dose levels, J, and the prevalence
+of the criteria. With J = 6, M = 3, a criteria prevalence of 50%, and only one covariate be-
+ing associated with toxicity, to achieve an average PCS of at least 50% across scenarios, the
+sample size would need to be at least 45. When more than one out of the three covariate is
+associated with toxicity, resulting in three or more subgroups with different MTDs, a sample
+
+A
+Scenario 1
+Scenario 2
+Scenario 3
+Scenario 4
+Scenario 5
+100+
+100-
+100-
+100-
+100-
+80.
+80.
+80.
+80.
+80-
+口
+0
+60口
+60
+60
+09
+PCS
+PCS
+PCS
+PCS
+40-
+40.
+40
+40.
+40-
+20
+20-
+20.
+20--
+20.
+0+
+0-
+0-
+0
+0
+30
+45 
+60
+72
+30
+45
+60
+72
+30
+45
+60
+72
+30
+45
+60
+72
+30
+45
+60
+72
+Nmax
+Nmax
+Nmax
+Z²=１
+Z² = O
+Z²=１
+Z² = O
+Z2 = O
+Z²=１
+Z2 = 0
+Entire Population
+P-CRM
+ One-sample CRM
+B
+Scenario 1
+Scenario 2
+Scenario 3
+Scenario 4
+Scenario 5
+100-
+100-
+100+
+100+
+100-
+1-0
+LLO
+口
+80/
+80.
+80.
+80.
+80
+O
+口
+60
+60
+60
+60/0
+09
+WPS
+WPS
+WPS
+WPS
+WPS
+0
+40-
+40-
+40.
+40
+40-
+O
+20-
+20-
+20-
+20-
+20-
+0+
+0-
+0-
+0-
+0-
+30
+45 
+60
+72
+30
+45 
+60
+72
+45 
+60
+72
+30
+45
+60
+72
+30
+45
+60
+72
+30
+Nmax
+Nmax
+Nmax
+Z²=１
+Z2 = 0
+Z²=１
+Z2 = 0
+○Z²=１
+Z2 = 0
+Z²=１
+Z2 = 0
+Entire Population
+P-CRM
+One-sample CRMPRECISION DOSE-FINDING DESIGN
+11
+size around 60 is needed (data not shown). When prevalence decreases, the overall PCS in-
+creases but the PCS of the less prevalent subgroup decreases, and a larger sample size will be
+needed to achieve a subgroup PCS of at least 50%. Given the same scenarios, when the num-
+ber of criteria to consider is only two, M = 2, a lower sample size around 35 would obtain an
+average PCS of at least 50% (data not shown). If only one criteria is being considered, then
+other existing methods should also be considered. That is, the particular advantage of this
+method is that it can consider multiple possible subpopulations at once and thus reduce the
+required sample size to find one or more MTDs. While learning about heterogeneity requires
+an increased sample size, conducting separate dose-finding trials for each potential subgroup
+using the one-sample CRM would require a substantially larger sample size since there are
+many possible patient subgroups (2M), some of which could have insufficient sample sizes
+due to low prevalence.
+4. Discussion.
+The precision dose-finding design, P-CRM, proposed in this paper ex-
+plores dose-finding when the patient population is suspected to be heterogeneous but it is
+unknown which patient covariates indeed affect toxicity, and thus must be accounted for in
+MTD determination. As eligibility criteria expand in cancer trials, safety profiles for newly
+eligible patients could differ given that many were originally included for safety and a method
+accounting for eligibility criteria is needed to ensure patient safety. One recommendation for
+how to deal with the broadening of eligibility criteria in early-stage trials is to use expansion
+cohorts after an initial traditional dose-finding trial [Malik and Lu (2019); Kim et al. (2017)].
+However, the number of potential cohorts could be large and the specific cohorts that should
+be included are unknown. The second stage of the P-CRM is comparable to an expansion
+cohort in that it accounts for potential subpopulations, but the P-CRM is also able to adjust
+the dose for those populations accurately through a continued dose-finding procedure and
+learn more about subpopulations that are truly heterogeneous.
+The P-CRM uses marginal logistic regression models to determine whether each covariate
+is associated with toxicity. Covariates are sequentially added to the dose-toxicity model if
+they demonstrate significant association with toxicity and are assessed for removal after pa-
+tient data from each cohort is obtained and other selected covariates are accounted for. Dose
+assignment and MTD recommendation are based on dose and any patient covariates selected
+in the model. When no patient covariates are added, the design reverts to the original CRM
+and assumes patient homogeneity. The design requires no assumption about which covariates
+differentiate the patient population and can screen for multiple potential covariates at a time.
+Given our potentially high dimensional model with a sparsity constraint, a natural approach
+might be the least absolute shrinkage and selection operator (LASSO) method. However, us-
+ing the LASSO method in the setting of dose-finding where the sample size is small and the
+data is sparse, led to high false positive rates in covariate selection and an overall smaller
+probability of correct MTD selection.
+Our simulation study found that the design can correctly identify the covariate associ-
+ated with toxicity while maintaining control over the false positive rate. Although the design
+performs most favorably when large patient heterogeneity exists, we find that across any pa-
+tient heterogeneity scenario, the P-CRM is still favorable over the naive approach of ignoring
+patient heterogeneity and MTD distinction is necessary. When the probability of correct co-
+variate selection was less favorable due to less heterogeneous subpopulations, the probability
+of choosing a desirable dose was still high since the true MTDs in that case are close together.
+Furthermore, the P-CRM did not as accurately choose the true MTD for the less prevalent
+subgroup, but it chose a dose closer to their respective true MTD than the true MTD of the
+more prevalent subgroup. Finally, the simulation study showed that ignoring patient charac-
+teristics can lead to overly toxic or sub-therapeutic dose recommendations for at least one but
+
+12
+possibly all subpopulations within the patient population. Thus, ignoring patient characteris-
+tics for a heterogeneous population is far worse than incorporating patient characteristics for
+a homogeneous population.
+Since our design is the first phase I design to consider expanded eligibility criteria and
+assume no prior knowledge on heterogeneity, we do not compare it to other methods that
+account for known patient heterogeneity. For example, the two-sample CRM proposed by
+O’Quigley et al. (1999) or the Hierarchical Bayesian CRM proposed by Morita et al. (2017)
+assume that the patient subpopulations are already known. The P-CRM must first identify the
+subpopulations and then determine the dose for each one.
+Although larger sample sizes are needed to accommodate the expansion of many patient
+criteria compared to traditional phase I trials, the advantage of accounting for heterogeneity
+even with a typical phase I sample size of 30 is already clear. Furthermore, with the broad-
+ening of eligibility, more patients will be available to recruit into the trial and benefit from a
+safe dose. Thus, it is important to weigh the desired number of criteria to be evaluated and
+the potential number of subgroups expected against the increase in sample size and the preva-
+lence of the criteria when applying the method in practice. Ongoing work involves including
+efficacy so that both heterogeneity in toxicity and efficacy can be accounted for to determine
+the optimal dose. While precision medicine approaches have been considered mainly for late
+stage trials and large sample size settings, our approach presents a precision dose-finding
+approach for early stage trials with a limited sample size.
+The proposed design, P-CRM, selects patient characteristics that are associated with tox-
+icity and determines a subpopulation-specific MTD in a simple manner. The design only
+requires use of logistic regression, so incorporation of precision medicine in this case does
+not necessitate specification of multiple priors. A more precise determination of MTD and
+identification of the source of patient heterogeneity can be an asset in the long term, leading
+to more successful phase II and III trials and being directly applicable to the true patient
+population.
+
+PRECISION DOSE-FINDING DESIGN
+13
+APPENDIX
+TABLE 2
+Probability of toxicity for each dose for Scenarios 1-5.
+Scenario
+Subpopulation
+D1
+D2
+D3
+D4
+D5
+D6
+1
+z2 = 1
+0.25
+0.45
+0.60
+0.75
+0.85
+0.90
+z2 = 0
+0.02
+0.25
+0.45
+0.60
+0.75
+0.85
+2
+z2 = 1
+0.05
+0.25
+0.45
+0.60
+0.75
+0.85
+z2 = 0
+0.02
+0.05
+0.25
+0.45
+0.60
+0.75
+3
+z2 = 1
+0.05
+0.25
+0.45
+0.60
+0.75
+0.85
+z2 = 0
+0.02
+0.05
+0.08
+0.25
+0.45
+0.60
+4
+z2 = 1
+0.05
+0.08
+0.25
+0.45
+0.60
+0.70
+z2 = 0
+0.01
+0.01
+0.02
+0.05
+0.08
+0.25
+5
+Entire Population
+0.08
+0.25
+0.45
+0.60
+0.70
+0.75
+TABLE 3
+Probability of Criteria Selection with three total covariates, M = 3,
+each with prevalence of 50% and 25% for Nmax = 60
+Prevalence = 50%
+Scenario
+No criteria
+Correct criterion
+Correct criterion with others
+Incorrect criteria
+1
+21
+57
+8
+14
+2
+24
+52
+7
+16
+3
+4
+76
+14
+6
+4
+3
+76
+19
+2
+5
+56
+NA
+NA
+44
+Prevalence = 25%
+Scenario
+No criteria
+Correct criterion
+Correct criterion with others
+Incorrect criteria
+1
+29
+50
+6
+16
+2
+31
+48
+5
+16
+3
+8
+73
+11
+8
+4
+2
+82
+14
+2
+5
+58
+NA
+NA
+42
+* Selection probabilities may not equal 100 due to rounding.
+
+14
+TABLE 4
+Probability of Criteria Selection with three total covariates, M = 3,
+each with prevalence of 50% and 25% for Nmax = 72
+Prevalence = 50%
+Scenario
+No criteria
+Correct criterion
+Correct criterion with others
+Incorrect criteria
+1
+18
+64
+8
+10
+2
+21
+58
+9
+13
+3
+2
+78
+16
+4
+4
+2
+77
+20
+1
+5
+56
+NA
+NA
+44
+Prevalence = 25%
+Scenario
+No criteria
+Correct criterion
+Correct criterion with others
+Incorrect criteria
+1
+26
+54
+6
+14
+2
+27
+52
+6
+14
+3
+6
+76
+13
+5
+4
+1
+83
+15
+1
+5
+55
+NA
+NA
+45
+* Selection probabilities may not equal 100 due to rounding.
+
+PRECISION DOSE-FINDING DESIGN
+15
+TABLE 5
+The proportion of dose selection, PCS, and WPS for each Nmax under the P-CRM
+when the criteria prevalence is 50%.
+P-CRM
+Scenario
+Nmax
+True MTD
+D1
+D2
+D3
+D4
+D5
+D6
+PCS
+WPS
+Scenario 1
+30
+1
+0.64
+0.30
+0.05
+0.01
+0.00
+0.00
+0.64
+0.87
+30
+2
+0.27
+0.46
+0.22
+0.04
+0.01
+0.00
+0.46
+0.79
+45
+1
+0.71
+0.25
+0.03
+0.00
+0.00
+0.00
+0.71
+0.90
+45
+2
+0.20
+0.53
+0.23
+0.03
+0.00
+0.00
+0.53
+0.82
+60
+1
+0.76
+0.22
+0.02
+0.00
+0.00
+0.00
+0.76
+0.92
+60
+2
+0.16
+0.62
+0.21
+0.02
+0.00
+0.00
+0.62
+0.86
+72
+1
+0.80
+0.19
+0.01
+0.00
+0.00
+0.00
+0.80
+0.94
+72
+2
+0.12
+0.69
+0.18
+0.01
+0.00
+0.00
+0.69
+0.89
+Scenario 2
+30
+2
+0.13
+0.49
+0.32
+0.05
+0.01
+0.00
+0.49
+0.81
+30
+3
+0.03
+0.28
+0.44
+0.20
+0.05
+0.00
+0.44
+0.76
+45
+2
+0.13
+0.58
+0.25
+0.04
+0.00
+0.00
+0.58
+0.85
+45
+3
+0.01
+0.22
+0.51
+0.22
+0.04
+0.00
+0.51
+0.79
+60
+2
+0.10
+0.65
+0.22
+0.02
+0.00
+0.00
+0.65
+0.88
+60
+3
+0.00
+0.19
+0.58
+0.20
+0.02
+0.00
+0.58
+0.82
+72
+2
+0.08
+0.72
+0.19
+0.01
+0.00
+0.00
+0.72
+0.90
+72
+3
+0.00
+0.17
+0.64
+0.17
+0.01
+0.00
+0.64
+0.85
+Scenario 3
+30
+2
+0.10
+0.49
+0.31
+0.08
+0.02
+0.00
+0.49
+0.76
+30
+4
+0.01
+0.13
+0.28
+0.35
+0.18
+0.05
+0.35
+0.63
+45
+2
+0.11
+0.62
+0.22
+0.04
+0.01
+0.00
+0.62
+0.83
+45
+4
+0.00
+0.06
+0.20
+0.48
+0.21
+0.04
+0.48
+0.70
+60
+2
+0.09
+0.70
+0.17
+0.03
+0.00
+0.00
+0.70
+0.87
+60
+4
+0.00
+0.03
+0.16
+0.61
+0.17
+0.03
+0.61
+0.78
+72
+2
+0.07
+0.76
+0.15
+0.01
+0.00
+0.00
+0.76
+0.90
+72
+4
+0.00
+0.02
+0.14
+0.66
+0.17
+0.01
+0.66
+0.81
+Scenario 4
+30
+3
+0.01
+0.10
+0.42
+0.32
+0.11
+0.04
+0.42
+0.69
+30
+6
+0.00
+0.01
+0.11
+0.18
+0.24
+0.45
+0.45
+0.56
+45
+3
+0.00
+0.12
+0.55
+0.26
+0.05
+0.01
+0.55
+0.79
+45
+6
+0.00
+0.00
+0.03
+0.07
+0.24
+0.65
+0.65
+0.74
+60
+3
+0.00
+0.11
+0.63
+0.23
+0.02
+0.00
+0.63
+0.83
+60
+6
+0.00
+0.00
+0.01
+0.04
+0.21
+0.74
+0.74
+0.81
+72
+3
+0.00
+0.10
+0.68
+0.20
+0.01
+0.00
+0.68
+0.86
+72
+6
+0.00
+0.00
+0.01
+0.03
+0.19
+0.78
+0.78
+0.84
+Scenario 5
+30
+2
+0.20
+0.57
+0.19
+0.04
+0.00
+0.00
+0.57
+0.83
+45
+2
+0.17
+0.63
+0.17
+0.02
+0.00
+0.00
+0.63
+0.85
+60
+2
+0.15
+0.69
+0.15
+0.01
+0.00
+0.00
+0.69
+0.88
+72
+2
+0.12
+0.73
+0.14
+0.01
+0.00
+0.00
+0.73
+0.90
+
+16
+TABLE 6
+The proportion of dose selection, PCS, and WPS for each Nmax under the one-sample CRM
+when the criteria prevalence is 50%
+One-sample CRM
+Scenario
+Nmax
+True MTD
+D1
+D2
+D3
+D4
+D5
+D6
+PCS
+WPS
+Scenario 1
+30
+1
+0.46
+0.52
+0.02
+0.00
+0.00
+0.00
+0.46
+0.83
+30
+2
+0.46
+0.52
+0.02
+0.00
+0.00
+0.00
+0.52
+0.82
+45
+1
+0.46
+0.54
+0.00
+0.00
+0.00
+0.00
+0.46
+0.83
+45
+2
+0.46
+0.54
+0.00
+0.00
+0.00
+0.00
+0.54
+0.82
+60
+1
+0.48
+0.52
+0.00
+0.00
+0.00
+0.00
+0.48
+0.84
+60
+2
+0.48
+0.52
+0.00
+0.00
+0.00
+0.00
+0.52
+0.82
+72
+1
+0.48
+0.52
+0.00
+0.00
+0.00
+0.00
+0.48
+0.84
+72
+2
+0.48
+0.52
+0.00
+0.00
+0.00
+0.00
+0.52
+0.82
+Scenario 2
+30
+2
+0.00
+0.52
+0.46
+0.02
+0.00
+0.00
+0.52
+0.84
+30
+3
+0.00
+0.52
+0.46
+0.02
+0.00
+0.00
+0.46
+0.78
+45
+2
+0.00
+0.52
+0.48
+0.00
+0.00
+0.00
+0.52
+0.84
+45
+3
+0.00
+0.52
+0.48
+0.00
+0.00
+0.00
+0.48
+0.79
+60
+2
+0.00
+0.52
+0.47
+0.00
+0.00
+0.00
+0.52
+0.84
+60
+3
+0.00
+0.52
+0.47
+0.00
+0.00
+0.00
+0.47
+0.79
+72
+2
+0.00
+0.52
+0.47
+0.00
+0.00
+0.00
+0.52
+0.84
+72
+3
+0.00
+0.52
+0.47
+0.00
+0.00
+0.00
+0.47
+0.79
+Scenario 3
+30
+2
+0.00
+0.30
+0.56
+0.14
+0.00
+0.00
+0.30
+0.68
+30
+4
+0.00
+0.30
+0.56
+0.14
+0.00
+0.00
+0.14
+0.55
+45
+2
+0.00
+0.25
+0.65
+0.10
+0.00
+0.00
+0.25
+0.67
+45
+4
+0.00
+0.25
+0.65
+0.10
+0.00
+0.00
+0.10
+0.54
+60
+2
+0.00
+0.23
+0.70
+0.07
+0.00
+0.00
+0.23
+0.67
+60
+4
+0.00
+0.23
+0.70
+0.07
+0.00
+0.00
+0.07
+0.53
+72
+2
+0.00
+0.22
+0.72
+0.05
+0.00
+0.00
+0.22
+0.67
+72
+4
+0.00
+0.22
+0.72
+0.05
+0.00
+0.00
+0.05
+0.52
+Scenario 4
+30
+3
+0.00
+0.01
+0.24
+0.52
+0.21
+0.03
+0.24
+0.58
+30
+6
+0.00
+0.01
+0.24
+0.52
+0.21
+0.03
+0.03
+0.19
+45
+3
+0.00
+0.00
+0.19
+0.57
+0.22
+0.01
+0.19
+0.56
+45
+6
+0.00
+0.00
+0.19
+0.57
+0.22
+0.01
+0.01
+0.18
+60
+3
+0.00
+0.00
+0.18
+0.61
+0.20
+0.01
+0.18
+0.56
+60
+6
+0.00
+0.00
+0.18
+0.61
+0.20
+0.01
+0.01
+0.17
+72
+3
+0.00
+0.00
+0.15
+0.66
+0.19
+0.00
+0.15
+0.55
+72
+6
+0.00
+0.00
+0.15
+0.66
+0.19
+0.00
+0.00
+0.18
+Scenario 5
+30
+2
+0.13
+0.76
+0.11
+0.00
+0.00
+0.00
+0.76
+0.91
+45
+2
+0.08
+0.86
+0.06
+0.00
+0.00
+0.00
+0.86
+0.95
+60
+2
+0.06
+0.90
+0.04
+0.00
+0.00
+0.00
+0.90
+0.96
+72
+2
+0.04
+0.92
+0.03
+0.00
+0.00
+0.00
+0.92
+0.97
+
+PRECISION DOSE-FINDING DESIGN
+17
+Acknowledgments.
+This publication was supported by the National Center for Ad-
+vancing Translational Sciences, National Institutes of Health, through Grant Numbers
+UL1TR001873 and TL1TR001875.
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+
diff --git a/mNE3T4oBgHgl3EQfiQps/content/tmp_files/load_file.txt b/mNE3T4oBgHgl3EQfiQps/content/tmp_files/load_file.txt
new file mode 100644
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+page_content=' LEE1,c  1Department of Biostatistics, Columbia University Mailman School of Public Health,  ars4025@cumc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='edu  2Herbert Irving Comprehensive Cancer Center, Columbia University, drdc2150@cumc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='columbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='edu  Broadening eligibility criteria in cancer trials has been advocated to rep-  resent the true patient population more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' While the advantages are  clear in terms of generalizability and recruitment, novel dose-finding designs  are needed to ensure patient safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' These designs should be able to recom-  mend precise doses for subpopulations if such subpopulations with different  toxicity profiles exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' While dose-finding designs accounting for patient het-  erogeneity have been proposed, all existing methods assume the source of  heterogeneity is known and thus pre-specify the subpopulations or only allow  inclusion of a few patient characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' We propose a precision dose-finding  design to address the setting of unknown patient heterogeneity in phase I can-  cer clinical trials where eligibility is expanded, and multiple eligibility criteria  could potentially lead to different optimal doses for patient subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The  design offers a two-in-one approach to dose-finding by simultaneously select-  ing patient criteria that differentiate the maximum tolerated dose (MTD) and  recommending the subpopulation-specific MTD if needed, using marginal  models to sequentially incorporate patient covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Our simulation study  compares the proposed design to the naive approach of assuming patient ho-  mogeneity and our design recommends multiple doses when heterogeneity  exists and a single dose when no heterogeneity exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The proposed dose-  finding design addresses the challenges of broadening eligibility criteria in  cancer trials and the desire for a more precise dose in the context of early  phase clinical trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The goal of a dose-finding phase I cancer clinical trial is to select a  safe dose of a new therapy for subsequent phase II and III trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The recommended dose is  often known as the maximum tolerated dose (MTD), assuming toxicity and efficacy of a treat-  ment increase with dosage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The majority of phase I trials recommend one MTD, assuming  all eligible patients have similar tolerability to the new treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Over the past decade, the  expansion of eligibility criteria for cancer trials which were initially designed for cytotoxic  chemotherapies has been advocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Eligibility criteria have been found to be too restrictive  and not reflective of the actual patient population that could benefit from a therapy [Fehren-  backer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Gerber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Malik and Lu (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Harvey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2021)], thus limiting the generalizability of study results and significantly slowing  down patient accrual [Malik and Lu (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' One study examining the ineligibility rates of  326 patients consecutively diagnosed with non-small cell lung cancer (NSCLC) found that  about 80% of patients are ineligible based on traditional eligibility criteria [Fehrenbacker et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2009)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Several authors have found that the inclusion of patients with brain metastases, HIV, or  prior cancers, does not affect trial outcomes and better reflects the actual patient population  Keywords and phrases: subpopulation-specific dose-finding, subgroup dose-finding, eligibility criteria, patient  heterogeneity, sequential design, phase I cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='04578v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='AP]  11 Jan 2023  2  [George (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2015, 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Beaver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Malik and Lu (2019)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In  2017, the American Society of Clinical Oncology (ASCO), Friends of Cancer Research, and  the US Food and Drug Administration published new recommendations for exclusion criteria  that should be reconsidered including brain metastases, HIV and AIDS, organ dysfunction,  prior or concurrent cancers, and a minimum age for enrollment [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Food and  Drug Administration (2020)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Moreover, eligibility criteria such as concomitant medications,  prior therapies, performance status, liver or renal dysfunction, and laboratory values such as  levels of bilirubin, platelets, hemoglobin and alkaline phosphatase, continue to be questioned  and evaluated by ASCO and Friends of Cancer Research [Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Magnuson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Osarogiagbon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2021)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  While the broadening of eligibility criteria has clear advantages, it also has to be done  cautiously to ensure patient safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Specifically, dose-finding methods should account for the  potential of heterogeneous populations where patient subpopulations could have different  tolerances and take a precision medicine approach to dose estimation by recommending a  subpopulation-specific MTD if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Methods have been proposed in the phase I setting  to account for a pre-specified patient covariate or subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' All these methods either  assume that the covariates of interest or subpopulations with different toxicity profiles are  known beforehand, or adaptively combine subgroups based on similar risk without the iden-  tification of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Therefore, these methods cannot be applied to address the challenges  in the setting of broadened eligibility criteria where it is not known beforehand whether  different MTDs are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Extensions of the Continual Reassessment Method (CRM) [Pi-  antadosi and Liu (1996)] and the Escalation with Overdose Control allow for the addition of a  patient-specific covariate to the dose-toxicity model [Babb and Rogatko (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  (2004)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Moreover, there are methods that allow for the pre-specification of subpopulations  including the two-sample CRM [O’Quigley, Shen, and Gamst (1999)], the two-sample CRM  for ordered groups [O’Quigley and Paoletti (2003)], and isotonic designs for multiple ordered  risk groups [Yuan and Chappell (2004)] and partially ordered risk groups [Conaway (2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  More recently, Morita, Thall, and Takeda (2017) proposed a hierarchical Bayesian approach  that uses subgroup-specific intercepts and Chapple and Thall (2018) proposed Spike and Slab  priors on subgroup parameters to account for multiple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  While precision medicine has been generally advocated for later stages in drug develop-  ment and with larger data sets, in our early stage of drug development, it would be ideal to  have a method that can screen for the expanded eligibility criteria, assess whether it is neces-  sary to account for them, and be able to recommend more than one MTD if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Beaver et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2017) propose that “a logical approach to defining eligibility could allow for detection of  safety signals in early clinical trials that use broad eligibility criteria”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' As eligibility criteria  in oncology trials expand, having a method that can learn and assess for signals of patient  heterogeneity in toxicity will be key to ensuring the safety of patients who have traditionally  been thought to be at increased risk of adverse outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  As a motivating example, we use the phase I trial evaluating intermittent dosing of a  mitogen-activated protein kinase kinase (MEK) inhibitor, Selumetinib, to treat patients with  uveal melanoma [Khan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2022)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The study evaluated six doses of Selumetinib (100, 125,  150, 175, 200, 225 mg) and estimated the MTD using the Time-to-Event CRM (TITE-CRM)  with a target toxicity level of 25% in 28 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Given the rarity of uveal melanoma, accrual  took over three years with three large sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The principal investigator of the trial identified  three criteria (creatinine levels, organ and marrow function, and the presence of known or  suspected brain metastases) that could have been loosened to allow more patients to benefit  from the trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For example, creatinine level captures renal function and between 12% and  53% percent of cancer patients have chronic kidney disease [Kitchlu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2018)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If eligi-  bility criteria were to expand, the design of this phase I trial would require accounting for  PRECISION DOSE-FINDING DESIGN  3  these three criteria, since knowledge about their association with toxicity is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Some of  these criteria may affect the probability of toxicity of Selumetinib while others may not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For  example, renal impairment has been associated with increased risk of toxicity, leading inves-  tigators to account for degree of renal dysfunction when determining dose [Lameire (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Leal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' LoRusso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2012)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' However, dose reductions have been found to not  always be necessary [Leal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2011)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If the loosened criteria are not accounted for and at  least one criterion is associated with increased risk of toxicity, the design could overestimate  the MTD for these patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Our proposed design, the Precision CRM (P-CRM), addresses the case where we want to  consider and learn about multiple patient characteristics, which contribute to a broader patient  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For example, if the eligibility criteria were expanded in the Selumetinib trial, we  would want to consider evaluating three patient criteria since the MTD could differ based on  one or more of these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Our method selects patient criterion to add to the dose-toxicity  model sequentially, based on their marginal correlation with toxicity after accounting for  dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' With each new enrollment cohort, given the additional information on toxicity and pa-  tient characteristics, the criteria are tested for inclusion or removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Dose assignment is based  on the selected patient criteria given the current data and the final subpopulation-specific  MTD is determined using the selected covariates in the final model after the total sample size  is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' This method provides a two-in-one approach to dose-finding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' the method learns  about the criteria and the need to differentiate the MTD and recommends a dose for each  patient subpopulation based on these criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The rest of the paper is outlined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The precision dose-finding method is described  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In Section 3, we present a simulation study based on the redesigned Selumetinib  trial and compare the performance of the P-CRM to the naive approach of not accounting for  any patient heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' We also discuss sample size considerations for the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Finally,  the significance and limitations of our proposed design are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Drug safety is most commonly measured by the occurrence of a severe  adverse event, or toxicity, in the first cycle of treatment, known as a dose-limiting toxicity  (DLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In model-based designs, a total of J doses, d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',dJ, are evaluated, and the proba-  bility of toxicity, or DLT, is estimated using a dose-toxicity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The MTD, defined as the  dose whose toxicity is closest to pT , the highest rate of toxicity that clinicians deem accept-  able, or target toxicity level, is estimated sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For all future notation, let Yi be the  binary toxicity outcome for the ith patient and Xi be the dose assigned to this patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Dose-toxicity Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Traditionally, it is assumed that the entire patient population  has the same probability of toxicity at each dose, and therefore has the same MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Assuming  a logistic model and complete patient homogeneity, π(dj,β0,β1) = P(Yi = 1|Xi = dj) is  modeled as  (1)  logit{π(dj,β0,β1)} = β0 + β1dj,j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  In the late 1990s, the patient homogeneity assumption in dose-finding models started being  questioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' A dose-toxicity model incorporating a patient covariate, z, was proposed, where  π(dj,zi,θ) = P(Yi = 1|Xi = dj,zi) is modeled as,  (2)  logit{π(dj,zi,θ)} = β0 + β1dj + γzi,  where θ = (β0,β1,γ)T [Piantadosi and Liu (1996)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Equation (2) represents an overly restric-  tive model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' it assumes that we know beforehand which covariate contributes to the hetero-  geneity and that heterogeneity is completely due to z, both of which are strong assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  4  In this paper, we consider a potentially large collection of patient covariates, z =  (z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',zM)T , which could be responsible for dose heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' We model π(dj,zi,θ) =  P(Yi = 1|Xi = dj,zi) as  (3)  logit{π(dj,zi,θ)} = β0 + β1dj +  M  �  m=1  γmzmi,  where θ = (β0,β1,γ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',γM)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In addition, we make the following sparsity assumption  M  �  m=1  1γm̸=0 = K ≪ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  (4)  In the setting of expanded eligibility criteria, Equation (3) represents a more realistic model  which is potentially high dimensional, but we assume that only a small subset of these criteria  are responsible for the heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Assumption (4) is consistent with clinicians’ belief that  there are only a few expanded criteria/covariates or none at all for which we need to actually  differentiate the MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Therefore, assuming there are M total patient covariates suspected to be associated with  toxicity of which a small subset actually differentiate the MTDs, our objective is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  One is to identify the subset of K covariates that are responsible for the heterogeneity in  MTDs, and the other is to implement a procedure that determines the subpopulation-specific  MTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Precision Dose-finding Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Let Γ = {γi,i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',M}, and Γ0 = {γi : γi ̸=  0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' By Equation (3), Γ0 is an unknown small subset of Γ that must be estimated because it is  responsible for the heterogeneity in MTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The estimation of Γ imposes a large challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Since K is unknown, and thus must be estimated, we need to search among all 2M subsets  of Γ for Γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Furthermore, in dose-finding studies, patients are assigned to doses in sequential  cohorts, not simultaneously to all doses as done in a parallel group design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' This means that  information concerning parameters in model (3) is to be accumulated gradually as more and  more toxicity data become available, suggesting that it would be too ambitious to estimate  Γ0 in a single effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Consistent with the sequential nature of dose finding studies, we propose to estimate Γ0  sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The proposed design, which we call the Precision CRM (P-CRM), comprises  of two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In the first stage, patient characteristic information is collected, but dose as-  signment is not patient-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In the second stage, once enough information is obtained, the  method sequentially selects patient covariates to add to the dose-toxicity model based on their  marginal correlation with toxicity after accounting for dose level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' With each new enrollment  cohort, given the additional information on toxicity and patient characteristics, the selected  covariate or covariates are tested for removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Please note these are enrollment cohorts and  not dose cohorts, and thus patients within a cohort are assigned to different dose levels de-  pending on their patient characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Dose assignment is then based on the selected patient  covariates given the current data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The final covariate-specific MTD is determined using the  selected covariates in the final model after the total sample size is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The dose-finding  stages of the P-CRM, in more detail, are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Stage I: For the first N1 patients, the regular one-sample CRM method based on working  model (1) is used [O’Quigley, Pepe, and Fisher (1990)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Covariate information is collected  but not used until Stage II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The skeleton can be elicited from physicians or calibrated through  the approach by Lee and Cheung (2009) after specifying a target toxicity level, pT , cohort  size, and initial guess of MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The dose label, or standardized dose, dj, for the jth dose is  determined through equation logit(p0j) = ˜β0 + ˜β1dj, where ˜β0 and ˜β1 are the prior guesses  PRECISION DOSE-FINDING DESIGN  5  of parameters β0 and β1 in Model (1), respectively, and p0j is the skeleton, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The  dose labels are updated using a new skeleton based on the posterior probabilities of toxicity  from N1 patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Stage II: After N1 patients have been assigned and dose and toxicity are observed, patient  covariates are sequentially assessed for inclusion after each patient enrollment cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In this stage, patients are enrolled in cohorts of size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Once the first cohort is enrolled, we start the selection of covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For each covariate  zm,m = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',M, we fit the following working model  (5)  logit{π(dj,zm,i,θm)} = β0m + β1mdj + γmzm,i  and denote the p-value associated with covariate zm as p1(zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Let m1 = arg minm p1(zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  If p1(zm1) ≥ α∗  1 for some prespecified α∗  1 > 0, then no covariate is added and dose as-  signment is conducted as in Stage I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If p1(zm1) < α∗  1, then covariate zm1 is selected, and  the (n + 1)-th cohort of patients is assigned to the dose closest to the target toxicity level  pT :  (6)  x[n+1] = argminx∈{d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',dJ}|π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='zm1,n+1, ˆθm1,n) − pT |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Once the data from a new enrollment cohort are collected, we proceed to select the next  covariate and consider removal of the selected covariate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For each of the remaining co-  variates zm, m ̸= m1, we fit working model (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Let p2(zm) be the p-value associated with  covariate zm,m ̸= m1, and let m2 = arg minm̸=m1 p2(zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Covariate zm2 is selected if  and only if p2(zm2) < α∗  2 for some prespecified α∗  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Similarly, given that q covariates zm1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',zmq are selected, to select the (q + 1)-th  covariate, the remaining M − q covariates are tested one by one under Model (5) where  m /∈ {m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',mq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Let pq+1(zm) be the p-value associated with covariate zm in Model  (5), and let mq+1 = arg minm/∈{m1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',mq} pq+1(zm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Then, covariate zmq+1 is selected if  and only if pq+1(zmq+1) < α∗  q+1 for some α∗  q+1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' To assess for removal of the selected covariates, supposing q covariates zm1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',zmq have  been selected, we fit a working model  (7)  logit{π(dj,zm,i,θm)} = β0m + β1mdj +  q  �  l=1  γmlzml,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Let m∗  q = arg maxm{pq(zm1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',pq(zmq)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Covariate zm∗  q is removed if pq(zm∗  q) >  α∗∗  q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Given q remaining covariates that have not been removed, dose assignment follows from  Equation (6) based on working model (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If after removal, no covariates remain, dose  assignment is conducted as in Stage I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Repeat Steps 3 to 5 until Nmax patients are assigned a recommended dose in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Once Nmax is reached, estimate the MTD as follows:  If no covariates remain selected, the MTD is estimated using the one-sample CRM and  a single MTD is recommended for all patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If k > 0 covariates zm1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',zmk remain in  the model, let z∗  k = (zm1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',zmk)T , then the MTD is  (8)  MTDz∗  k = argminx∈{d1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',dJ}|π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='z∗  k, ˆθm,Nmax) − pT |,  where π(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='z∗  k, ˆθm,Nmax) comes from the final model  (9)  logit{π(dj,z∗  k,i,θm)} = β0m + β1mdj +  k  �  l=1  γmlzml,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  6  Regarding the choice of cohort size, L, we recommend using three to five patients de-  pending on the total sample size and number of criteria being considered, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The inclusion  thresholds α∗  q and exclusion thresholds α∗∗  q ,q = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',M, should depend on the number of  marginal models that are fit for a new cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If M is the total number of covariates being  tested in the trial and q is the number of covariates already selected in the model, we recom-  mend using α∗  q = α(M − q)/M for inclusion and α∗∗  q = α/q for exclusion, with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Our suggested choice of the inclusion threshold α∗  q is a decreasing function in q, indicat-  ing that we put more penalty for including more covariates, in consistency with the sparsity  assumption (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Numerical Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Simulation Setting: Redesigning the Selumetinib study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  We performed a simulation  study to examine the operating characteristics of the P-CRM dose-finding design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' We re-  designed the Selumetinib trial to consider expanding three patient criteria and evaluated the  need for a subpopulation-specific MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The design was compared to the one-sample CRM  to demonstrate how ignoring patient heterogeneity if it exists affects MTD recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  We tested three binary patient criteria (M = 3), and examined five scenarios of dose-  toxicity relationships with six dose levels (J = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For each scenario, we considered each  criterion having a prevalence of 50% and 25%, P(zm = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25, for m = 1,2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The dose-toxicity relationships for each scenario are illustrated in Figure 1 and the exact  probability of DLT at each dose is provided in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' None of the scenarios were  generated from the dose-toxicity models specified previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The scenarios include cases when zero or one of the three eligibility criteria are strongly  associated with toxicity, leading to one or two subgroups with different dose-toxicity rela-  tionships, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In Scenarios 1 through 4, z2 is associated with toxicity and the MTD  for newly eligible patients satisfying the loosened criterion, z2 = 1, differs from the MTD for  patients who meet the originally published criterion, z2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In both Scenarios 1 and 2, the  MTDs of the subpopulations differ by one dose but in Scenario 1, the lower MTD is dose  level one while in Scenario 2 the lower MTD is dose level two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' These scenarios explore how  the method performs when the MTDs exist on the boundary of the range of dose levels versus  inside the range of dose levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In Scenarios 3 and 4, the MTDs of the subpopulations dif-  fer by two and three dose levels, respectively, representing increasingly more heterogeneous  subpopulations and a greater need to differentiate the MTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Lastly, in Scenario 5, no crite-  ria are associated with toxicity so no heterogeneity exists and the MTD is the same for all  patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  In Stage I of the P-CRM and in the one-sample CRM, we started the CRM at dose level  two and a target toxicity level, pT , of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25, as in the original trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' We used the logistic  dose-toxicity model with a fixed intercept of 3 and assumed a normal prior of mean 0 and  variance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='34 on the dose covariate, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The dose-toxicity model was calibrated to select a  dose that yields between a 17% and 33% DLT rate [Lee and Cheung (2009)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The cohort  size was three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In the P-CRM, we set N1 = 15 and after N1 patients were assigned a dose  using the CRM, we obtained the updated scaled dose as dj = log(  p∗  0j  1−p∗  0j ) − 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',J,  where p∗  0j is the posterior probability of toxicity at dose j, assuming a dose effect coefficient  of 1 and intercept of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In Stage II, we used a fixed-intercept of 3 in each marginal logistic  regression model and a cohort size of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Although three patients were added at a time,  dose-assignment depended on their respective covariate pattern and the covariates chosen  in the dose-finding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Additionally, we did not allow skipping a dose that had not  yet been assigned to a patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' We ran 2,000 simulations of the Selumetinib trial with four  different total sample sizes of Nmax = 30,45,60,72 to evaluate the impact of sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  PRECISION DOSE-FINDING DESIGN  7  FIG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Dose-toxicity relationships for Scenarios 1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Scenarios 1, 2, 3, and 4 have one criterion associated with  toxicity, differentiating the MTD by one, two, or three doses levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In Scenario 5, no criteria are associated with  toxicity so there is one MTD for the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Horizontal dotted line represents the target toxicity level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Star-shaped point represents the covariate-specific MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  For each scenario and criteria prevalence, we were interested in three performance mea-  sures: the probability of correct covariate selection, the probability of correct MTD selection  (PCS), and the weighted probability of MTD selection (WPS) [Morita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The  probability of correct covariate selection was defined as the frequency in which the k co-  variates remaining in the final model from Equation 9 were the true criteria associated with  toxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For Scenarios 1 through 4, the correct final model includes one additional covariate  besides dose, z2, and for Scenario 5 it includes no additional covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The PCS was ob-  tained by estimating the MTD for the observed patients based on the covariates chosen in the  final model at Nmax = 30,45,60,72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For each observed patient covariate pattern, we used  the final model obtained from the dose-finding procedure to estimate the covariate-specific  probability of toxicity at each dose and estimated the covariate-specific MTD as discussed in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If no covariates were chosen, the one-sample CRM was used to estimate the MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  To obtain a weighted probability of selection, we used the measure defined by Morita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  (2017), which accounts for dose selection close to the MTD using a weight characterized by  the distance in probability of toxicity at the selected dose and the target toxicity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Let  rj,k be the true probability of DLT at dose level j, j = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',6 for the kth subgroup, where  subgroups are defined by the true MTD as shown in Figure 1 such that k = 1,2 for Scenarios  1 through 4, and k = 1 for Scenario 5, ordered by increasing true MTD, and pT is the target  toxicity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Then the WPS for subgroup k, WPSk is defined as  WPSk =  J  �  j=1  wj,kP(dj is selected as the MTD for subgroup k),  where  wj,k =  (maxj′=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',J |rj′,k − pT |) − |rj,k − pT |  (maxj′=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',J |rj′,k − pT |) − (minj′=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=',J |rj′,k − pT |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The WPS gives no weight to the least desirable dose and relatively higher weight for doses  yielding true toxicity rates closer to the target toxicity level, while the PCS gives no weight to  all doses except the true MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The subgroup PCS and WPS from the P-CRM were compared  to the subgroup PCS and WPS from the one-sample CRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Scenario 1  Scenario 2  Scenario 3  Scenario 4  Scenario 5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50 -  0  α  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25 +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25 +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25 +  101  101  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='0/0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='0  D2  D3  D5  D6  D2  D3  D4  D5  D6  D1  D3  D4  D6  D3  D6  D1  D2  D3  D4  D5  01  D4  D3  D2  D5  01  D  Dose level  Dose level  Dose level  Dose level  Dose level  Z2 = 0  Z2= 0  Z²= 0  Z2 =  Z2= 0  Entire Population  Z2 = 18  TABLE 1  Probability of Criteria Selection with three total covariates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' M = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='each with prevalence of 50% and 25% for Nmax = 45  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Prevalence = 50%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Scenario  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='No criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion with others  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Incorrect criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='48  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='73  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='56  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Prevalence = 25%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Scenario  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='No criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion with others  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Incorrect criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='38  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='39  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='41  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='67  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='79  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='63  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='37  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Selection probabilities may not equal 100 due to rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Simulation Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The probability of criteria selection for the P-CRM when the  prevalence of each criterion is 50% and 25% is given in Table 1 for Nmax = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The PCS and  WPS for the P-CRM and one-sample CRM across total sample size for prevalence of each  criterion of 50% and 25% are shown in Figures 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Additional simulation  results for criteria selection for Nmax = 60,72 and probability of selection of each dose under  the P-CRM and one-sample CRM for a prevalence of 50% can be found in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Table 1 gives the probability of criteria selection when P(zm = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25,m =  1,2,3 and Nmax = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The probability of correct criteria selection is in bold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' in Scenarios  1 through 4, only one criterion, z2, should be selected, and in Scenario 5, no criteria should  be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The P-CRM demonstrates higher criteria selection when the dose-toxicity rela-  tionships between subpopulations are more heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For example, for a prevalence of  50%, when the MTDs are at least two dose levels apart, the design has a high selection rate  of the true criterion, z2, of around 70%, but in the more challenging case of Scenarios 1 and  2, where the MTDs are only one dose apart, the method selects z2 between 40 to 50% of the  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In these cases, when the method does not correctly select the true criterion, it will most  often select no criteria, defaulting to the naive method of not including any covariates in the  dose-toxicity model and assuming a homogeneous population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' In Scenario 5, when no crite-  ria are associated with toxicity, the method selects none of them about 56% of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' When  the prevalence of each criterion is 25%, the percentage of correct criterion selection remains  similar, slightly decreasing for Scenarios 1 through 3 by an average of 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For larger sample  sizes of Nmax = 60 and Nmax = 72, the probability of correct criterion selection improves,  particularly for the cases of only slight heterogeneity, shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Given that the design has no prior assumptions about which criterion are associated with  toxicity, the design selects the correct criterion at a desirable rate for heterogeneous popu-  lations where the optimal doses differ by more than one dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The design has a controlled  probability of incorrect criteria selection or false positive rate (FPR), except for Scenario 5  when the population is homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If the investigator wishes to lower the FPR, they could  consider a more conservative adjustment of α, but the design will be less likely to screen  for patient heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' When the prevalence decreases by half, the method performs al-  most as well in selecting criteria as when the criteria prevalence is 50%, implying the correct  selection rate will only fall substantially when the criteria are very rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Figure 2 (A and B) compare the subpopulation-specific PCS and WPS, respectively, be-  tween the P-CRM and one-sample CRM when prevalence of each criterion is 50% across  Nmax = 30,45,60,72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' As heterogeneity of the subgroups grows and as total sample size  PRECISION DOSE-FINDING DESIGN  9  FIG 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' A) Proportion of Correct Selection of MTD (PCS) and B) Weighted Probability of Selection (WPS) for  each subgroup under the P-CRM and one-sample CRM across Nmax = 30,45,60,72 for criteria prevalence of  50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  grows, the P-CRM demonstrates substantially improved PCS and WPS over the one-sample  CRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Performance of the one-sample CRM indicates that when there is more than one true  MTD, ignoring patient criteria will lead to incorrectly dosing at least one subgroup, and often  both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' When the subgroups differ in true MTD by at least two doses, the one-sample CRM  most often assigns a potentially sub-therapeutic dose to one subgroup and unsafe dose to the  other, misidentifying each true respective MTD, whereas the P-CRM most often assigns the  true subgroup MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Under the P-CRM, the PCS increases more over Nmax for more het-  erogeneous populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For scenarios where heterogeneity exists, the average PCS exceeds  60% by Nmax = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For a homogeneous population, the PCS is close to 60% by Nmax = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The high WPS for all scenarios indicates that the P-CRM rarely assigns a dose that is more  than one dose away from the true MTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Detail of selection rates across each dose for the  P-CRM and one-sample CRM are given in Appendix Tables 3 and 4 for criteria prevalence  of 50%, showing that the subgroup with a lower MTD is more likely to be recommended a  dose lower than the subgroup without the expanded eligibility criterion when the exact MTD  assignment in incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Figure 3 (A and B) compare the PCS and WPS, respectively, between the P-CRM and the  one-sample CRM when the prevalence of each criterion is 25% across Nmax = 30,45,60,72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Compared to when the prevalence is 50%, the PCS for the subgroups that satisfy the broad-  ened eligibility criteria (zi = 1,i = 1,2) is lower on average by 14%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' However, even though  estimating the exact subpopulation-specific MTD for this subgroup is more challenging, the  design is more likely to assign a lower dose to the less tolerant subgroup and therefore rec-  ommend a dose closer to the true MTD, as evidenced by the still favorable WPS compared  A  Scenario 1  Scenario 2  Scenario 3  Scenario 4  Scenario 5  100+  100-  100-  100-  100-  80-  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  101  60  60  60  60  60  PCS  PCS  PCS  10  10  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  40  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  40-  20-  20-  20-  20  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  0-  0-  0+  0-  0  45  60  72  30  45  60  72  30  45  60  72  30  45  60  72  45  60  72  30  30  Nmax  Nmax  Nmax  Z²=１  Z2 = 0  Z²=１  Z² = O  Z2 = 0  Z²=１  Z2 = 0  Entire Population  P-CRM  One-sample CRM  B  Scenario 1  Scenario 2  Scenario 3  Scenario 4  Scenario 5  100-  100-  100+  100+  100-  0  口  O  Q  80  80  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  口  80  80  O:  0  口  09  09  60  60  09  WPS  10  WPS  WPS  WPS  WPS  40-  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  40-  40-  40-  20-  20-  20-  20-  0+  0-  0-  0-  0-  30  45  60  72  30  45  60  72  30  45  60  72  30  45  60  72  30  45  60  72  Nmax  Nmax  Nmax  Nmax  Z²=１  Z2 = 0  Z²=１  Z2 = 0  2=１  Z2 = 0  Z²=１  Z2 = 0  Entire Population  P-CRM  One-sample CRM10  FIG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' A) Proportion of Correct Selection of MTD (PCS) and B) Weighted Probability of Selection (WPS) for  each subgroup under the P-CRM and one-sample CRM across Nmax = 30,45,60,72 for criteria prevalence of  25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  to the one-sample CRM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Under the one-sample CRM when the criteria prevalence is 25%,  the subgroup satisfying z2 = 1 is completely ignored and for any heterogeneity where the  subgroups have MTDs more than one dose apart, the PCS is close to 0% with a decreasing  WPS as heterogeneity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Overall, the P-CRM performs well when there is one criterion differentiating the optimal  dose for the patient population when the probability of toxicity at each subgroup’s MTD  differs by at least 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The design identifies the true criterion more often for more heteroge-  neous populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Regarding dose assignment, we find that even when the design does not  correctly identify the true MTD, particularly when the prevalence of the criteria is low, the  design tends to recommend a dose that is close to the true MTD rather than assign the same  dose as the one recommended for the more prevalent subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Finally, the poor performance  of the one-sample CRM highlights how assuming patient homogeneity and ignoring patient  criteria leads to overly toxic or sub-therapeutic dose recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Sample Size Considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The total sample size required for the design depends  on the number of eligibility criteria that are being considered, M, the number of criteria for  which we need to differentiate the MTD, K, the number of dose levels, J, and the prevalence  of the criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' With J = 6, M = 3, a criteria prevalence of 50%, and only one covariate be-  ing associated with toxicity, to achieve an average PCS of at least 50% across scenarios, the  sample size would need to be at least 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' When more than one out of the three covariate is  associated with toxicity, resulting in three or more subgroups with different MTDs, a sample  A  Scenario 1  Scenario 2  Scenario 3  Scenario 4  Scenario 5  100+  100-  100-  100-  100-  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80-  口  0  60口  60  60  09  PCS  PCS  PCS  PCS  40-  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  40  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  40-  20  20-  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  20--  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  0+  0-  0-  0  0  30  45  60  72  30  45  60  72  30  45  60  72  30  45  60  72  30  45  60  72  Nmax  Nmax  Nmax  Z²=１  Z² = O  Z²=１  Z² = O  Z2 = O  Z²=１  Z2 = 0  Entire Population  P-CRM  One-sample CRM  B  Scenario 1  Scenario 2  Scenario 3  Scenario 4  Scenario 5  100-  100-  100+  100+  100-  1-0  LLO  口  80/  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  80  O  口  60  60  60  60/0  09  WPS  WPS  WPS  WPS  WPS  0  40-  40-  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  40  40-  O  20-  20-  20-  20-  20-  0+  0-  0-  0-  0-  30  45  60  72  30  45  60  72  45  60  72  30  45  60  72  30  45  60  72  30  Nmax  Nmax  Nmax  Z²=１  Z2 = 0  Z²=１  Z2 = 0  Z²=１  Z2 = 0  Z²=１  Z2 = 0  Entire Population  P-CRM  One-sample CRMPRECISION DOSE-FINDING DESIGN  11  size around 60 is needed (data not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' When prevalence decreases, the overall PCS in-  creases but the PCS of the less prevalent subgroup decreases, and a larger sample size will be  needed to achieve a subgroup PCS of at least 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Given the same scenarios, when the num-  ber of criteria to consider is only two, M = 2, a lower sample size around 35 would obtain an  average PCS of at least 50% (data not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' If only one criteria is being considered, then  other existing methods should also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' That is, the particular advantage of this  method is that it can consider multiple possible subpopulations at once and thus reduce the  required sample size to find one or more MTDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' While learning about heterogeneity requires  an increased sample size, conducting separate dose-finding trials for each potential subgroup  using the one-sample CRM would require a substantially larger sample size since there are  many possible patient subgroups (2M), some of which could have insufficient sample sizes  due to low prevalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The precision dose-finding design, P-CRM, proposed in this paper ex-  plores dose-finding when the patient population is suspected to be heterogeneous but it is  unknown which patient covariates indeed affect toxicity, and thus must be accounted for in  MTD determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' As eligibility criteria expand in cancer trials, safety profiles for newly  eligible patients could differ given that many were originally included for safety and a method  accounting for eligibility criteria is needed to ensure patient safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' One recommendation for  how to deal with the broadening of eligibility criteria in early-stage trials is to use expansion  cohorts after an initial traditional dose-finding trial [Malik and Lu (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2017)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  However, the number of potential cohorts could be large and the specific cohorts that should  be included are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The second stage of the P-CRM is comparable to an expansion  cohort in that it accounts for potential subpopulations, but the P-CRM is also able to adjust  the dose for those populations accurately through a continued dose-finding procedure and  learn more about subpopulations that are truly heterogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The P-CRM uses marginal logistic regression models to determine whether each covariate  is associated with toxicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Covariates are sequentially added to the dose-toxicity model if  they demonstrate significant association with toxicity and are assessed for removal after pa-  tient data from each cohort is obtained and other selected covariates are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Dose  assignment and MTD recommendation are based on dose and any patient covariates selected  in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' When no patient covariates are added, the design reverts to the original CRM  and assumes patient homogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The design requires no assumption about which covariates  differentiate the patient population and can screen for multiple potential covariates at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Given our potentially high dimensional model with a sparsity constraint, a natural approach  might be the least absolute shrinkage and selection operator (LASSO) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' However, us-  ing the LASSO method in the setting of dose-finding where the sample size is small and the  data is sparse, led to high false positive rates in covariate selection and an overall smaller  probability of correct MTD selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Our simulation study found that the design can correctly identify the covariate associ-  ated with toxicity while maintaining control over the false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Although the design  performs most favorably when large patient heterogeneity exists, we find that across any pa-  tient heterogeneity scenario, the P-CRM is still favorable over the naive approach of ignoring  patient heterogeneity and MTD distinction is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' When the probability of correct co-  variate selection was less favorable due to less heterogeneous subpopulations, the probability  of choosing a desirable dose was still high since the true MTDs in that case are close together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Furthermore, the P-CRM did not as accurately choose the true MTD for the less prevalent  subgroup, but it chose a dose closer to their respective true MTD than the true MTD of the  more prevalent subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Finally, the simulation study showed that ignoring patient charac-  teristics can lead to overly toxic or sub-therapeutic dose recommendations for at least one but  12  possibly all subpopulations within the patient population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Thus, ignoring patient characteris-  tics for a heterogeneous population is far worse than incorporating patient characteristics for  a homogeneous population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Since our design is the first phase I design to consider expanded eligibility criteria and  assume no prior knowledge on heterogeneity, we do not compare it to other methods that  account for known patient heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' For example, the two-sample CRM proposed by  O’Quigley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (1999) or the Hierarchical Bayesian CRM proposed by Morita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2017)  assume that the patient subpopulations are already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The P-CRM must first identify the  subpopulations and then determine the dose for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Although larger sample sizes are needed to accommodate the expansion of many patient  criteria compared to traditional phase I trials, the advantage of accounting for heterogeneity  even with a typical phase I sample size of 30 is already clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Furthermore, with the broad-  ening of eligibility, more patients will be available to recruit into the trial and benefit from a  safe dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Thus, it is important to weigh the desired number of criteria to be evaluated and  the potential number of subgroups expected against the increase in sample size and the preva-  lence of the criteria when applying the method in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Ongoing work involves including  efficacy so that both heterogeneity in toxicity and efficacy can be accounted for to determine  the optimal dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' While precision medicine approaches have been considered mainly for late  stage trials and large sample size settings, our approach presents a precision dose-finding  approach for early stage trials with a limited sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  The proposed design, P-CRM, selects patient characteristics that are associated with tox-  icity and determines a subpopulation-specific MTD in a simple manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' The design only  requires use of logistic regression, so incorporation of precision medicine in this case does  not necessitate specification of multiple priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' A more precise determination of MTD and  identification of the source of patient heterogeneity can be an asset in the long term, leading  to more successful phase II and III trials and being directly applicable to the true patient  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  PRECISION DOSE-FINDING DESIGN  13  APPENDIX  TABLE 2  Probability of toxicity for each dose for Scenarios 1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Scenario  Subpopulation  D1  D2  D3  D4  D5  D6  1  z2 = 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='90  z2 = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='85  2  z2 = 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='85  z2 = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75  3  z2 = 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='85  z2 = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  4  z2 = 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='70  z2 = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  5  Entire Population  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='75  TABLE 3  Probability of Criteria Selection with three total covariates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' M = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='each with prevalence of 50% and 25% for Nmax = 60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Prevalence = 50%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Scenario  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='No criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion with others  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Incorrect criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='57  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='76  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='76  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='56  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Prevalence = 25%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Scenario  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='No criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion with others  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Incorrect criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='29  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='31  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='48  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='73  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='82  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='58  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='42  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Selection probabilities may not equal 100 due to rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  14  TABLE 4  Probability of Criteria Selection with three total covariates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' M = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='each with prevalence of 50% and 25% for Nmax = 72  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Prevalence = 50%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Scenario  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='No criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion with others  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Incorrect criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='56  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='NA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='44  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Prevalence = 25%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Scenario  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='No criteria  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='Correct criterion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content='97  PRECISION DOSE-FINDING DESIGN  17  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  This publication was supported by the National Center for Ad-  vancing Translational Sciences, National Institutes of Health, through Grant Numbers  UL1TR001873 and TL1TR001875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  REFERENCES  BABB, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' AND ROGATKO, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Patient specific dosing in a cancer phase I clinical trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Statistics in  Medicine 20, 2079–2090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  BEAVER, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=', ISON, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=', AND PAZDUR, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Reevaluating Eligibility Criteria — Balancing Patient  Protection and Participation in Oncology Trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' New England Journal of Medicine 16, 1504–1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  CHAPPLE, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' AND THALL, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Subgroup-specific dose finding in phase I clinical trials based on  time to toxicity allowing adaptive subgroup combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Pharmaceutical statistics 17, 734–749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  CHENG, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content=' Dose-escalating and pharmacological study of bortezomib in adult cancer patients with impaired renal  function: a National Cancer Institute Organ Dysfunction Working Group Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content=' Modernizing Clinical Trial Eligibility Criteria: Recommendations of the ASCO-Friends of Cancer  Research Performance Status Work Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Clinical Cancer Research: An Official Journal of the American  Association for Cancer Research 27, 2424–2429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+page_content=' Enhancing the Diversity of Clinical Trial Pop-  ulations  —  Eligibility  Criteria,  Enrollment  Practices,  and  Trial  Designs  Guidance  for  Industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='fda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='gov/regulatory-information/search-fda-guidance-documents/enhancing-diversity-clinical-  trial-populations-eligibility-criteria-enrollment-practices-and-trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  YUAN, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' AND CHAPPELL, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content=' Isotonic designs for phase I cancer clinical trials with multiple risk groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
+page_content='  Clinical Trials (London, England) 1, 499–508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE3T4oBgHgl3EQfiQps/content/2301.04578v1.pdf'}
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+Prepared for submission to JHEP
+Simplified likelihoods using linearized systematic
+uncertainties
+N. Bergera
+aLAPP, Univ. Savoie Mont Blanc, CNRS/IN2P3, Annecy
+E-mail: nicolas.berger@cern.ch
+Abstract: This paper presents a simplified likelihood framework designed to facilitate the
+reuse, reinterpretation and combination of LHC experimental results. The framework is
+based on the same underlying structure as the widely used HistFactory format, but with
+systematic uncertainties considered at linear order only. This simplification leads to large
+gains in computing performance for likelihood evaluation and maximization, compared to
+the original experimental likelihoods. The framework accurately describes non-Gaussian
+effects from low event counts, as well as correlated uncertainties in combinations. While
+primarily targeted towards binned descriptions of the data, it is also applicable to unbinned
+models.
+1Corresponding author.
+arXiv:2301.05676v1  [hep-ex]  13 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+Simplified likelihood formalism
+3
+2.1
+The HistFactory framework
+3
+2.2
+Simplified likelihoods with linearized systematics
+3
+2.3
+Implementation and storage format
+4
+2.4
+Example
+5
+3
+Application to an ATLAS search for new phenomena
+6
+3.1
+Full likelihood
+6
+3.2
+Simplified likelihood
+7
+4
+Simplified likelihoods for unbinned models
+8
+4.1
+Binned description of unbinned models
+8
+4.2
+Full model example
+10
+4.3
+Simplified likelihood
+11
+5
+Discussion
+13
+6
+Conclusion
+15
+A Linearization procedure
+16
+B Binned approximation to an unbinned PDF
+16
+1
+Introduction
+The likelihoods describing experimental measurements are a key component of many LHC
+data analyses. Consisting of the probability distribution function (PDF) of the measure-
+ment together with the observed dataset, they are used to compute the final experimental
+results — e.g. confidence intervals for model parameters, or significance values for possible
+excesses over background — often through the use of frequentist profile-likelihood ratio
+(PLR) methods [1]. They can also be utilized to make further use of the measurement
+information, for instance in combinations with other results, or as reinterpretations in the
+context of alternative signal models.
+Despite this central role, likelihoods are not systematically made available as part of
+experimental publications. This is partly for technical reasons: first, they are often complex,
+with up to O(104) parameters in some cases [2]. A single maximization the likelihood,
+which is needed to compute the PLR, can therefore require up to several hours or days of
+– 1 –
+
+computation time. Another limitation is the fact that the likelihoods of LHC measurements
+are typically implemented within formats and tools not widely used in other fields, such as
+the ROOT framework [3].
+The information provided in publications, such as the best-fit value of the parameters
+of interest (POIs) and the covariance matrix of their measurement, typically allow a partial
+reconstruction of the experimental likelihood. However this is only possible under addi-
+tional assumptions, in particular Gaussian approximations that do not allow an accurate
+description of data taken in the Poisson regime with low expected event counts. In cases
+where full PLR scans are published, the description of systematic uncertainties also does
+not typically allow a full separation of the different sources of uncertainty, so that corre-
+lations across different measurements cannot be properly accounted for when performing
+their combination.
+For these reasons, recent efforts have encouraged the publication of faithful represen-
+tations of the experimental likelihoods under FAIR (Findable, Accessible, Interoperable,
+Reusable) principles [4], in particular with a view towards reinterpretations targeting alter-
+nate signal models [5, 6]. This objective can be realized in particular through the publication
+of full experimental likelihood in open formats. Some recent progress has been achieved in
+this direction, such as the publication of full likelihoods by the ATLAS collaboration using
+the pyhf [7] framework. These cases however remain rare so far, in particular due to the
+limitations described above.
+Simplified likelihoods offer compromise solutions that aim to provide less complex de-
+scriptions of the experimental likelihoods that remain more accurate than Gaussian models.
+Several approaches have been proposed [8–12]. This work describes a simplified likelihood
+format in which the dependence on the POIs of the measurement is treated exactly, but
+the remaining nuisance parameters (NPs) are considered at linear order only. This allows
+the likelihood maximization with respect to the NPs (usually denoted as profiling the like-
+lihood) to be performed in closed form using matrix algebra techniques. This in turn can
+significantly decreases the computing times of the PLR computation since the NPs, which
+are used to describe in particular systematic uncertainties, typically form a large fraction
+of the model parameters. The structure of the simplified model, in terms of the POIs, NPs,
+measurement regions and event samples, remains faithful to the original likelihood. The
+models are stored in plain text, and computations are performed using python-based tools.
+The method is applicable to both binned and unbinned descriptions of the experimental
+data, with unbinned models treated in a binned approximation. This flavor of simplified
+likelihoods is denoted as Simplified Likelihoods with Linearized Systematics (SLLS) in the
+rest of this paper to avoid confusion with other simplified likelihood formats.
+The paper is organized as follows: the SLLS formalism is presented in detail in Section 2;
+Section 3 shows a realistic application to a ATLAS search for supersymmetric particles. An
+application to an unbinned model is presented in Section 4, and Sections 5 and 6 present a
+discussion of these results and conclusions.
+– 2 –
+
+2
+Simplified likelihood formalism
+2.1
+The HistFactory framework
+The simplified likelihoods described in this work are based on the HistFactory frame-
+work [13], which is widely used in LHC experiments and implemented within both ROOT
+and pyhf. It describes measurements with multiple counting bins as a set of channels,
+each corresponding to a measurement region and itself consisting of one or several bins. In
+each bin, a counting experiment is described using a Poisson distribution. Each expected
+event yield is expressed as a sum of contributions from several samples, representing both
+signal(s) and background(s), and each is a function of the POIs and NPs of the model.
+Systematic uncertainties are represented as nuisance parameters that are constrained by
+external information described by a constraint PDF. This constraint is a representation of
+a separate auxiliary experiment, sensitive to the value of the NP through the measurement
+of an auxiliary observable. The full PDF is written as
+P(n; µ, θ) =
+Nchannels
+�
+c=1
+Nbins,c
+�
+b=1
+Pois
+�
+�ncb,
+Nsamples,c
+�
+s=1
+νcbs(µ, θ)
+�
+�
+Nconstraints
+�
+l=1
+Cl(˜θl, θl)
+(2.1)
+where the index c runs over the Nchannels measurement channels, b runs over the Nbins,c
+bins in channel c, and s over the Nsamples,c samples. The observed event yield in bin b of
+channel c, denoted by ncb, is described by the Poisson PDF Pois in terms of the expected
+yields νcbs(µ, θ) for each sample s. The µ and θ refer collectively to the POIs and the NPs,
+respectively, and the index l runs over the Nconstraints constrained nuisance parameters θl
+and their respective auxiliary observables ˜θl. The constraints Cl are in principle arbitrary
+but in practice either Poisson or Gaussian forms are used, depending on the properties of
+the associated systematic uncertainty.
+2.2
+Simplified likelihoods with linearized systematics
+The SLLS formalism introduced in this paper brings two simplifications to the HistFactory
+description. Firstly, the dependence of the νcbs on the NPs is described at linear order only,
+as
+νcbs(µ, θ) = νnom
+cbs (µ)
+�
+1 +
+NNP
+�
+k=1
+∆cbsk(θk − θnom
+k
+)
+�
+.
+(2.2)
+The νnom
+cbs (µ) are the expected event yields computed at the nominal values θnom
+k
+of the
+NPs. The ∆cbsk are linear coefficients specifying the impact of θk on νcbs, for each of the
+NNP nuisance parameters θk. As noted above, the dependence of the νcbs(µ, θ) on the µ is
+described exactly, while the dependence on the θ is described at linear order only. Secondly,
+the constraints Cl are all assumed to be Gaussian, and are collectively represented as a single
+multivariate Gaussian PDF with central value ˜θ and inverse covariance matrix Γ.
+With these assumptions, the profiled value ˆˆθk(µ) = arg maxθk L(µ, θ) of the parameter
+θk at a given value µ of the POIs can be computed in closed form as
+– 3 –
+
+ˆˆθk(µ) = θnom
+k
++
+�
+k′
+�
+(Γ + P(µ))−1�
+kk′
+��
+k′′
+Γk′k′′(˜θk′′ − θnom
+k′′ ) − Qk′(µ)
+�
+.
+(2.3)
+with the vector Q(µ) and the matrix P(µ) given by
+Qk(µ) =
+Nchannels
+�
+c=1
+Nbins,c
+�
+b=1
+(νnom
+cb
+(µ) − ncb)
+Nsamples,c
+�
+s=1
+νnom
+cbs (µ)
+νnom
+cb
+(µ)∆cbsk
+(2.4)
+Pkk′(µ) =
+Nchannels
+�
+c=1
+Nbins,c
+�
+b=1
+ncb
+Nsamples,c
+�
+s,s′=1
+νnom
+cbs (µ)νnom
+cbs′ (µ)
+[νnom
+cb
+(µ)]2
+∆cbsk∆cbs′k′
+(2.5)
+where νnom
+cb
+(µ) = �
+s νnom
+cbs (µ).
+Using these relations, the profiling of the NPs at a given µ can be performed using
+simple matrix algebra. The sizes of the matrices is given by the number of NPs which can
+be fairly large – in some cases up to O(104) – but building these matrices and perform-
+ing multiplication and inversion operations is nevertheless far quicker than the non-linear
+maximization of the full likelihood using e.g. a gradient descent algorithm.
+While the form given in Eq. 2.2 is used to profile the nuisance parameters, the evaluation
+of the likelihood uses instead the alternative form
+νcbs(µ, θ) = νnom
+cbs (µ) exp
+�NNP
+�
+k=1
+∆cbsk(θk − θnom
+k
+)
+�
+,
+(2.6)
+which matches Eq. 2.2 at leading order in the θk but guarantees that νcbs(µ, θ) ≥ 0 for all
+θ as required for the expected event yield of a Poisson PDF.
+2.3
+Implementation and storage format
+A python implementation of the SLLS formalism is provided in the fastprof public pack-
+age1. It includes tools for likelihood evaluation, profiling and maximum-likelihood fits, as
+well as for higher-level computations such for hypothesis testing, limit-setting and confi-
+dence interval estimation. Other tools are provided to validate the simplified models and
+perform other operations such as combining models or pruning parameters and channels.
+The computations make use of the linear algebra routines included in numpy [14] and the
+minimization routines provided by scipy [15].
+The models are stored in a plain-text format using the JSON markup language. The
+format specifies the POIs, NPs, auxiliary observables, and model channels. Each channel
+is described as a list of samples, specified as a set of nominal expected bin yield Nnom
+cbs ,
+the linear impacts ∆cbsk of each NP on the expected yields, and an optional normalization
+factor K(µ) that can be an arbitrary function of the µ. The expected yields are then
+expressed as in Eq. 2.2, with νnom
+cbs (µ) = K(µ)/K(µnom)Nnom
+cbs , where µnom is the value of
+the POIs for which the nominal yields Nnom
+cbs
+are provided.
+The format also includes a specification for observed data, in terms of the observed
+counts for each bin of each channel and the observed values of the auxiliary observables.
+1https://github.com/fastprof-hep/fastprof
+– 4 –
+
+2.4
+Example
+As an illustration, we consider a simple example measurement consisting of a single-bin
+counting experiment in the presence of both signal and background contributions. The
+expected background yield is b0 = 2, with a relative uncertainty ϵ = 10%. The background
+yield is treated as a nuisance parameter in the fit, associated with a Gaussian constraint
+(as would occur in the case where the background is determined from a control region with
+a sufficiently large number of events). The observed yield is n = 3. The JSON specification
+of the model is given in Figure 1.
+1
+{
+2
+"model": {
+3
+"name": " simple_bkg_uncertainty ",
+4
+"POIs": [
+5
+{ "name" : "xs_signal", "unit" : "fb", "min_value": 0, "max_value": 10, " initial_value ": 1 }
+6
+],
+7
+"NPs": [
+8
+{ "name": "np_bkg", " nominal_value ": 0, "constraint": 1, "aux_obs": "aux_bkg" }
+9
+],
+10
+"aux_obs": [
+11
+{ "name": "aux_bkg", "min_value": -5, "max_value": 5 }
+12
+],
+13
+"channels": [
+14
+{
+15
+"name": " measurement_region ",
+16
+"type": "bin",
+17
+"samples": [
+18
+{
+19
+"name": "Signal",
+20
+"norm": "xs_signal",
+21
+" nominal_yields ": [ 1 ]
+22
+},
+23
+{
+24
+"name": "Background",
+25
+" nominal_yields ": [ 2 ],
+26
+"impacts": {
+27
+"np_bkg": { "+1":
+0.10, "-1":
+-0.10 }
+28
+}
+29
+}
+30
+]
+31
+}
+32
+]
+33
+},
+34
+35
+"data": {
+36
+"channels": [
+37
+{ "name" : " measurement_region ", "counts": 3 }
+38
+],
+39
+"aux_obs": [
+40
+{ "name": "aux_bkg", "value": 0 }
+41
+]
+42
+}
+43
+}
+Figure 1: Specification for the example SLLS model described in the text
+The results for the signal yield s are computed using its maximum likelihood estimator
+(MLE) ˆs and the profile-likelihood ratio
+Λ(s) = −2 log L(s,ˆˆb(s); n)
+L(ˆs,ˆb; n)
+(2.7)
+where L(s, b, ; n) is the measurement likelihood, ˆb is the MLE of b and ˆˆb(s) its conditional
+MLE at a fixed value s of the signal yield. The values of Λ(s) can then be used to derive
+– 5 –
+
+−1
+0
+1
+2
+3
+s
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Λ(s)
+Λ(s)
+Λref(s)
+(a)
+−1
+0
+1
+2
+3
+s
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+0.4
+ˆb(s)
+ˆb(s)
+ˆbref(s)
+(b)
+Figure 2: Values of (a) Λ(s) and (b) the conditional MLE pull (ˆˆb(s) − b0)/ϵ for a range of
+values of the signal yield s, computed from the model described in the text. In each plot,
+the simplified likelihood result (solid blue) is compared to an exact closed-form expression
+of the same quantity (dashed red), showing very close agreement.
+results such as confidence intervals on s or a discovery significance for the presence of the
+signal.
+Figure 2 shows the values of Λ(s) and ˆˆb(s) computed from the model given in Figure 1
+for a range of values of s. In this simple case both results can also be computed in closed
+form as
+ˆˆb(s) = 1
+2
+��
+(s + b0 − b2
+0ϵ2)2 + 4b2
+0ϵ2n − (s − b0 + b2
+0ϵ2)
+�
+(2.8)
+Λ(s) = 2(s − ˆs + ˆˆb(s) − ˆb) − 2n log
+�
+s + ˆˆb(s)
+ˆs + ˆb
+�
+,
+(2.9)
+and excellent agreement is observed between these expressions and the numerical results.
+The asymmetric shape of Λ(s) is driven by the Poisson nature of the measurement, and the
+good agreement in this case is due to the fact that this feature is accounted for exactly in
+the simplified likelihood. Using a Gaussian approximation would yield a parabolic shape
+for Λ(s) that would provide a less accurate description. While the systematic uncertainty
+on the background yield plays only a small role in this example, the good agreement in
+the profiled values ˆˆb(s) of the corresponding NP shows that these effects are also described
+accurately within the linear approximation.
+3
+Application to an ATLAS search for new phenomena
+3.1
+Full likelihood
+This section presents a realistic application of the SLLS framework to a search for new
+phenomena by the ATLAS collaboration [16] for which the full experimental likelihood
+– 6 –
+
+has been published [17]. The search targets supersymmetric particles in final states with
+at least three charged leptons originating from the chargino decay ˜χ+
+1 → Zℓ → 3ℓ. The
+analysis considers three signal regions (SRs), targeting signatures with 3 leptons (3ℓ), 4
+leptons (4ℓ) and 4 leptons with a fully reconstructed W, Z or H boson (FR). Each signal
+region is divided into 16 bins of the invariant mass mZℓ of the trilepton system. Three
+single-bin control regions are also included to provide data-driven estimates of the main
+backgrounds, from the Standard Model production of a WZ boson pair, a ZZ pair, or a t¯t
+pair accompanied by a Z boson (t¯tZ).
+The full likelihood of the analysis was published by the ATLAS collaboration as a
+pyhf model available in the HEPData repository [17]. In this example we consider the
+example case of a chargino with a mass of 500 GeV with branching ratios to W, Z and H
+bosons of respectively 20%, 60% and 20%, and equal branching ratios to e, µ and τ for the
+accompanying lepton.
+3.2
+Simplified likelihood
+A SLLS likelihood model is constructed from the pyhf model using as reference the param-
+eter values obtained in a fit to the observed data under the background-only hypothesis.
+The conversion is performed using an automated tool included in the fastprof package.
+The measurement regions of the analysis as implemented in the simplified model are shown
+in Figure 3.
+The profile likelihood scan of the signal strength parameter µsignal using the simplified
+likelihood is shown in Figure 4a. A reference scan computed using the full likelihood is also
+presented for comparison, and shows that the simplified likelihood provides an adequate
+description of the full result. A simple Gaussian model, using the best-fit value µsignal in the
+observed data computed by pyhf and the corresponding parabolic error, is also displayed
+and shows worse agreement. The 95% CLs limit on µsignal computed using the simplified
+model is 0.126, in good agreement with the value of 0.124 obtained using the full model.
+The Gaussian model yields a value of 0.114.
+The fits to the SLLS likelihood with fixed µsignal take about 50 ms on a laptop computer
+equipped with a 16-core Intel i7-10875H CPU. The fit with free µsignal, which relies on non-
+linear rather than linear minimization for this parameter (since POIs are treated exactly),
+takes about 0.5 s. A full-likelihood fit performed with pyhf require approximately 10 min
+on the same computing platform, a factor ≈ 1000 longer. These fits are performed with
+numpy as the numerical backend to pyhf, the same as used the fastprof implementation
+of SLLS likelihoods. Better performance can however likely be achieved using other pyhf
+backends interfacing to the tensorflow [18] or pytorch [19] packages.
+To validate the SLLS linear profiling technique, the profiled values for some of the
+model NPs are shown in Figures 4b and 4c for both the SLLS and the full likelihood
+models. Good agreement is seen between the two cases, illustrating that the full likelihood
+is modeled to good approximation at the level of individual NPs. As a further illustration,
+the exclusion contour presented in Figure 9 of the original publication is recomputed using
+the SLLS simplified likelihoods and compared to the full-likelihood results. The results are
+shown in Figure 5, and good agreement is again observed. An exclusion contour based on
+– 7 –
+
+100
+200
+300
+400
+500
+600
+700
+mZl [GeV]
+10−2
+10−1
+100
+101
+Events / 1 GeV
+SR3l
+Full model
+gaugino 500
+Diboson3l
+Diboson4l
+ttZ
+Other
+Triboson
+Higgs
+Fakes all
+Data
+(a)
+100
+200
+300
+400
+500
+600
+700
+mZl [GeV]
+10−2
+10−1
+100
+101
+Events / 1 GeV
+SR4l
+Full model
+gaugino 500
+Diboson4l
+ttZ
+Other
+Triboson
+Higgs
+Fakes all
+Data
+(b)
+100
+200
+300
+400
+500
+600
+700
+mZl [GeV]
+10−4
+10−3
+10−2
+10−1
+100
+101
+Events / 1 GeV
+SRFR
+Full model
+gaugino 500
+Diboson4l
+ttZ
+Other
+Triboson
+Higgs
+Fakes all
+Data
+(c)
+103
+104
+Events
+CRWZ all cuts
+101
+102
+103
+Events
+CRZZ all cuts
+102
+Events
+CRttZ all cuts
+(d)
+Figure 3: Expected and observed event counts in the SR3l, SR4l and SRFR signal regions
+of the analysis of Ref [16], shown respectively in panels (a), (b) and (c). Panel (d) shows
+the analysis control regions. The signal regions are binned in the mZl observable, while the
+control regions each use a single inclusive event count. The observed data (black points) is
+overlaid with stacked histograms (filled areas) representing the gaugino signal (dark blue)
+and the main background contributions.
+Gaussian models built as described above is also presented for comparison and shows similar
+agreement in this case, in part due to the fact that the signal production cross-sections vary
+rapidly with the chargino mass.
+4
+Simplified likelihoods for unbinned models
+4.1
+Binned description of unbinned models
+The previous examples used a binned description of the experimental measurement, which
+employs only two types of PDFs: Poisson distributions to describe the counting experiments
+in each bin, and Gaussian distributions for the constraints. Another common modeling
+option is unbinned likelihoods, in which the model describes the continuous probability
+distribution of the measurement observables. It is used for instance to study the H → γγ
+decay of the Higgs boson at the LHC [20, 21], as well as in many results published by
+LHCb. It requires support for arbitrary PDF forms, as needed to describe each measure-
+ment, and therefore more general and flexible tools than for binned likelihoods. For LHC
+measurements, this functionality is usually provided by the roofit package [22] distributed
+– 8 –
+
+0.00
+0.05
+0.10
+0.15
+0.20
+µsig
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Λ(µsig)
+fastprof
+pyhf
+Gaussian
+(a)
+0.00
+0.05
+0.10
+0.15
+0.20
+µsig
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+ˆˆθ(µsig)/σθ
+ϵelec
+ϵmuon
+ϵb-tag
+MET scale
+(b)
+0.00
+0.05
+0.10
+0.15
+0.20
+µsig
+−0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+( ˆˆθ(µsig) − 1)/σθ
+µ3l
+µ4l
+µttZ
+(c)
+Figure 4: Comparison between the SLLS simplified model (solid lines) and the full model
+(dotted lines) for (a) the PLR Λ(µsignal) as a function of µsignal, (b) the profiled values
+of selected NPs describing systematic uncertainties and (c) the profiled values of NPs de-
+scribing scale factors applied to the normalization of the main analysis backgrounds. The
+profiled values are shown as deviations from the nominal value of the parameters (0 for
+systematic uncertainties, 1 for background scaling factors), divided by the uncertainty on
+the parameter in the full-model fit to the observed data with free µsignal. The SLLS results
+are computed using the fastprof tool, and the full-model results with the pyhf tool. Panel
+(a) also shows the PLR scan computed using a Gaussian model as described in the text.
+Good agreement is observed overall between the SLLS and full-model results. The largest
+deviation is seen in the scale factor for the t¯tZ background, corresponding to about 30% of
+the fit uncertainty.
+as part of ROOT, but this and other similar tools are not widely used outside the high-energy
+– 9 –
+
+200
+400
+600
+800
+1000
+Chargino mass
+0
+20
+40
+60
+80
+100
+Branching fraction to Z
+pyhf
+fastprof
+Gaussian
+Figure 5: Exclusion plot in the plane of chargino mass and its branching ratio to Z bosons,
+assuming equal branching ratios to W and H and to all lepton flavors. The computation
+from SLLS simplified likelihoods (solid blue) is compared with a reference (dashed red)
+taken from the top-left panel of Figure 9 in Ref. [16] and good agreement is observed.
+Gaussian models computed from the full likelihood as described in the text (dotted black)
+also show good agreement in this case.
+physics experimental community. While there are some recent new efforts to provide more
+portable alternatives, none is currently in wide use.
+A possible way forward is based on the observation that unbinned models can be
+approximated by binned models with a sufficiently fine binning (see Appendix B). While
+this approach typically runs into practical difficulties for full likelihoods due to the large
+number of bins required, it is feasible for simplified likelihoods which are quick to evaluate
+even for relatively large bin numbers. In the rest of this section, we present the application
+of the SLLS likelihood framework to an unbinned model loosely inspired by an ATLAS
+H → γγ measurement.
+4.2
+Full model example
+We consider a simple example based on the ATLAS H → γγ analysis of Ref. [20]. The
+analysis uses an unbinned model based on the distribution the invariant mass mγγ of the
+two photons in the range 105 < mγγ < 160 GeV. The Higgs boson signal manifests itself
+as a sharp peak in the mγγ distribution, with a position close to the Higgs boson mass
+and a width of 1.1–2.1 GeV depending on event kinematics. The background contributions
+follow smoothly falling shapes. Several signal regions (referred to as categories in the rest
+of this section) are defined according to the properties of the signal photons and other
+reconstructed objects.
+– 10 –
+
+The example uses a simplified description of the 33 categories defined in Ref. [20] to
+study Higgs boson production in the gluon-fusion process. The signal and background dis-
+tributions are represented respectively by Gaussian and exponential distributions, instead
+of the more complex shapes used in Ref. [20]. The peak position and width of the Gaus-
+sian, as well as the expected signal and background yields are taken from Ref. [20], while
+the exponential slope of the background is assumed to be −0.02 GeV−1 in all categories.
+The background normalizations and exponential slopes are free to vary in the fit, except
+for the slopes in five low-statistics categories which are kept fixed to avoid unstable fits.
+Five nuisance parameters are used to describe the leading systematic uncertainties: the
+uncertainty on the integrated luminosity of the dataset; on the reference cross-section for
+the gluon-fusion production process; on the effect of parton shower modeling on the signal
+yields; on the H → γγ reconstruction efficiency; and on the photon energy resolution. This
+last uncertainty leads to a change in the width of the signal peak and therefore induces
+highly non-linear effects in the per-bin signal yields in a binned description of the likeli-
+hood. Systematic uncertainties on the background model are implemented using separate
+nuisance parameters in each category, following the "spurious signal" method described in
+Ref. [20]. The values of the uncertainties listed above are all taken from Ref. [20]. In total,
+99 NPs are defined. The single POI is the Higgs boson signal strength µ, applied as a
+scaling factor to the expected signal yield in all categories. A dataset of events randomly
+generated from the model PDF is used as the "observed" data in this example. The model
+and data are stored in a roofit workspace.
+4.3
+Simplified likelihood
+The SLLS likelihood is built as a binned approximation to the full likelihood. A fine binning
+is required to obtain an accurate description of the signal peak. In this example a uniform
+bin width of 0.1 GeV is used, leading to 18150 bins in total for the 33 categories2. The mγγ
+distributions for two selected categories (the first and last in the order used in Ref. [20])
+are shown in Figure 6.
+The conversion is performed using an automated tool distributed as part of the fastprof
+package. All the nuisance parameters are retained, and their effect is described in terms of
+their linear impact on the event yield in each measurement bin, following the SLLS pro-
+cedure. In some cases, in particular normalization parameters such as the one shown on
+Figure 7a, impacts are linear by construction. By contrast, the photon energy resolution
+systematic shown in Figure 7b exhibits non-linear behavior since it induces a change in
+the width of the signal peak which does not propagate linearly to the bin contents. Non-
+linearities become larger as moves towards the tail of the signal peak, but with a smaller
+impact on the results due to lower signal yields.
+The linear approximation remains in
+any case typically accurate for small deviations of the NP from the nominal.
+Figure 8a
+shows the profile likelihood scan for the signal strength parameter µ obtained with the lin-
+earized model. The reference result obtained with the full unbinned likelihood, computed
+2A variable-width binning with wider bins away from the peak can also be considered, but a uniform
+binning was chosen in this example for simplicity.
+– 11 –
+
+110
+120
+130
+140
+150
+160
+mγγ [GeV]
+0
+20
+40
+60
+80
+100
+Events / 0.1 GeV
+000
+Full model
+Sig 000
+Bkg 000
+Data
+110
+120
+130
+140
+150
+160
+mγγ [GeV]
+10−3
+10−2
+10−1
+100
+Events / 0.1 GeV
+032
+Full model
+Sig 032
+Bkg 032
+Data
+Figure 6:
+Distributions of the observable mγγ for two selected categories, labeled 0-jet,
+pH
+T < 10 GeV (top) and pH
+T ≥ 650 GeV (bottom) in Ref. [20]. The signal and background
+contributions (blue and green histograms respectively) are shown together with the example
+dataset (black points), for the best-fit value of the model parameters.
+using the RooFitUtils package3, is also shown for comparison and excellent agreement
+is observed. The resulting 68% CL likelihood intervals are µ = 1.082+0.117
+−0.093 for the full
+model and µ = 1.082+0.113
+−0.093 for the simplified model. A fully Gaussian approximation, as
+described in the previous section, yields µ = 1.082 ± 0.098. The fits take about 15 min to
+perform on the full model, compared to about 50 ms and 1 s for simplified likelihood fits
+with respectively a free and floating µ.
+To better compare the treatment of systematic effects, the profiled values of the five
+nuisance parameters describing the leading systematic uncertainties are shown in Figure 8b.
+The agreement between the simplified and the full model is found to be accurate to about
+10% of the parameter uncertainties. This agreement is crucial to the description of this
+example since the uncertainty on µ is dominated by systematic effects. In particular, the
+profiling of the photon energy resolution systematic (which represents about 20% of the
+total uncertainty) shows good agreement with the full model in spite of the non-linear
+effects highlighted in Figure 7b. Figure 8c shows the difference between the profiled values
+of the other NPs in the simplified and full models, normalized to their fit uncertainty. This
+difference is below 10% of the fit uncertainty for about 80% of the parameters.
+3https://gitlab.cern.ch/cburgard/RooFitUtils
+– 12 –
+
+−4
+−2
+0
+2
+4
+Normalized parameter value
+−0.03
+−0.02
+−0.01
+0.00
+0.01
+0.02
+0.03
+Relative impact on bin yield
+nBkg 000
+Full model impact
+Linear impact
+Non-linear impact
+(a)
+−4
+−2
+0
+2
+4
+Normalized parameter value
+−0.2
+−0.1
+0.0
+0.1
+0.2
+Relative impact on bin yield
+npPER
+Full model impact
+Linear impact
+Non-linear impact
+(b)
+Figure 7: Relative change in expected bin yields as a function of the normalized parameter
+value for two cases: (a) the impact of the background normalization parameter nBkg_000
+on the expected background yield; and (b) the impact of the photon energy resolution
+parameter npPER on the expected signal yield. In both cases, the bin is located at mγγ ≈
+127 GeV in the first category of the model, which lies about 0.7σ above the signal peak.
+The impacts computed from the full model (dots) are compared with the linear impacts
+computed from Eq. 2.2 (dotted red line) and the non-linear impacts from Eq. 2.6 (solid
+blue line).
+5
+Discussion
+As observed in the examples shown in this paper, linearized NP impacts provide a generally
+adequate approximation of their behavior in the full model, in particular in the description
+of systematic effects.
+It can be noted that the approximation coincides with the exact
+description in the case where the impact of the NP is naturally linear, such as the case
+shown in Figure 7a. Discrepancies are expected in cases which deviate from this ideal case,
+in particular for:
+• Large non-linear systematic uncertainties, with effects that are not fully accounted
+for in the linear approximation, such as the one shown in Figure 7b.
+• Asymmetric systematic uncertainties, with different impacts for NPs above or be-
+low their nominal value. These effects cannot be included in the profiling described
+in Section 2.2, although they can be taken into account for the evaluation of the
+likelihood.
+• Low expected event yields, leading to Poisson counts that are not well-described by
+Gaussian distributions. While the Poisson distribution itself is described exactly in
+the SLLS formalism, non-linearities can occur for systematics on the expected event
+yield, since the Poisson PDF does not depend linearly on its expected yield.
+These situations are partially covered in the examples described in this paper, and it is
+encouraging that in these cases at least, the linear description seems adequate. However
+– 13 –
+
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+µ
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Λ(µ)
+fastprof
+pyhf
+(a)
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5
+µ
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+1.5
+ˆˆθ(µ)/σθ
+θLumi
+θggF
+θϵ
+θUPS
+θPER
+(b)
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+µ
+−1.00
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+∆ ˆˆθ(µ)/σθ
+(c)
+Figure 8: Comparison between the SLLS model (solid lines) and the full likelihood (dotted
+lines) for (a) the profile likelihood Λ(µ) as a function of the parameter of interest µ; (b)
+the profiled values of a selection of nuisance parameters describing systematic uncertainties;
+and (c) the difference between the profiled values of each NP obtained from the SLLS model
+and the full likelihood, divided by the uncertainty on the parameter in the full-likelihood
+fit to the data with free µ. Good agreement is observed overall between the linear and
+full-model results.
+simplified likelihoods should be carefully validated against the full likelihood in each case
+nevertheless. Tools to perform these checks are included in the fastprof package, using
+methods similar to those shown in this paper.
+Another limitation to take into consideration is the memory footprint of the ∆cbsk coef-
+ficients which encode the linear impacts: their number is given by NNPs × Nbins × Nsamples,
+which can reach O(108) or more for complex models. For models with a large numbers
+– 14 –
+
+of bins and nuisance parameters, such as converted unbinned models, memory constraints
+can be more stringent than those related to computation times, since these computations
+mainly involves matrix operations that are quite efficient even for large models.
+6
+Conclusion
+Simplified likelihoods provide a convenient setting for the use and reuse of experimental
+results, and functionality that is complementary to that of full likelihoods and Gaussian
+approximations.
+The SLLS framework based on a linear description of nuisance parameter impacts pro-
+vides an approximation with several benefits: it describes the Poisson behavior of counting
+measurements exactly, and it also preserves the nuisance parameters of the full model and
+therefore a fully granular description of its sources of systematic uncertainties. Both of these
+aspects make it well-suited to the description of LHC measurements, where systematic un-
+certainties and non-Gaussian effects from low event counts (e.g. in tails of distributions)
+both play important roles. The preservation of the nuisance parameter structure allows
+in particular a proper treatment of correlated systematic effects when performing com-
+binations of measurements, by identifying parameters associated with identical sources of
+uncertainty in the combination inputs. Since the parameters of interest of the original model
+are also preserved, reinterpretations of the simplified likelihood can also be performed by
+re-expressing them in terms of the parameters of an alternative models, as can be done for
+full likelihoods.
+SLLS models can be built automatically from binned likelihood implemented using the
+HistFactory formalism within the ROOT and pyhf framework, or from unbinned likelihood
+using binned approximations. An implementation of the SLLS framework is provided in
+the fastprof package at https://github.com/fastprof-hep/fastprof. The models are
+stored in a plain-text JSON format, and computations and other operations are performed
+using python tools based on the widely available numpy and scipy libraries.
+Together with other full and simplified likelihood formats with complementary func-
+tionality, it is hoped that this framework will encourage the further publication of detailed
+experimental likelihoods by LHC experiments and beyond.
+Acknowledgments
+The author would like to thank Nick Wardle providing the code for the simplified likelihoods
+of Ref. [9], and Tetiana Hryn’ova for valuable feedback. Plots in this paper were produced
+with matplotlib using the SciencePlots style package [23]. This research was funded, in
+whole or in part, by l’Agence Nationale de la Recherche (ANR), project ANR-22-CE31-
+0022.
+– 15 –
+
+A
+Linearization procedure
+This section provides a sketch of the derivation of the profile value ˆˆθk(µ) of the NP θk given
+by Eq 2.3. Starting from the likelihood in Eq. 2.1 and applying the linearization procedure
+described in Section 2.2, we obtain the negative log-likelihood
+λ(µ, θ) =
+Nchannels
+�
+c=1
+Nbins,c
+�
+b=1
+[νcb(µ, θ) − ncb log νcb(µ, θ)] + 1
+2(θ − ˜θ)Γ(θ − ˜θ)
+(A.1)
+up to a an additive constant. In the expression above, indices c, b and s run respectively
+over measurement channels, bins within each channel, and event samples. The νcb(µ, θ) =
+�
+s νcbs(µ, θ) are the total expected events yields for the corresponding channel bin, and the
+per-sample yields νcbs(µ, θ) are given by Eq 2.2. The Gaussian constraints on the nuisance
+parameters are parameterized using the auxiliary observables ˜θ and the inverse covariance
+matrix Γ.
+The derivative of λ(µ, θ) with respect to the nuisance parameters θ is
+∂λ
+∂θ(µ, θ) =
+Nchannels
+�
+c=1
+Nbins,c
+�
+b=1
+��
+s
+νnom
+cbs (µ)∆cbs
+�
+1 −
+ncb
+νcb(µ, θ)
+��
++ Γ(θ − ˜θ).
+(A.2)
+where ∆cbs is the vector with components ∆cbsk, the linear impacts of the parameter θk
+on νcbs. The linear approximation of nuisance parameters impacts can be applied to the
+denominator as
+ncb
+νcb(µ, θ) ≈
+ncb
+νnom
+cb
+(µ)
+�
+1 −
+�
+s
+νnom
+cbs (µ)
+νnom
+cb
+(µ)∆cbs(θ − θnom)
+�
+.
+(A.3)
+and one finally obtains
+∂λ
+∂θ(µ, θ) = Q(µ) + P(µ) [θ − θnom] + Γ
+�
+θ − ˜θ
+�
+(A.4)
+with Q(µ) and P(µ) defined by Eq. 2.5, and the profile values ˆˆθ(µ) defined by ∂λ/∂θ(µ, ˆˆθ(µ)) =
+0 are therefore given by Eq. 2.3.
+B
+Binned approximation to an unbinned PDF
+We consider an extended unbinned PDF for an observable x,
+P(x; θ)
+n
+�
+i=1
+dxi = e−N(θ)
+n!
+N(θ)n
+n
+�
+i=1
+f(xi; θ)dxi
+(B.1)
+where f(x, θ) is the PDF for one observation of x, the dataset consists of the values x1 · · · xn,
+θ are the model parameters, and the expected number of observations is N(θ). We include
+the infinitesimal volume elements dxi in the expression, as these will be useful below.
+– 16 –
+
+We introduce a set of bins Ba, a = 1 · · · Nbins that span the allowed range of x. In the
+spirit of finite-element analysis, we approximate f(x) by a form that is constant over each
+bin as
+f(x, θ) =
+�
+i
+fa(θ)Ia(x)
+(B.2a)
+fa(θ) = 1
+wa
+�
+Ba
+f(x, θ)dx
+(B.2b)
+where the indicator Ia(x) is 1 if x ∈ Ba and 0 otherwise, and wa =
+�
+Ba Ia(x)dx is the
+measure of bin Ba.
+The value fa(θ) is the average of f(x, θ) over the bin Ba, so that for a sufficiently fine
+binning and smooth f(x, θ), f(x, θ) ≈ fa(θ) for x ∈ Ba.
+One can remove the explicit dependence on the xi by integrating them out of the
+likelihood. The integration of the product term of Eq. B.1 can be written as
+�
+n
+�
+i=1
+f(xi, θ)dxi =
+n
+�
+i=1
+�
+i
+fa(θ)
+�
+Ia(xi)dxi =
+n
+�
+i=1
+fai(θ)wai =
+Nbins
+�
+a=1
+[fa(θ)wa]na
+(B.2c)
+where ai is the index of the bin to which xi belongs, na is the number of observations that
+fall in bin Ba, and we have used the fact that the xi are independent to propagate the
+integral through the product. Returning to the full expression of Eq. B.1, we can write the
+likelihood as a function of the n as
+P(n; θ) =
+e−N(θ)
+n1! · · · nNbins!N(θ)n
+Nbins
+�
+a=1
+[wafa(θ)]na ,
+(B.2d)
+after including an additional multiplicative factor (n, n1, n2, · · · nNbins) to account for the
+number of different orderings of the xi that can yield a given set of na. One can introduce
+the per-bin expected yields
+Na(θ) = wafa(θ)N(θ)
+(B.2e)
+and note that since
+1 =
+�
+f(x, θ)dx =
+Nbins
+�
+a=1
+�
+Ba
+f(x, θ)dx =
+Nbins
+�
+a=1
+wafa(θ)
+one has N(θ) = �
+a Na(θ) as expected. One can finally rewrite
+P(n; θ) =
+Nbins
+�
+a=1
+e−Na(θ)
+na!
+N(θ)na(wafa(θ))na =
+Nbins
+�
+a=1
+e−Na(θ)
+na!
+Na(θ)na.
+(B.2f)
+This takes the usual form of a binned likelihood, with a Poisson distribution in each mea-
+surement bin with expected yields given by Eq. B.2e.
+– 17 –
+
+References
+[1] G. Cowan, K. Cranmer, E. Gross, and O. Vitells, Asymptotic formulae for likelihood-based
+tests of new physics, The European Physical Journal C 71 (feb, 2011).
+[2] CMS Collaboration, A portrait of the Higgs boson by the CMS experiment ten years after the
+discovery, Nature 607 (2022), no. 7917 60–68, [arXiv:2207.00043].
+[3] R. Brun and F. Rademakers, Root - an object oriented data analysis framework, in
+AIHENP’96 Workshop, Lausane, vol. 389, pp. 81–86, 1996.
+[4] M. D. Wilkinson et al., The fair guiding principles for scientific data management and
+stewardship, Scientific data 3 (2016).
+[5] W. Abdallah et al., Reinterpretation of LHC results for new physics: Status and
+recommendations after run 2, SciPost Physics 9 (aug, 2020).
+[6] K. Cranmer et al., Publishing statistical models: Getting the most out of particle physics
+experiments, SciPost Physics 12 (jan, 2022).
+[7] L. Heinrich, M. Feickert, G. Stark, and K. Cranmer, pyhf: pure-python implementation of
+histfactory statistical models, Journal of Open Source Software 6 (2021), no. 58 2823.
+[8] A. Buckley, M. Citron, S. Fichet, S. Kraml, W. Waltenberger, and N. Wardle, The Simplified
+Likelihood Framework, JHEP 04 (2019) 064, [arXiv:1809.05548].
+[9] CMS Collaboration, Simplified likelihood for the re-interpretation of public CMS results, tech.
+rep., CERN, Geneva, 2017.
+[10] A. Coccaro, M. Pierini, L. Silvestrini, and R. Torre, The DNNLikelihood: enhancing likelihood
+distribution with Deep Learning, Eur. Phys. J. C 80 (2020), no. 7 664, [arXiv:1911.03305].
+[11] S. Fichet, Taming systematic uncertainties at the LHC with the central limit theorem, Nucl.
+Phys. B 911 (2016) 623–637, [arXiv:1603.03061].
+[12] K. Cranmer, S. Kreiss, D. Lopez-Val, and T. Plehn, Decoupling Theoretical Uncertainties
+from Measurements of the Higgs Boson, Phys. Rev. D 91 (2015), no. 5 054032,
+[arXiv:1401.0080].
+[13] K. Cranmer, G. Lewis, L. Moneta, A. Shibata, and W. Verkerke, HistFactory: A tool for
+creating statistical models for use with RooFit and RooStats, tech. rep., New York U., New
+York, 2012. https://cds.cern.ch/record/1456844.
+[14] C. R. Harris et al., Array programming with NumPy, Nature 585 (Sept., 2020) 357–362.
+[15] P. Virtanen et al., SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,
+Nature Methods 17 (2020) 261–272.
+[16] ATLAS Collaboration, Search for trilepton resonances from chargino and neutralino pair
+production in √s = 13 TeV pp collisions with the ATLAS detector, Phys. Rev. D 103 (2021),
+no. 11 112003, [arXiv:2011.10543].
+[17] ATLAS Collaboration, Full likelihood of Search for trilepton resonances from chargino and
+neutralino pair production in √s = 13 TeV pp collisions with the ATLAS detector (Version
+2), 2021. https://doi.org/10.17182/hepdata.99806.v2/r2.
+[18] M. Abadi et al., TensorFlow: Large-scale machine learning on heterogeneous systems, 2015.
+Software available from tensorflow.org.
+– 18 –
+
+[19] A. Paszke et al., Pytorch: An imperative style, high-performance deep learning library, in
+Advances in Neural Information Processing Systems 32, pp. 8024–8035. Curran Associates,
+Inc., 2019.
+[20] ATLAS Collaboration, Measurement of the properties of higgs boson production at √s = 13
+tev in the h → γγ channel using 139 fb−1 of pp collision data with the atlas experiment, 2022.
+[21] CMS Colllaboration, Measurements of Higgs boson properties in the diphoton decay channel
+in proton-proton collisions at √s = 13 TeV, JHEP 11 (2018) 185, [arXiv:1804.02716].
+[22] W. Verkerke and D. Kirkby, The roofit toolkit for data modeling, physics/0306116.
+[23] J. D. Garrett, garrettj403/SciencePlots, . http://doi.org/10.5281/zenodo.4106649.
+– 19 –
+
diff --git a/sdE5T4oBgHgl3EQfmA-1/content/tmp_files/load_file.txt b/sdE5T4oBgHgl3EQfmA-1/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..da80f3cd85dbf626725366674b99625092b0d730
--- /dev/null
+++ b/sdE5T4oBgHgl3EQfmA-1/content/tmp_files/load_file.txt
@@ -0,0 +1,685 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf,len=684
+page_content='Prepared for submission to JHEP  Simplified likelihoods using linearized systematic  uncertainties  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Bergera  aLAPP, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Savoie Mont Blanc, CNRS/IN2P3, Annecy  E-mail: nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='berger@cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='ch  Abstract: This paper presents a simplified likelihood framework designed to facilitate the  reuse, reinterpretation and combination of LHC experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The framework is  based on the same underlying structure as the widely used HistFactory format, but with  systematic uncertainties considered at linear order only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This simplification leads to large  gains in computing performance for likelihood evaluation and maximization, compared to  the original experimental likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The framework accurately describes non-Gaussian  effects from low event counts, as well as correlated uncertainties in combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' While  primarily targeted towards binned descriptions of the data, it is also applicable to unbinned  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  1Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='05676v1  [hep-ex]  13 Jan 2023  Contents  1  Introduction  1  2  Simplified likelihood formalism  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  The HistFactory framework  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  Simplified likelihoods with linearized systematics  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  Implementation and storage format  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='4  Example  5  3  Application to an ATLAS search for new phenomena  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  Full likelihood  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  Simplified likelihood  7  4  Simplified likelihoods for unbinned models  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  Binned description of unbinned models  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  Full model example  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  Simplified likelihood  11  5  Discussion  13  6  Conclusion  15  A Linearization procedure  16  B Binned approximation to an unbinned PDF  16  1  Introduction  The likelihoods describing experimental measurements are a key component of many LHC  data analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Consisting of the probability distribution function (PDF) of the measure-  ment together with the observed dataset, they are used to compute the final experimental  results — e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' confidence intervals for model parameters, or significance values for possible  excesses over background — often through the use of frequentist profile-likelihood ratio  (PLR) methods [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' They can also be utilized to make further use of the measurement  information, for instance in combinations with other results, or as reinterpretations in the  context of alternative signal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Despite this central role, likelihoods are not systematically made available as part of  experimental publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This is partly for technical reasons: first, they are often complex,  with up to O(104) parameters in some cases [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' A single maximization the likelihood,  which is needed to compute the PLR, can therefore require up to several hours or days of  – 1 –  computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Another limitation is the fact that the likelihoods of LHC measurements  are typically implemented within formats and tools not widely used in other fields, such as  the ROOT framework [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The information provided in publications, such as the best-fit value of the parameters  of interest (POIs) and the covariance matrix of their measurement, typically allow a partial  reconstruction of the experimental likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' However this is only possible under addi-  tional assumptions, in particular Gaussian approximations that do not allow an accurate  description of data taken in the Poisson regime with low expected event counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In cases  where full PLR scans are published, the description of systematic uncertainties also does  not typically allow a full separation of the different sources of uncertainty, so that corre-  lations across different measurements cannot be properly accounted for when performing  their combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  For these reasons, recent efforts have encouraged the publication of faithful represen-  tations of the experimental likelihoods under FAIR (Findable, Accessible, Interoperable,  Reusable) principles [4], in particular with a view towards reinterpretations targeting alter-  nate signal models [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This objective can be realized in particular through the publication  of full experimental likelihood in open formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Some recent progress has been achieved in  this direction, such as the publication of full likelihoods by the ATLAS collaboration using  the pyhf [7] framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' These cases however remain rare so far, in particular due to the  limitations described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Simplified likelihoods offer compromise solutions that aim to provide less complex de-  scriptions of the experimental likelihoods that remain more accurate than Gaussian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Several approaches have been proposed [8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This work describes a simplified likelihood  format in which the dependence on the POIs of the measurement is treated exactly, but  the remaining nuisance parameters (NPs) are considered at linear order only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This allows  the likelihood maximization with respect to the NPs (usually denoted as profiling the like-  lihood) to be performed in closed form using matrix algebra techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This in turn can  significantly decreases the computing times of the PLR computation since the NPs, which  are used to describe in particular systematic uncertainties, typically form a large fraction  of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The structure of the simplified model, in terms of the POIs, NPs,  measurement regions and event samples, remains faithful to the original likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The  models are stored in plain text, and computations are performed using python-based tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The method is applicable to both binned and unbinned descriptions of the experimental  data, with unbinned models treated in a binned approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This flavor of simplified  likelihoods is denoted as Simplified Likelihoods with Linearized Systematics (SLLS) in the  rest of this paper to avoid confusion with other simplified likelihood formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The paper is organized as follows: the SLLS formalism is presented in detail in Section 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Section 3 shows a realistic application to a ATLAS search for supersymmetric particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' An  application to an unbinned model is presented in Section 4, and Sections 5 and 6 present a  discussion of these results and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  – 2 –  2  Simplified likelihood formalism  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  The HistFactory framework  The simplified likelihoods described in this work are based on the HistFactory frame-  work [13], which is widely used in LHC experiments and implemented within both ROOT  and pyhf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' It describes measurements with multiple counting bins as a set of channels,  each corresponding to a measurement region and itself consisting of one or several bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In  each bin, a counting experiment is described using a Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Each expected  event yield is expressed as a sum of contributions from several samples, representing both  signal(s) and background(s), and each is a function of the POIs and NPs of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Systematic uncertainties are represented as nuisance parameters that are constrained by  external information described by a constraint PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This constraint is a representation of  a separate auxiliary experiment, sensitive to the value of the NP through the measurement  of an auxiliary observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The full PDF is written as  P(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' µ, θ) =  Nchannels  �  c=1  Nbins,c  �  b=1  Pois  �  �ncb,  Nsamples,c  �  s=1  νcbs(µ, θ)  �  �  Nconstraints  �  l=1  Cl(˜θl, θl)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1)  where the index c runs over the Nchannels measurement channels, b runs over the Nbins,c  bins in channel c, and s over the Nsamples,c samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The observed event yield in bin b of  channel c, denoted by ncb, is described by the Poisson PDF Pois in terms of the expected  yields νcbs(µ, θ) for each sample s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The µ and θ refer collectively to the POIs and the NPs,  respectively, and the index l runs over the Nconstraints constrained nuisance parameters θl  and their respective auxiliary observables ˜θl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The constraints Cl are in principle arbitrary  but in practice either Poisson or Gaussian forms are used, depending on the properties of  the associated systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  Simplified likelihoods with linearized systematics  The SLLS formalism introduced in this paper brings two simplifications to the HistFactory  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Firstly, the dependence of the νcbs on the NPs is described at linear order only,  as  νcbs(µ, θ) = νnom  cbs (µ)  �  1 +  NNP  �  k=1  ∆cbsk(θk − θnom  k  )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2)  The νnom  cbs (µ) are the expected event yields computed at the nominal values θnom  k  of the  NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The ∆cbsk are linear coefficients specifying the impact of θk on νcbs, for each of the  NNP nuisance parameters θk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' As noted above, the dependence of the νcbs(µ, θ) on the µ is  described exactly, while the dependence on the θ is described at linear order only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Secondly,  the constraints Cl are all assumed to be Gaussian, and are collectively represented as a single  multivariate Gaussian PDF with central value ˜θ and inverse covariance matrix Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  With these assumptions, the profiled value ˆˆθk(µ) = arg maxθk L(µ, θ) of the parameter  θk at a given value µ of the POIs can be computed in closed form as  – 3 –  ˆˆθk(µ) = θnom  k  +  �  k′  �  (Γ + P(µ))−1�  kk′  ��  k′′  Γk′k′′(˜θk′′ − θnom  k′′ ) − Qk′(µ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3)  with the vector Q(µ) and the matrix P(µ) given by  Qk(µ) =  Nchannels  �  c=1  Nbins,c  �  b=1  (νnom  cb  (µ) − ncb)  Nsamples,c  �  s=1  νnom  cbs (µ)  νnom  cb  (µ)∆cbsk  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='4)  Pkk′(µ) =  Nchannels  �  c=1  Nbins,c  �  b=1  ncb  Nsamples,c  �  s,s′=1  νnom  cbs (µ)νnom  cbs′ (µ)  [νnom  cb  (µ)]2  ∆cbsk∆cbs′k′  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5)  where νnom  cb  (µ) = �  s νnom  cbs (µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Using these relations, the profiling of the NPs at a given µ can be performed using  simple matrix algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The sizes of the matrices is given by the number of NPs which can  be fairly large – in some cases up to O(104) – but building these matrices and perform-  ing multiplication and inversion operations is nevertheless far quicker than the non-linear  maximization of the full likelihood using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' a gradient descent algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  While the form given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2 is used to profile the nuisance parameters, the evaluation  of the likelihood uses instead the alternative form  νcbs(µ, θ) = νnom  cbs (µ) exp  �NNP  �  k=1  ∆cbsk(θk − θnom  k  )  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='6)  which matches Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2 at leading order in the θk but guarantees that νcbs(µ, θ) ≥ 0 for all  θ as required for the expected event yield of a Poisson PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  Implementation and storage format  A python implementation of the SLLS formalism is provided in the fastprof public pack-  age1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' It includes tools for likelihood evaluation, profiling and maximum-likelihood fits, as  well as for higher-level computations such for hypothesis testing, limit-setting and confi-  dence interval estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Other tools are provided to validate the simplified models and  perform other operations such as combining models or pruning parameters and channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The computations make use of the linear algebra routines included in numpy [14] and the  minimization routines provided by scipy [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The models are stored in a plain-text format using the JSON markup language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The  format specifies the POIs, NPs, auxiliary observables, and model channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Each channel  is described as a list of samples, specified as a set of nominal expected bin yield Nnom  cbs ,  the linear impacts ∆cbsk of each NP on the expected yields, and an optional normalization  factor K(µ) that can be an arbitrary function of the µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The expected yields are then  expressed as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2, with νnom  cbs (µ) = K(µ)/K(µnom)Nnom  cbs , where µnom is the value of  the POIs for which the nominal yields Nnom  cbs  are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The format also includes a specification for observed data, in terms of the observed  counts for each bin of each channel and the observed values of the auxiliary observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='com/fastprof-hep/fastprof  – 4 –  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='4  Example  As an illustration, we consider a simple example measurement consisting of a single-bin  counting experiment in the presence of both signal and background contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The  expected background yield is b0 = 2, with a relative uncertainty ϵ = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The background  yield is treated as a nuisance parameter in the fit, associated with a Gaussian constraint  (as would occur in the case where the background is determined from a control region with  a sufficiently large number of events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The observed yield is n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The JSON specification  of the model is given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  1  {  2  "model": {  3  "name": " simple_bkg_uncertainty ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  4  "POIs": [  5  { "name" : "xs_signal",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' "unit" : "fb",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' "min_value": 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' "max_value": 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' " initial_value ": 1 }  6  ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  7  "NPs": [  8  { "name": "np_bkg",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' " nominal_value ": 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' "constraint": 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' "aux_obs": "aux_bkg" }  9  ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  10  "aux_obs": [  11  { "name": "aux_bkg",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' "min_value": -5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' "max_value": 5 }  12  ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  13  "channels": [  14  {  15  "name": " measurement_region ",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  16  "type": "bin",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  17  "samples": [  18  {  19  "name": "Signal",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  20  "norm": "xs_signal",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  21  " nominal_yields ": [ 1 ]  22  },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  23  {  24  "name": "Background",' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  25  " nominal_yields ": [ 2 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  26  "impacts": {  27  "np_bkg": { "+1":  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='10, "-1":  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='10 }  28  }  29  }  30  ]  31  }  32  ]  33  },  34  35  "data": {  36  "channels": [  37  { "name" : " measurement_region ", "counts": 3 }  38  ],  39  "aux_obs": [  40  { "name": "aux_bkg", "value": 0 }  41  ]  42  }  43  }  Figure 1: Specification for the example SLLS model described in the text  The results for the signal yield s are computed using its maximum likelihood estimator  (MLE) ˆs and the profile-likelihood ratio  Λ(s) = −2 log L(s,ˆˆb(s);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' n)  L(ˆs,ˆb;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' n)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='7)  where L(s, b, ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' n) is the measurement likelihood, ˆb is the MLE of b and ˆˆb(s) its conditional  MLE at a fixed value s of the signal yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The values of Λ(s) can then be used to derive  – 5 –  −1  0  1  2  3  s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  Λ(s)  Λ(s)  Λref(s)  (a)  −1  0  1  2  3  s  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='4  ˆb(s)  ˆb(s)  ˆbref(s)  (b)  Figure 2: Values of (a) Λ(s) and (b) the conditional MLE pull (ˆˆb(s) − b0)/ϵ for a range of  values of the signal yield s, computed from the model described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In each plot,  the simplified likelihood result (solid blue) is compared to an exact closed-form expression  of the same quantity (dashed red), showing very close agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  results such as confidence intervals on s or a discovery significance for the presence of the  signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Figure 2 shows the values of Λ(s) and ˆˆb(s) computed from the model given in Figure 1  for a range of values of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In this simple case both results can also be computed in closed  form as  ˆˆb(s) = 1  2  ��  (s + b0 − b2  0ϵ2)2 + 4b2  0ϵ2n − (s − b0 + b2  0ϵ2)  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='8)  Λ(s) = 2(s − ˆs + ˆˆb(s) − ˆb) − 2n log  �  s + ˆˆb(s)  ˆs + ˆb  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='9)  and excellent agreement is observed between these expressions and the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The asymmetric shape of Λ(s) is driven by the Poisson nature of the measurement, and the  good agreement in this case is due to the fact that this feature is accounted for exactly in  the simplified likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Using a Gaussian approximation would yield a parabolic shape  for Λ(s) that would provide a less accurate description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' While the systematic uncertainty  on the background yield plays only a small role in this example, the good agreement in  the profiled values ˆˆb(s) of the corresponding NP shows that these effects are also described  accurately within the linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  3  Application to an ATLAS search for new phenomena  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  Full likelihood  This section presents a realistic application of the SLLS framework to a search for new  phenomena by the ATLAS collaboration [16] for which the full experimental likelihood  – 6 –  has been published [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The search targets supersymmetric particles in final states with  at least three charged leptons originating from the chargino decay ˜χ+  1 → Zℓ → 3ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The  analysis considers three signal regions (SRs), targeting signatures with 3 leptons (3ℓ), 4  leptons (4ℓ) and 4 leptons with a fully reconstructed W, Z or H boson (FR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Each signal  region is divided into 16 bins of the invariant mass mZℓ of the trilepton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Three  single-bin control regions are also included to provide data-driven estimates of the main  backgrounds, from the Standard Model production of a WZ boson pair, a ZZ pair, or a t¯t  pair accompanied by a Z boson (t¯tZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The full likelihood of the analysis was published by the ATLAS collaboration as a  pyhf model available in the HEPData repository [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In this example we consider the  example case of a chargino with a mass of 500 GeV with branching ratios to W, Z and H  bosons of respectively 20%, 60% and 20%, and equal branching ratios to e, µ and τ for the  accompanying lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  Simplified likelihood  A SLLS likelihood model is constructed from the pyhf model using as reference the param-  eter values obtained in a fit to the observed data under the background-only hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The conversion is performed using an automated tool included in the fastprof package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The measurement regions of the analysis as implemented in the simplified model are shown  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The profile likelihood scan of the signal strength parameter µsignal using the simplified  likelihood is shown in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' A reference scan computed using the full likelihood is also  presented for comparison, and shows that the simplified likelihood provides an adequate  description of the full result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' A simple Gaussian model, using the best-fit value µsignal in the  observed data computed by pyhf and the corresponding parabolic error, is also displayed  and shows worse agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The 95% CLs limit on µsignal computed using the simplified  model is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='126, in good agreement with the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='124 obtained using the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The Gaussian model yields a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The fits to the SLLS likelihood with fixed µsignal take about 50 ms on a laptop computer  equipped with a 16-core Intel i7-10875H CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The fit with free µsignal, which relies on non-  linear rather than linear minimization for this parameter (since POIs are treated exactly),  takes about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' A full-likelihood fit performed with pyhf require approximately 10 min  on the same computing platform, a factor ≈ 1000 longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' These fits are performed with  numpy as the numerical backend to pyhf, the same as used the fastprof implementation  of SLLS likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Better performance can however likely be achieved using other pyhf  backends interfacing to the tensorflow [18] or pytorch [19] packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  To validate the SLLS linear profiling technique, the profiled values for some of the  model NPs are shown in Figures 4b and 4c for both the SLLS and the full likelihood  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Good agreement is seen between the two cases, illustrating that the full likelihood  is modeled to good approximation at the level of individual NPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' As a further illustration,  the exclusion contour presented in Figure 9 of the original publication is recomputed using  the SLLS simplified likelihoods and compared to the full-likelihood results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The results are  shown in Figure 5, and good agreement is again observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
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+page_content='Events  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='CRZZ all cuts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='Events  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='CRttZ all cuts  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='(d)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='Figure 3: Expected and observed event counts in the SR3l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' SR4l and SRFR signal regions  of the analysis of Ref [16],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' shown respectively in panels (a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Panel (d) shows  the analysis control regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The signal regions are binned in the mZl observable, while the  control regions each use a single inclusive event count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The observed data (black points) is  overlaid with stacked histograms (filled areas) representing the gaugino signal (dark blue)  and the main background contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Gaussian models built as described above is also presented for comparison and shows similar  agreement in this case, in part due to the fact that the signal production cross-sections vary  rapidly with the chargino mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  4  Simplified likelihoods for unbinned models  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  Binned description of unbinned models  The previous examples used a binned description of the experimental measurement, which  employs only two types of PDFs: Poisson distributions to describe the counting experiments  in each bin, and Gaussian distributions for the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Another common modeling  option is unbinned likelihoods, in which the model describes the continuous probability  distribution of the measurement observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' It is used for instance to study the H → γγ  decay of the Higgs boson at the LHC [20, 21], as well as in many results published by  LHCb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' It requires support for arbitrary PDF forms, as needed to describe each measure-  ment, and therefore more general and flexible tools than for binned likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' For LHC  measurements, this functionality is usually provided by the roofit package [22] distributed  – 8 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='20  µsig  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  Λ(µsig)  fastprof  pyhf  Gaussian  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
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+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='20  µsig  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
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+page_content='3  ˆˆθ(µsig)/σθ  ϵelec  ϵmuon  ϵb-tag  MET scale  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='20  µsig  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  ( ˆˆθ(µsig) − 1)/σθ  µ3l  µ4l  µttZ  (c)  Figure 4: Comparison between the SLLS simplified model (solid lines) and the full model  (dotted lines) for (a) the PLR Λ(µsignal) as a function of µsignal, (b) the profiled values  of selected NPs describing systematic uncertainties and (c) the profiled values of NPs de-  scribing scale factors applied to the normalization of the main analysis backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The  profiled values are shown as deviations from the nominal value of the parameters (0 for  systematic uncertainties, 1 for background scaling factors), divided by the uncertainty on  the parameter in the full-model fit to the observed data with free µsignal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The SLLS results  are computed using the fastprof tool, and the full-model results with the pyhf tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Panel  (a) also shows the PLR scan computed using a Gaussian model as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Good agreement is observed overall between the SLLS and full-model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The largest  deviation is seen in the scale factor for the t¯tZ background, corresponding to about 30% of  the fit uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  as part of ROOT, but this and other similar tools are not widely used outside the high-energy  – 9 –  200  400  600  800  1000  Chargino mass  0  20  40  60  80  100  Branching fraction to Z  pyhf  fastprof  Gaussian  Figure 5: Exclusion plot in the plane of chargino mass and its branching ratio to Z bosons,  assuming equal branching ratios to W and H and to all lepton flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The computation  from SLLS simplified likelihoods (solid blue) is compared with a reference (dashed red)  taken from the top-left panel of Figure 9 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [16] and good agreement is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Gaussian models computed from the full likelihood as described in the text (dotted black)  also show good agreement in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  physics experimental community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' While there are some recent new efforts to provide more  portable alternatives, none is currently in wide use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  A possible way forward is based on the observation that unbinned models can be  approximated by binned models with a sufficiently fine binning (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' While  this approach typically runs into practical difficulties for full likelihoods due to the large  number of bins required, it is feasible for simplified likelihoods which are quick to evaluate  even for relatively large bin numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In the rest of this section, we present the application  of the SLLS likelihood framework to an unbinned model loosely inspired by an ATLAS  H → γγ measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  Full model example  We consider a simple example based on the ATLAS H → γγ analysis of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The  analysis uses an unbinned model based on the distribution the invariant mass mγγ of the  two photons in the range 105 < mγγ < 160 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The Higgs boson signal manifests itself  as a sharp peak in the mγγ distribution, with a position close to the Higgs boson mass  and a width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1 GeV depending on event kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The background contributions  follow smoothly falling shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Several signal regions (referred to as categories in the rest  of this section) are defined according to the properties of the signal photons and other  reconstructed objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  – 10 –  The example uses a simplified description of the 33 categories defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20] to  study Higgs boson production in the gluon-fusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The signal and background dis-  tributions are represented respectively by Gaussian and exponential distributions, instead  of the more complex shapes used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The peak position and width of the Gaus-  sian, as well as the expected signal and background yields are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20], while  the exponential slope of the background is assumed to be −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='02 GeV−1 in all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The background normalizations and exponential slopes are free to vary in the fit, except  for the slopes in five low-statistics categories which are kept fixed to avoid unstable fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Five nuisance parameters are used to describe the leading systematic uncertainties: the  uncertainty on the integrated luminosity of the dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' on the reference cross-section for  the gluon-fusion production process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' on the effect of parton shower modeling on the signal  yields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' on the H → γγ reconstruction efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' and on the photon energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This  last uncertainty leads to a change in the width of the signal peak and therefore induces  highly non-linear effects in the per-bin signal yields in a binned description of the likeli-  hood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Systematic uncertainties on the background model are implemented using separate  nuisance parameters in each category, following the "spurious signal" method described in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The values of the uncertainties listed above are all taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In total,  99 NPs are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The single POI is the Higgs boson signal strength µ, applied as a  scaling factor to the expected signal yield in all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' A dataset of events randomly  generated from the model PDF is used as the "observed" data in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The model  and data are stored in a roofit workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  Simplified likelihood  The SLLS likelihood is built as a binned approximation to the full likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' A fine binning  is required to obtain an accurate description of the signal peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In this example a uniform  bin width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1 GeV is used, leading to 18150 bins in total for the 33 categories2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The mγγ  distributions for two selected categories (the first and last in the order used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20])  are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The conversion is performed using an automated tool distributed as part of the fastprof  package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' All the nuisance parameters are retained, and their effect is described in terms of  their linear impact on the event yield in each measurement bin, following the SLLS pro-  cedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In some cases, in particular normalization parameters such as the one shown on  Figure 7a, impacts are linear by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' By contrast, the photon energy resolution  systematic shown in Figure 7b exhibits non-linear behavior since it induces a change in  the width of the signal peak which does not propagate linearly to the bin contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Non-  linearities become larger as moves towards the tail of the signal peak, but with a smaller  impact on the results due to lower signal yields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The linear approximation remains in  any case typically accurate for small deviations of the NP from the nominal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Figure 8a  shows the profile likelihood scan for the signal strength parameter µ obtained with the lin-  earized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The reference result obtained with the full unbinned likelihood, computed  2A variable-width binning with wider bins away from the peak can also be considered, but a uniform  binning was chosen in this example for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  – 11 –  110  120  130  140  150  160  mγγ [GeV]  0  20  40  60  80  100  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1 GeV  000  Full model  Sig 000  Bkg 000  Data  110  120  130  140  150  160  mγγ [GeV]  10−3  10−2  10−1  100  Events / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1 GeV  032  Full model  Sig 032  Bkg 032  Data  Figure 6:  Distributions of the observable mγγ for two selected categories, labeled 0-jet,  pH  T < 10 GeV (top) and pH  T ≥ 650 GeV (bottom) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The signal and background  contributions (blue and green histograms respectively) are shown together with the example  dataset (black points), for the best-fit value of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  using the RooFitUtils package3, is also shown for comparison and excellent agreement  is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The resulting 68% CL likelihood intervals are µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='082+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='117  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='093 for the full  model and µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='082+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='113  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='093 for the simplified model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' A fully Gaussian approximation, as  described in the previous section, yields µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='082 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The fits take about 15 min to  perform on the full model, compared to about 50 ms and 1 s for simplified likelihood fits  with respectively a free and floating µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  To better compare the treatment of systematic effects, the profiled values of the five  nuisance parameters describing the leading systematic uncertainties are shown in Figure 8b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The agreement between the simplified and the full model is found to be accurate to about  10% of the parameter uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This agreement is crucial to the description of this  example since the uncertainty on µ is dominated by systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In particular, the  profiling of the photon energy resolution systematic (which represents about 20% of the  total uncertainty) shows good agreement with the full model in spite of the non-linear  effects highlighted in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Figure 8c shows the difference between the profiled values  of the other NPs in the simplified and full models, normalized to their fit uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This  difference is below 10% of the fit uncertainty for about 80% of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  3https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='ch/cburgard/RooFitUtils  – 12 –  −4  −2  0  2  4  Normalized parameter value  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='03  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='03  Relative impact on bin yield  nBkg 000  Full model impact  Linear impact  Non-linear impact  (a)  −4  −2  0  2  4  Normalized parameter value  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  Relative impact on bin yield  npPER  Full model impact  Linear impact  Non-linear impact  (b)  Figure 7: Relative change in expected bin yields as a function of the normalized parameter  value for two cases: (a) the impact of the background normalization parameter nBkg_000  on the expected background yield;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' and (b) the impact of the photon energy resolution  parameter npPER on the expected signal yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In both cases, the bin is located at mγγ ≈  127 GeV in the first category of the model, which lies about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='7σ above the signal peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The impacts computed from the full model (dots) are compared with the linear impacts  computed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2 (dotted red line) and the non-linear impacts from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='6 (solid  blue line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  5  Discussion  As observed in the examples shown in this paper, linearized NP impacts provide a generally  adequate approximation of their behavior in the full model, in particular in the description  of systematic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  It can be noted that the approximation coincides with the exact  description in the case where the impact of the NP is naturally linear, such as the case  shown in Figure 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Discrepancies are expected in cases which deviate from this ideal case,  in particular for:  Large non-linear systematic uncertainties, with effects that are not fully accounted  for in the linear approximation, such as the one shown in Figure 7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Asymmetric systematic uncertainties, with different impacts for NPs above or be-  low their nominal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' These effects cannot be included in the profiling described  in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2, although they can be taken into account for the evaluation of the  likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Low expected event yields, leading to Poisson counts that are not well-described by  Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' While the Poisson distribution itself is described exactly in  the SLLS formalism, non-linearities can occur for systematics on the expected event  yield, since the Poisson PDF does not depend linearly on its expected yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  These situations are partially covered in the examples described in this paper, and it is  encouraging that in these cases at least, the linear description seems adequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' However  – 13 –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  µ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  Λ(µ)  fastprof  pyhf  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  µ  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5  ˆˆθ(µ)/σθ  θLumi  θggF  θϵ  θUPS  θPER  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
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+page_content='3  µ  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='00  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='75  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='50  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='00  ∆ ˆˆθ(µ)/σθ  (c)  Figure 8: Comparison between the SLLS model (solid lines) and the full likelihood (dotted  lines) for (a) the profile likelihood Λ(µ) as a function of the parameter of interest µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' (b)  the profiled values of a selection of nuisance parameters describing systematic uncertainties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  and (c) the difference between the profiled values of each NP obtained from the SLLS model  and the full likelihood, divided by the uncertainty on the parameter in the full-likelihood  fit to the data with free µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Good agreement is observed overall between the linear and  full-model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  simplified likelihoods should be carefully validated against the full likelihood in each case  nevertheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Tools to perform these checks are included in the fastprof package, using  methods similar to those shown in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Another limitation to take into consideration is the memory footprint of the ∆cbsk coef-  ficients which encode the linear impacts: their number is given by NNPs × Nbins × Nsamples,  which can reach O(108) or more for complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' For models with a large numbers  – 14 –  of bins and nuisance parameters, such as converted unbinned models, memory constraints  can be more stringent than those related to computation times, since these computations  mainly involves matrix operations that are quite efficient even for large models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  6  Conclusion  Simplified likelihoods provide a convenient setting for the use and reuse of experimental  results, and functionality that is complementary to that of full likelihoods and Gaussian  approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The SLLS framework based on a linear description of nuisance parameter impacts pro-  vides an approximation with several benefits: it describes the Poisson behavior of counting  measurements exactly, and it also preserves the nuisance parameters of the full model and  therefore a fully granular description of its sources of systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Both of these  aspects make it well-suited to the description of LHC measurements, where systematic un-  certainties and non-Gaussian effects from low event counts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' in tails of distributions)  both play important roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The preservation of the nuisance parameter structure allows  in particular a proper treatment of correlated systematic effects when performing com-  binations of measurements, by identifying parameters associated with identical sources of  uncertainty in the combination inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Since the parameters of interest of the original model  are also preserved, reinterpretations of the simplified likelihood can also be performed by  re-expressing them in terms of the parameters of an alternative models, as can be done for  full likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  SLLS models can be built automatically from binned likelihood implemented using the  HistFactory formalism within the ROOT and pyhf framework, or from unbinned likelihood  using binned approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' An implementation of the SLLS framework is provided in  the fastprof package at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='com/fastprof-hep/fastprof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The models are  stored in a plain-text JSON format, and computations and other operations are performed  using python tools based on the widely available numpy and scipy libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Together with other full and simplified likelihood formats with complementary func-  tionality, it is hoped that this framework will encourage the further publication of detailed  experimental likelihoods by LHC experiments and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Acknowledgments  The author would like to thank Nick Wardle providing the code for the simplified likelihoods  of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' [9], and Tetiana Hryn’ova for valuable feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Plots in this paper were produced  with matplotlib using the SciencePlots style package [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' This research was funded, in  whole or in part, by l’Agence Nationale de la Recherche (ANR), project ANR-22-CE31-  0022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  – 15 –  A  Linearization procedure  This section provides a sketch of the derivation of the profile value ˆˆθk(µ) of the NP θk given  by Eq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Starting from the likelihood in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1 and applying the linearization procedure  described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2, we obtain the negative log-likelihood  λ(µ, θ) =  Nchannels  �  c=1  Nbins,c  �  b=1  [νcb(µ, θ) − ncb log νcb(µ, θ)] + 1  2(θ − ˜θ)Γ(θ − ˜θ)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1)  up to a an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In the expression above, indices c, b and s run respectively  over measurement channels, bins within each channel, and event samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The νcb(µ, θ) =  �  s νcbs(µ, θ) are the total expected events yields for the corresponding channel bin, and the  per-sample yields νcbs(µ, θ) are given by Eq 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The Gaussian constraints on the nuisance  parameters are parameterized using the auxiliary observables ˜θ and the inverse covariance  matrix Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The derivative of λ(µ, θ) with respect to the nuisance parameters θ is  ∂λ  ∂θ(µ, θ) =  Nchannels  �  c=1  Nbins,c  �  b=1  ��  s  νnom  cbs (µ)∆cbs  �  1 −  ncb  νcb(µ, θ)  ��  + Γ(θ − ˜θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2)  where ∆cbs is the vector with components ∆cbsk, the linear impacts of the parameter θk  on νcbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The linear approximation of nuisance parameters impacts can be applied to the  denominator as  ncb  νcb(µ, θ) ≈  ncb  νnom  cb  (µ)  �  1 −  �  s  νnom  cbs (µ)  νnom  cb  (µ)∆cbs(θ − θnom)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3)  and one finally obtains  ∂λ  ∂θ(µ, θ) = Q(µ) + P(µ) [θ − θnom] + Γ  �  θ − ˜θ  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='4)  with Q(µ) and P(µ) defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='5, and the profile values ˆˆθ(µ) defined by ∂λ/∂θ(µ, ˆˆθ(µ)) =  0 are therefore given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  B  Binned approximation to an unbinned PDF  We consider an extended unbinned PDF for an observable x,  P(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' θ)  n  �  i=1  dxi = e−N(θ)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  N(θ)n  n  �  i=1  f(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' θ)dxi  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1)  where f(x, θ) is the PDF for one observation of x, the dataset consists of the values x1 · · · xn,  θ are the model parameters, and the expected number of observations is N(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' We include  the infinitesimal volume elements dxi in the expression, as these will be useful below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  – 16 –  We introduce a set of bins Ba, a = 1 · · · Nbins that span the allowed range of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' In the  spirit of finite-element analysis, we approximate f(x) by a form that is constant over each  bin as  f(x, θ) =  �  i  fa(θ)Ia(x)  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2a)  fa(θ) = 1  wa  �  Ba  f(x, θ)dx  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2b)  where the indicator Ia(x) is 1 if x ∈ Ba and 0 otherwise, and wa =  �  Ba Ia(x)dx is the  measure of bin Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  The value fa(θ) is the average of f(x, θ) over the bin Ba, so that for a sufficiently fine  binning and smooth f(x, θ), f(x, θ) ≈ fa(θ) for x ∈ Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  One can remove the explicit dependence on the xi by integrating them out of the  likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' The integration of the product term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1 can be written as  �  n  �  i=1  f(xi, θ)dxi =  n  �  i=1  �  i  fa(θ)  �  Ia(xi)dxi =  n  �  i=1  fai(θ)wai =  Nbins  �  a=1  [fa(θ)wa]na  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2c)  where ai is the index of the bin to which xi belongs, na is the number of observations that  fall in bin Ba, and we have used the fact that the xi are independent to propagate the  integral through the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Returning to the full expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='1, we can write the  likelihood as a function of the n as  P(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' θ) =  e−N(θ)  n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' · · · nNbins!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='N(θ)n  Nbins  �  a=1  [wafa(θ)]na ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2d)  after including an additional multiplicative factor (n, n1, n2, · · · nNbins) to account for the  number of different orderings of the xi that can yield a given set of na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' One can introduce  the per-bin expected yields  Na(θ) = wafa(θ)N(θ)  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2e)  and note that since  1 =  �  f(x, θ)dx =  Nbins  �  a=1  �  Ba  f(x, θ)dx =  Nbins  �  a=1  wafa(θ)  one has N(θ) = �  a Na(θ) as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' One can finally rewrite  P(n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' θ) =  Nbins  �  a=1  e−Na(θ)  na!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  N(θ)na(wafa(θ))na =  Nbins  �  a=1  e−Na(θ)  na!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  Na(θ)na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2f)  This takes the usual form of a binned likelihood, with a Poisson distribution in each mea-  surement bin with expected yields given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  – 17 –  References  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Cowan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Cranmer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Gross, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content=' Vitells, Asymptotic formulae for likelihood-based  tests of new physics, The European Physical Journal C 71 (feb, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
+page_content='  [2] CMS Collaboration, A portrait of the Higgs boson by the CMS experiment ten years after the  discovery, Nature 607 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdE5T4oBgHgl3EQfmA-1/content/2301.05676v1.pdf'}
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+Robust Consensus Clustering and its Applications for Advertising Forecasting
+Deguang Kong*, Miao Lu, Konstantin Shmakov and Jian Yang
+Yahoo Research, San Jose, California, U.S.A, 94089
+doogkong@gmail.com, ml4ey@virginia.edu, kshmakov@yahooinc.com, jianyang@yahooinc.com
+Abstract
+Consensus clustering aggregates partitions in order to find
+a better fit by reconciling clustering results from different
+sources/executions. In practice, there exist noise and outliers
+in clustering task, which, however, may significantly degrade
+the performance. To address this issue, we propose a novel
+algorithm – robust consensus clustering that can find com-
+mon ground truth among experts’ opinions, which tends to
+be minimally affected by the bias caused by the outliers. In
+particular, we formalize the robust consensus clustering prob-
+lem as a constraint optimization problem, and then derive an
+effective algorithm upon alternating direction method of mul-
+tipliers (ADMM) with rigorous convergence guarantee. Our
+method outperforms the baselines on benchmarks. We apply
+the proposed method to the real-world advertising campaign
+segmentation and forecasting tasks using the proposed con-
+sensus clustering results based on the similarity computed
+via Kolmogorov-Smirnov Statistics. The accurate clustering
+result is helpful for building the advertiser profiles so as to
+perform the forecasting.
+Introduction
+Consensus clustering reconciles clustering information from
+different sources (or different executions) to be better fit
+than the existing clustering results. The consensus cluster-
+ing results, however can be biased due to different types of
+features, different executions of algorithms, different defi-
+nitions of distance metrics and even different parameter set-
+tings, as is sharply observed by existing studies (Jain, Murty,
+and Flynn 1999). All these factors may lead to disparity
+in clustering results. The major consensus clustering algo-
+rithms include Hypergraph partitioning (Strehl and Ghosh
+2003), (Fern and Brodley 2004), voting approach (Dudoit
+and Fridlyand 2003), (Roth et al. 2003), mutual information
+(Wang, She, and Cao 2013) (Topchy, Jain, and Punch 2004),
+co-association approach (Xu and Wunsch 2005), mixture
+model (Topchy, Jain, and Punch 2004)
+(Azimi and Fern
+*This work was done during the authors were working at
+Yahoo Research. The correspondence should be addressed to
+doogkong@gmail.com. The views and conclusions contained in
+this document are those of the author(s) and should not be inter-
+preted as representing the official policies, either expressed or im-
+plied, of any companies.
+Copyright © 2019, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+2009), correlation consensus (Wang et al. 2014), ensemble
+clustering (Zhou et al. 2015), (Gao 2016), (Liu, Latecki,
+and Yan 2010), (Huang, Lai, and Wang 2016), etc.
+One key observation is that in consensus clustering
+(Strehl and Ghosh 2003), if there exist noise and outliers
+in one source of features in any execution of algorithm, the
+clustering result might be significantly affected due to the
+least square loss function used in most of the clustering
+methods (such as k-means, Gaussian mixture model), be-
+cause the errors are squared. Even worse, most of the time,
+the users have little prior knowledge of noise, which makes
+the clustering result more unstable and much harder to inter-
+pret with different initializations and parameter settings. For
+example, if one information source that we used for consen-
+sus clustering is not accurate, when we align the consen-
+sus clustering results against this “inaccurate” source, we
+will suffer from these inaccurate annotations. The inaccu-
+rate common characteristics extracted from the samples due
+to the biased clustering results, are in fact, however, less gen-
+eralizable to those unseen ones.
+To address these issues, this paper proposes a robust con-
+sensus clustering schema that is minimally affected by the
+outliers/noise. In particular, we combine multiple experts’
+opinions on data clustering using the robust ℓ1-loss func-
+tion that aims to find the maximum consistency on the ex-
+perts’ opinions with minimum conflict. Our work can be
+viewed as an effective method of data clustering from het-
+erogeneous information sources. The proposed method is
+practically feasible because it is independent of parameter
+settings of each clustering algorithm before aggregating the
+experts’ opinions. Driven by advertising applications, we ap-
+ply the concensus clustering algorithm to cluster advertiser
+profiles (Mahdian and Wang 2009) into different clusters so
+as to accurately perform performance (e.g.. Click, Conver-
+sions) forecasting.
+The main contribution of this paper is summarized as fol-
+lows.
+• To address the issue of consensus clustering performance
+degradation in existence of noise and outliers, we rigor-
+ously formulate the problem of robust consensus cluster-
+ing as an optimization problem using the robust loss func-
+tion.
+• To find the best solution for robust consensus clustering,
+arXiv:2301.00717v1  [cs.LG]  27 Dec 2022
+
+we develop an effective algorithm upon ADMM to de-
+rive the optimal solution, whose convergence can be rig-
+orously proved.
+• We experimentally evaluate our proposed method on both
+benchmarks and real-world advertiser segmentation and
+forecasting tasks, and show that our method is effective
+and efficient in producing the better clustering results. As
+an application, the proposed algorithm is applied for ad-
+vertising forecasting tasks with more stable and trustful
+solutions.
+Robust consensus clustering Model
+Problem setting Assume we have n data points, each data
+point can generate features from different views (total view
+number=V). For example, an image can generate features
+using different descriptors, such as SIFT1, HOG2, CNN fea-
+tures (Girshick et al. 2013). More formally, let xv
+i ∈ ℜpv
+be v-th (1 ≤ v ≤ V ) view/modal feature of a data point i,
+pv is the dimensionality of feature extracted from v-th view.
+Consider all data points,
+Xv = [xv
+1, xv
+2, · · · , xv
+n],
+where each data column vector is xv
+i ∈ ℜpv×1. Each data
+point has the ground-truth label yi ∈ ℜK. Simple concate-
+nation of all data views gives
+X =
+�
+���
+X1
+X2
+...
+XV
+�
+��� = [x1, x2, · · · , xn],
+xi =
+�
+�
+�
+�
+x1
+i
+x2
+i...
+xV
+i
+�
+�
+�
+� .
+Suppose we are given the clustering/partition results (P)
+for data points {X1, X2, · · · , XV } from V different views,
+i.e., P = {G1, G2, · · · , GV }, where Gv ∈ {0, 1}n×K (for
+K clusters) is the clustering result from view v. The clus-
+ters/partitions assignment might be different for different
+views. Let the connectivity matrix (a.k.a co-association ma-
+trix) M ∈ {0, 1}n×n, where Mij(GV ) for view v between
+data point i, j is:
+Mij(Gv) =
+�
+1;
+(i,j) belongs
+to
+the same
+clustering
+0.
+otherwise
+Co-association consensus clustering
+Standard consensus
+clustering looks for a consensus clustering (based on major-
+ity agreement) G∗ ∈ {0, 1}n×K, such that G∗ is closest to
+the all the given partitions, i.e.,
+min
+G∗ J(G∗) = 1
+V
+V
+�
+v=1
+n
+�
+i=1
+n
+�
+j=1
+D(Mij(Gv) − Mij(G∗)),
+(1)
+where D(.) is a distance function between the optimal solu-
+tion and the solution from each view. Generally, least square
+1https://en.wikipedia.org/wiki/
+Scale-invariant feature transform
+2https://en.wikipedia.org/wiki/
+Histogram of oriented gradients
+loss is used for function D(.):
+min
+G∗ J(G∗) = 1
+V
+V
+�
+v=1
+n
+�
+i=1
+n
+�
+j=1
+�
+Mij(Gv) − Mij(G∗)
+�2
+,
+(2)
+Notice that Mij(Gv) = 0, 1, therefore, Eq.(2) can be
+equivalently written as:
+min
+G∗ J2(G∗) = 1
+V
+V
+�
+v=1
+n
+�
+i=1
+n
+�
+j=1
+|Mij(Gv) − Mij(G∗)|
+= 1
+V
+V
+�
+v=1
+∥M(Gv) − M(G∗)∥1,
+(3)
+with ∥X∥1 := �
+i,j |Xij|.
+Let
+Gℓ2 = arg min
+G∗
+J(G∗)
+(4)
+Gℓ1 = arg min
+G∗
+J2(G∗).
+(5)
+Clearly,
+Gℓ2 = Gℓ1.
+(6)
+Analysis Given little (or no) prior information of cluster-
+ing result from each view, one natural way to compute the
+associate between data points i and j is to get the expected
+value of average association ˜
+M = [ ˜
+Mij], i.e.,
+˜
+Mij = 1
+V
+V
+�
+v=1
+Mij(Gv).
+(7)
+Back to model of Eq.(3), the upper bound of J2 is given by:
+J2 ≤ 1
+V
+V
+�
+v=1
+∥M(Gv) − ˜M∥1 + ∥ ˜M − M(G∗)∥1
+= C + ∥ ˜M − M(G∗)∥1
+= C + J3(G∗)
+(8)
+where
+J3(G∗) = ∥ ˜M − M(G∗)∥1
+(9)
+C = 1
+V
+V
+�
+v=1
+∥M(Gv) − ˜M∥1
+(10)
+Note the first term is a constant C because it measures the
+average difference between the final clustering assignment
+and the average consensus association3
+˜M. The smaller
+value of this term gives the closer distance between the final
+clustering assignment and current partition from a specific
+view.
+On the other hand,
+J3 ≤ 1
+V
+V
+�
+v=1
+�
+∥M(Gv) − M(G∗)∥1 + ∥ ˜M − M(Gv)∥1
+�
+= J2(G∗) + C (11)
+3The weighted average using attention can be adopted as well
+with learned weights.
+
+This analysis indicates that the optimization of J2 can be
+achieved by J3.
+Clustering indicator One way to optimize J3 of ∥ ˜M −
+M(G∗)∥1 of Eq.(9) is to assign the optimal clustering indi-
+cator G ∈ {0, 1}n×K, with the constraint that �
+k Gik = 1,
+i.e., there is only one ‘1’ in each row of G, and the rest
+are zeros. The connection between the averaged connectiv-
+ity matrix M(G∗) and G is:
+M(G∗) = G∗G∗⊺.
+(12)
+Therefore the objective of Eq.(9) becomes J4, i.e.,
+minG J4(G) = ∥ ˜M − GG⊺∥1
+s.t.,
+G ∈ {0, 1}n×K, �
+k Gik = 1
+(13)
+Relaxation in continuous domain
+However, the major
+difficulty of solving Eq.(13) is that is involves the dis-
+crete clustering indicator G and the algorithm is NP-
+complete (Filkov and Skiena 2004). Also the objective in-
+volves non-smooth ℓ1 norm, which generally, is hard to han-
+dle. Thus, as most of the spectral clustering solvers, we do
+normalization on cluster indicators. In particular, we use
+H ∈ ℜn×K to denote the new clustering indicator, i.e.,
+H = G(G⊺G)− 1
+2 .
+Notice that
+G⊺G =
+�
+�
+�
+�
+�
+n1
+n2
+...
+...
+nk
+�
+�
+�
+�
+� ,
+(14)
+where nk is the number of data points that falls in category
+k. Let diagonal matrix D ∈ ℜk×k, i.e.,
+D := G⊺G.
+then
+GG⊺ = HDH⊺;
+(15)
+H⊺H = G(G⊺G)−1 = Ik,
+(16)
+where Ik is an identity matrix with size kxk. Therefore the
+objective function to be solved becomes:
+min
+H,D J5(H, D) = ∥ ˜M − HDH⊺∥1
+s.t.,
+H⊺H = IK, D ≥ 0; D is diagonal, (17)
+where G = HD
+1
+2 can be obtained in continous domain.
+In practice, the solution obtained from ˆM may not be ac-
+curate due to noise or outliers. For this reason, we term our
+method as “robust” because we use ℓ1 distance to measure
+the differences between the consensus optimal solution and
+the solution computed from each view. The errors are not
+squared. Therefore, our model is capable of handling any
+noises/outliers in some view/execution in data clustering.
+Further, we have:
+Lemma 1 Set ˆM to be the normalized pairwise similarity
+matrix ˆ
+SS = D− 1
+2 SSD− 1
+2 (SS ∈ ℜn×n and Sij is the
+similarity between data point i and j), using robust ℓ1-norm
+as the loss function, i.e.,
+min
+H J0(H) = ∥ ˆ
+SS − HH⊺∥2
+F
+s.t.,
+H⊺H = Ik.
+(18)
+This is identical to normalized cut spectral clustering (Shi
+and Malik 2000).
+Proof
+Recall that in standard normalized cut spectral
+clustering, the objective is:
+min
+H Tr(H⊺(I − ˆ
+SS)H)
+s.t.,
+H⊺H = Ik,
+(19)
+where ˆ
+SS = D− 1
+2 SSD− 1
+2 . This is equivalent to optimiz-
+ing:
+max
+H Tr(H⊺ ˆ
+SSH)
+s.t.,
+H⊺H = Ik.
+(20)
+Notice that
+∥ ˆ
+SS − HH⊺∥2
+F = Tr( ˆ
+SS
+⊺ ˆ
+SS + HH⊺H⊺H − 2 ˆ
+SS
+⊺HH⊺),
+Tr(HH⊺H⊺H)
+=
+const, and therefore minH ∥ ˆ
+SS −
+HH⊺∥2
+F is equivalent to maxH Tr(H⊺ ˆ
+SSH). This com-
+pletes the proof.
+In this paper next, we discuss how to solve the robust con-
+sensus clustering model of Eq.(17).
+Optimization Algorithm
+Eq.(17) seems difficult to solve since it involves non-smooth
+loss and high-order matrix optimization. We show how to
+apply alternating direction method of multipliers (ADMM)
+to solve it. ADMM method decomposes a large optimization
+problem by breaking them into smaller pieces, each of which
+are then easier to handle. ADMM combines the benefits of
+dual decomposition and augmented Lagrangian method for
+constrained optimization problem (Bertsekas 1996) and has
+been applied in many applications. The problem solved by
+ADMM method usually has the following general forms4,
+i.e.,
+min
+X,Z f(X) + g(Z),
+s.t.
+AX + BZ = C(21)
+After enforcing the Lagrangian multiplier while introducing
+more variables, the problem can be solved alternatively, i.e.,
+ℓ(X, Z, µ) = f(X) + g(Z) + Ω⊤(AX + BZ − C)
++µ
+2 ||AX + BZ − C||2
+F ,
+(22)
+Xt+1 := arg min
+X
+ℓ(X, Zt, µt),
+Zt+1 := arg min
+Z
+ℓ(Xt+1, Z, µt),
+Ωt+1 := Ωt + µ(AXt+1 + BXt+1 − C).
+and Ω is the augmented Lagrangian multiplier, and µ is the
+non-negative step size.
+4Here X, Z can be vectors or matrices.
+
+Algorithm 1 Solving robust consensus clustering model of
+Eq.(17)
+Input: clustering results from different views ˜
+M, parameter ρ >
+1.
+Output: final clustering indicator H.
+Procedure:
+1: Initialize E0, H0, D0, Ω0, µ0 > 0, t = 0
+2: while Not converge do
+3:
+Update E via Eq.(26)
+4:
+Update H via Eq.(28)
+5:
+Update D via Eq.(29)
+6:
+µt+1 := ρµt
+7:
+Ω := Ω + µ(E − ( ˜
+M − HDH
+⊺))
+8:
+t := t + 1
+9: end while
+Optimization Algorithm to solve Eq.(17)
+According to ADMM algorithm, by imposing constraint
+variable
+E = ˜M − HDH
+⊺,
+the problem of Eq.(17) is equivalent to solving,
+min
+E, H, D ∥E∥1,
+s.t.
+E − ( ˜M − HDH⊺) = 0;
+s.t., H
+⊺ ˜H = IK, D ≥ 0; D is diagonal (23)
+In ADMM algorithm, to solve f(E) = ∥E∥1, under the con-
+straint
+h(E) = E − ( ˜M − HDHT ),
+the ADMM function can be formulated as follows,
+ℓ(E, H, D, Ω, µ) = f(E) + Ω⊤h(E) + µ
+2 ||h(E)||2
+F .
+where Lagrange multiplier is Ω and µ is the penalty constant.
+We solve a sequence of subproblems
+min
+E,H,D ∥E∥1 + Ω
+⊤(E − ( ˜M − HDH
+⊺))
++µ
+2 ∥E − ( ˜M − HDH
+⊺)∥2
+F .
+(24)
+with Ω and µ updated in a specified pattern:
+Ω ← Ω + µ(E − ( ˜M − HDH
+⊺)),
+µ ← ρµ.
+To solve Eq.(24), we search for optimal E, H, D itera-
+tively until the algorithm converges. Now we discuss how to
+solve E, H, D in each step. Alg.1 summarizes the complete
+algorithm.
+Update E
+To update the error matrix E, we derive Eq.(25)
+with fixed H, D and obtain the following form:
+min
+E
+µ
+2 ||E − A||2
+F + ||E||1
+(25)
+where
+A = HDHT − ˜M + Ω
+µ ,
+It is well-known that the solution to the above LASSO type
+problem (Wright et al. 2009) is given by
+Eij = sign(Aij) max(|Aij| − 1
+µ, 0).
+(26)
+Table 1: dataset descriptions
+datasets
+#data points
+# feature
+#class
+CSTR
+475
+1000
+4
+Glass
+214
+9
+7
+Ionosphere
+351
+34
+2
+Iris
+150
+4
+3
+Reuters
+2900
+1000
+10
+Soybean
+47
+35
+4
+Wine
+178
+13
+3
+Zoo
+101
+18
+7
+Update H, D
+To update H, D while fixing E, we mini-
+mize the relevant part of Eq.(24) which is
+min
+H,D
+µ
+2 ∥B − HDHT ∥2
+F ,
+s.t., HT H = IK, D ≥ 0; D is diagonal (27)
+where
+B = ˜M − E + Ω
+µ .
+It is easy to see the optimal solution H is given by the
+k-largest eigenvectors of B, i.e., H = [h1, h2, · · ·, hk],
+Bhk = λkhk,
+(28)
+where λk is the associated eigen-value with respect to eigen-
+vector hk, and Λ is a diagonal matrix, and
+Λ = diag([λ1, λ2, · · ·, λk]).
+Then the optimal solution of D is given by
+D = Λ.
+(29)
+The completes the algorithm for solving Eq.(17).
+Time Complexity Analysis Since Eq.(17) is not a convex
+problem, in each iteration, given µ, Σ, ρ, the algorithm will
+find its local solution. The convergence of ADMM algorithm
+has been proved and widely discussed in (Bertsekas 1996).
+The overall time cost for the algorithm depends on itera-
+tions and time cost for each variable updating. The compu-
+tation of E takes O(nk2) time. The major burden is from the
+updating of D and H, which requires computation of top k
+eigen-vector of B,i.e., O(n3) time. Overall, the cost of the
+algorithm 1 is O(T(n3 + nk2)), where T is the iteration
+number before convergence. To make the solution scalable
+for large-scale dataset, We can accelerate this via Hessen-
+berg reduction and QR iteration with multi-thread solver us-
+ing GPU acceleration (Gates, Haidar, and Dongarra 2014)
+and also use divide-and-conquer (Talwalkar et al. 2013) to
+accelerate the execution.
+Experimental Results on Benchmarks
+In order to validate the effectiveness of our method, we
+perform experiments on benchmark datasets. In particular,
+we use six datasets downloaded from UCI machine learn-
+ing repository5, including Glass, Ionosphere, Iris, Soybean,
+Wine, zoo. We also adopted two widely used text datasets
+for document clustering, including CSTR6 and Reuters. The
+5https://archive.ics.uci.edu/ml/datasets.html
+6https://github.com/franrole/cclust package/
+tree/master/datasets
+
+Table 2: Accuracy on 8 benchmark datasets
+datasets
+k-means
+KC
+CSPA
+HPGA
+NMFC
+WC
+L2CC
+RCC (our method)
+ES
+CorC
+CSTR
+0.45
+0.38
+0.50
+0.62
+0.56
+0.64
+0.61
+0.65
+0.64
+0.61
+Glass
+0.38
+0.45
+0.43
+0.40
+0.49
+0.49
+0.49
+0.52
+0.52
+0.50
+Ionosphere
+0.70
+0.71
+0.68
+0.52
+0.71
+0.71
+0.71
+0.72
+0.71
+0.70
+Iris
+0.83
+0.72
+0.86
+0.69
+0.89
+0.89
+0.86
+0.89
+0.88
+0.86
+Reuters
+0.45
+0.44
+0.43
+0.44
+0.43
+0.44
+0.43
+0.45
+0.44
+0.43
+Soybean
+0.72
+0.82
+0.70
+0.81
+0.89
+0.91
+0.76
+1.00
+0.98
+0.95
+Wine
+0.68
+0.68
+0.69
+0.52
+0.70
+0.72
+0.65
+0.96
+0.84
+0.89
+Zoo
+0.61
+0.59
+0.56
+0.58
+0.62
+0.70
+0.80
+0.84
+0.79
+0.82
+Table 3: Clustering accuracy on three multi-view datasets
+datasets
+LBP
+HOG
+GIST
+Classemes
+FC
+MVKmean
+AP
+LCC
+RCC (Our method)
+MSRC-v1
+0.4731
+0.6367
+0.6283
+0.5431
+0.7423
+0.7871
+0.5369
+0.7542
+0.8017
+Caltech-7
+0.5236
+0.5561
+0.5473
+0.4983
+0.6123
+0.6640
+0.5359
+0.6643
+0.6819
+Caltech-20
+0.3378
+0.3679
+0.3925
+0.3660
+0.5489
+0.5619
+0.3421
+0.6287
+0.6374
+features we adopt are represented using vector space model
+after removing the stop words and unnecessary tags and
+headers. In Reuter dataset, we use the ten most frequent cat-
+egories.
+In our experiment, the consensus co-association matrix
+is obtained by running k-means algorithm for 40 times and
+make an average of results using different initializations. All
+datasets have the ground-truth of clustering labels. We also
+compare against the following baseline methods:
+• k-means on original feature space
+• KC: k-means on the consensus similarity matrix;
+• NMFCC (Li, Ding, and Jordan 2007): NMF-based con-
+sensus clustering;
+• CSPA (Domeniconi and Al-Razgan 2009): cluster based
+similarity partition algorithm
+• HPGA (Strehl and Ghosh 2003): Hypergraph partitioning
+algorithm
+• WCC (Li and Ding 2008): weighted consensus clustering
+• L2CC: using standard ℓ2 loss for solving the consensus
+clustering on the consensus matrix;
+• RCC: proposed robust consensus clustering algorithm
+• Correlation Consensus (CorC) (Wang et al. 2014): Corre-
+lation based consensus clustering by exploiting the rank-
+ing of different views;
+• Ensemble Consensus(ES) (Zhou et al. 2015): ensemble
+consensus via exploiting Kullback-Leibler divergence of
+different aspects.
+Evaluation metrics As the other clustering tasks, we use
+the clustering accuracy as the metric to evaluate the per-
+formance of our algorithms. Clustering accuracy(ACC) is
+defined as,
+ACC =
+�n
+i=1 δ(li, map(ci))
+n
+,
+(30)
+where li is the true class label and ci is the obtained cluster
+label of xi, map(.) is the best mapping function, δ(x, y) is the
+delta function where δ(x, y) = 1 if x = y, and δ(x, y) = 0
+otherwise. The mapping function map(.) matches the true
+class label and the obtained clustering label, where the best
+mapping is solved by Hungarian algorithm. The larger, the
+better.
+Experiment result analysis We report clustering results
+in Table. 2. Clearly, our method is very robust and outper-
+forms the other methods. The accuracy gain is very signif-
+icant especially on dataset Soybean, Wine and zoo. In
+some cases, the clustering result from one execution might
+be severely biased. This can be addressed by running clus-
+tering multiple times to reduce the variance. The Correla-
+tion Consensus algorithm (Wang et al. 2014) is designed for
+multi-modal data, which did not show very promising per-
+formance for the multiple execution of datasets with differ-
+ent initializations.
+Experiment results on multi-view datasets
+Dataset Our algorithm can be easily extended for pro-
+cessing multi-view data when
+˜M of Eq.(17) is com-
+puted using multi-view features. We evaluate this over sev-
+eral public datasets: Caltech101 (Fei-Fei, Fergus, and Per-
+ona 2007) and Microsoft Research Cambridge Volume 1
+(MSRCV1) (Winn and Jojic 2005). MSRCV1 has images
+from 7 classes, i.e., airplane, bicycle, building, car, cow,
+face, and tree. Each class has 30 images. Caltech-101 is an
+image dataset of 8677 images and 101 object classes. We
+follow the same experimental procedure as (Dueck and Frey
+2007) and extract 7 and 20 classes for each experimental
+setup. We extract 1984-dimensional LBP (OJA 1996) fea-
+tures, 4000-dimensional GIST (Oliva and Torralba 2001)
+features, 768-dimensional HOG (Dalal and Triggs 2005)
+features, 2659-dimensional classemes (Torresani, Szummer,
+and Fitzgibbon 2010) features from each image.
+Comparison results We compare our methods against
+clustering using each view of features, i.e., LBP, HOG, GIST
+and Classemes, respectively. We also compare several multi-
+view clustering methods: Spectral clustering using simple
+multi-view feature concatenation (denoted as FC), multi-
+
+Figure 1: (left) eCPM cost c.d.f vs. eCPM bid; (right) win rate
+c.d.f vs. eCPM bid.
+view k-means (denoted as MVKmean) (Cai, Nie, and Huang
+2013), affinity propagation using multi-view features ( de-
+noted as AP) (Dueck and Frey 2007), low-rank consen-
+sus clustering (denoted as LCC) (Tao et al. 2017). Table. 3
+presents the clustering accuracy results, which demonstrates
+the superiority of using our method for solving multi-view
+clusering problems. Moreover, our method is flexible for in-
+corporating any view of features after properly feature ex-
+traction. Essentially, our method learns the low-rank sub-
+space using eigen-decomposition using co-association ma-
+trix from multi-view data, which plays the similar role as
+those in (Tao et al. 2017), (Cai, Nie, and Huang 2013).
+Applications in Advertising Segmentation
+Computational Advertising refers to serving the most rele-
+vant advertisement (ads for short) matching to the particular
+context in internet webs. One import problem is to segment
+advertisers into different segmentations and provide person-
+alized service for targeting selection (i.e., provide the best
+target audiences) and forecasting (i.e., predict clicks and
+conversions for advertisers in winning auctions).
+To cluster advertisers into different segmentations, we
+have three views of information: (i) advertisers’ textual de-
+scriptions; (ii) advertisers’ win-rate cumulative distribution
+function (i.e., c.d.f for short); (iii) advertisers’ eCPM cost
+c.d.f. In advertising, win rate is a percentage metric in
+programmatic media marketing that measures the number
+of impressions won over the number of impressions bid,
+and win rate cumulative distribution function (c.d.f) mea-
+sures the percentages of impressions won blow certain bids.
+eCPM cost is the effective cost per thousand impressions
+given the bid, and eCPM cost cumulative distribution func-
+tion (c.d.f) measures the cost of impressions below cer-
+tain bids. Fig.1 give an example of advertisers’ win-rate
+c.d.f and eCPM cost c.d.f (a.k.a bid landscape model
+that allows one to estimate performance of advertising cam-
+paigns under different bidding scenarios). We extract adver-
+tisers’ textual description for clustering purpose.
+Connection to the proposed robust clustering method
+Our method consists of two key steps: (i) generating clus-
+tering results using each view information and obtain the
+co-association matrix of ˜
+M of Eq.(7) (ii) using the robust
+consensus clustering model of Eq.(17) to generate the con-
+sensus clustering results.
+Similarity Computation using KS Statistics Given the
+pairwise similarity in each view, a natural way is to segment
+different them into different disjoint K subsets where each
+subset denotes a clustering using graph cut type algorithm
+(e.g., Normalized Cut). In order to generate the clustering
+results using win-rate c.d.f and eCPM cost c.d.f, we
+refer to Kolmogorov- Smirnov (KS for short) statistics (Kol-
+mogorov 1933), (Smirnov 1948), which offers a nonpara-
+metric way to quantify the distances between the empiri-
+cal distribution functions. The traditional way is to compare
+the mean or median of two distributions with parametric
+tests (using Z-test or t-test), or non-parametric tests (Mann-
+Whitney or Kruskal-Wallis test). However, these methods
+only consider the domain differences of two distributions,
+not the scale. Moreover, some parametric methods impose
+strong normal distribution assumption, which is not realistic
+for ad campaigns.
+Suppose we have independent identical distributed
+(i.i.d) samples X1, · · · , Xm of size m with cumulative
+distribution function (c.d.f) F(x) and the second i.i.d
+sample Y1, ..., Yn of size n with c.d.f G(x). One wants
+to test the null hypothesis (H0 : F = G) vs. the alternative
+(H1 : F ̸= G). Fm(x) and Gn(x) are the corresponding
+c.d.fs, where
+Fm(x) = 1
+m
+m
+�
+i=1
+I(Xi ≤ x),
+Gn(x) = 1
+n
+n
+�
+i=1
+I(Yi ≤ x).
+The KS test statistic is defined as the maximum value of the
+absolute difference between two c.d.f.s, i.e.,
+Dmn =
+� mn
+m + n sup
+x |Fm(x) − Gn(x)|
+(31)
+The distribution of KS test statistic doesn’t depend on the
+known distribution. In addition, it converge to KS distribu-
+tion, i.e.
+lim
+m,n→∞ P(Dmn ≤ z) = 1 − 2
+∞
+�
+i=1
+(−1)i−1exp(−2i2z2)
+(32)
+This nice property enables the wide application of KS-
+statistics for solving real-world problems. In the context of
+advertiser clustering, let fm(i) be the winning rate function
+c.d.f for advertiser i generated from m observations, and
+fn(j) be the win rate function c.d.f for advertiser j gen-
+erated from n observations, then distance dist(i, j) between
+i and j is defined as:
+dist(i, j|m, n) =
+� mn
+m + n sup
+x |fm(BFi) − fn(BFj)|.
+(33)
+Therefore the similarity (denoted as Swin
+ij ) of pairwise adver-
+tisers i, j is computed based on
+Swin
+ij
+=
+�
+1;
+if
+dist(i, j|m, n) ≤ Cα
+0;
+otherwise,
+(34)
+where Cα is the confidence value set corresponding to p-
+value. In particular, if one chooses a critical value Cα such
+
+eCPM cost vs. eCPM bid
+wining rate vs. eCPM bio
+1.0
+2.5
+0.8
+2.0
+0.6
+eCPM cost
+ rate
+1.5
+wining I
+0.4
+1.0
+0.2
+0.5
+0.0
+0.0
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+eCPM bid
+eCPM bidthat
+Pr(dist(i, j|m, n) > Cα) = α,
+then a band of width Cα around the distribution of
+dist(i, j|m, n) will entirely contain the estimated value with
+probability (1−α). eCPM cost similarity is computed using
+the same way.
+Experimental Results
+The ad campaign data are collected from a major web por-
+tal. In this study, we consider 1,268 advertisers over a week.
+As is introduced in Kolmogorov-Smirnov statistics, we set
+p-value = {0.01, 0.05, 0.1} (p = 0.01 lower bound and
+p = 0.1 upper bound). Table 4 shows the clustering number
+using the win rate c.d.f and eCPM cost c.d.f features
+respectively. Clearly, when p-value increases, the critical
+Cα-value for determination of similar campaigns decreases
+because there are smaller chances for advertisers falling in
+the range of P(dist(i, j|m, n)) > Cα using Eq.(34).
+Table 4: # of clusterings at different p
+Category
+p = 0.01
+p = 0.05
+p = 0.1
+win rate
+65
+47
+32
+eCPM
+73
+51
+40
+Consistency of clustering For any pairwise advertisers,
+if both win rate c.d.f and eCPM cost c.d.f cluster them
+to the same group (or the different group), then they are
+viewed as true consistency, otherwise, if one method clus-
+ters them in the same group while the other clusters them in
+different groups, then viewed as inconsistency (FPN). More
+mathematically, let Ci, Cj be the cluster label obtained us-
+ing eCPM cost for i, j and let ℓi, ℓj be the cluster using win
+rate for i and j , then
+TPN =
+�
+i<j
+��
+ICi=Cj
+� � �
+Iℓi=ℓj
+�� � ��
+ICi̸=Cj
+� � �
+Iℓi̸=ℓj
+��
+FPN =
+�
+i<j
+��
+ICi=Cj
+� � �
+Iℓi̸=ℓj
+�� � ��
+ICi̸=Cj
+� � �
+Iℓi=ℓj
+��
+Table 5: Consistency of clustering using win rate c.d.f
+and eCPM cost c.d.f
+Category
+Percentage %
+TPN
+74.21%
+FPN
+25.79%
+Table 5 shows the consensus clustering results using these
+two views of features. Clearly, around three fourth advertis-
+ers share very similar results even using different features.
+Clustering performance Given the fact that we do not
+have ground-truth for advertiser clustering results, we can-
+not directly evaluate the performance of our method. In-
+stead, we view the robust consensus clustering result as the
+advertiser segmentation result, and use it to re-generate the
+Table 6: Forecast impression error (the smaller, the better)
+Feature type
+# forecasting error
+win c.d.f
+18.10%
+eCPM c.d.f
+20.31%
+textual
+30.98%
+consensus clustering
+17.87%
+propose method
+15.96%
+Table 7: Forecast spend error (the smaller, the better)
+Feature type
+# forecasting error
+win c.d.f
+12.34%
+eCPM c.d.f
+13.56%
+textual
+25.78%
+consensus clustering
+12.78%
+propose method
+9.64%
+eCPM cost and win-rate c.d.ds using all the information
+from the same advertiser clusters. With the updated eCPM
+cost (Cui et al. 2011) and win-rate c.d.f.s, we forecast
+the clicks and spend, and compute the relative percentage er-
+rors to validate the performance of updated eCPM cost and
+win-rate distributions. In particular, if the clustering result
+is better, then the re-generated eCPM and win-rate distribu-
+tions are more accurate.
+# Impression = Total supply × win-rate
+# Spend
+= # Impression ×
+eCPM cost
+which is to say, the forecasted impression (# Impression) is
+equal to the total number of supplies by applying the win-
+rate, and the forecasted spend is equal to the forecasted im-
+pressions multiplied by eCPM cost for per impression. The
+accurate estimation of win-rate and eCPM will make the
+forecasted impressions and spend more accurate.
+Therefore, we re-generate the eCPM and win-rate distri-
+butions using the following several consensus clustering re-
+sults: (i) only ads textual; (ii) only win-rate; (iii) only eCPM
+cost; (iv) proposed robust consensus clustering; (v) consen-
+sus clustering using ℓ2 distance; and compare their perfor-
+mances. The mean of absolute percentage error is defined
+as: MAPE = 1
+n
+�n
+i=1
+|yi−ˆyi|
+yi
+, where yi is the ground-truth
+for advertiser i and ˆyi is the forecasted impression (or spend)
+for advertiser i. The smaller of these values, the better per-
+formance of clustering. Tables 6, 7 show the forecasted im-
+pression and spend errors, respectively. Thanks to the pro-
+posed robust consensus clustering algorithm, the forecast-
+ing error has been dropped significantly for both forecasted
+clicks and spend.
+Conclusion
+This paper presents a novel approach for consensus cluster-
+ing. The new clustering objective is more robust to noise
+and outliers. We formulate the problem rigorously and show
+that the optimal solution can be derived using ADMM algo-
+rithm. We apply our method to solve real-world advertiser
+segmentation problem, where the consensus clustering pro-
+duces better forecasting results.
+
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf,len=866
+page_content='Robust Consensus Clustering and its Applications for Advertising Forecasting  Deguang Kong*, Miao Lu, Konstantin Shmakov and Jian Yang  Yahoo Research, San Jose, California, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='A, 94089  doogkong@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='com, ml4ey@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='edu, kshmakov@yahooinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='com, jianyang@yahooinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='com  Abstract  Consensus clustering aggregates partitions in order to find  a better fit by reconciling clustering results from different  sources/executions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In practice, there exist noise and outliers  in clustering task, which, however, may significantly degrade  the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' To address this issue, we propose a novel  algorithm – robust consensus clustering that can find com-  mon ground truth among experts’ opinions, which tends to  be minimally affected by the bias caused by the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In  particular, we formalize the robust consensus clustering prob-  lem as a constraint optimization problem, and then derive an  effective algorithm upon alternating direction method of mul-  tipliers (ADMM) with rigorous convergence guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Our  method outperforms the baselines on benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We apply  the proposed method to the real-world advertising campaign  segmentation and forecasting tasks using the proposed con-  sensus clustering results based on the similarity computed  via Kolmogorov-Smirnov Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The accurate clustering  result is helpful for building the advertiser profiles so as to  perform the forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Introduction  Consensus clustering reconciles clustering information from  different sources (or different executions) to be better fit  than the existing clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The consensus cluster-  ing results, however can be biased due to different types of  features, different executions of algorithms, different defi-  nitions of distance metrics and even different parameter set-  tings, as is sharply observed by existing studies (Jain, Murty,  and Flynn 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' All these factors may lead to disparity  in clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The major consensus clustering algo-  rithms include Hypergraph partitioning (Strehl and Ghosh  2003), (Fern and Brodley 2004), voting approach (Dudoit  and Fridlyand 2003), (Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2003), mutual information  (Wang, She, and Cao 2013) (Topchy, Jain, and Punch 2004),  co-association approach (Xu and Wunsch 2005), mixture  model (Topchy, Jain, and Punch 2004)  (Azimi and Fern  This work was done during the authors were working at  Yahoo Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The correspondence should be addressed to  doogkong@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The views and conclusions contained in  this document are those of the author(s) and should not be inter-  preted as representing the official policies, either expressed or im-  plied, of any companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Copyright © 2019, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  2009), correlation consensus (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2014), ensemble  clustering (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2015), (Gao 2016), (Liu, Latecki,  and Yan 2010), (Huang, Lai, and Wang 2016), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  One key observation is that in consensus clustering  (Strehl and Ghosh 2003), if there exist noise and outliers  in one source of features in any execution of algorithm, the  clustering result might be significantly affected due to the  least square loss function used in most of the clustering  methods (such as k-means, Gaussian mixture model), be-  cause the errors are squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Even worse, most of the time,  the users have little prior knowledge of noise, which makes  the clustering result more unstable and much harder to inter-  pret with different initializations and parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' For  example, if one information source that we used for consen-  sus clustering is not accurate, when we align the consen-  sus clustering results against this “inaccurate” source, we  will suffer from these inaccurate annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The inaccu-  rate common characteristics extracted from the samples due  to the biased clustering results, are in fact, however, less gen-  eralizable to those unseen ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  To address these issues, this paper proposes a robust con-  sensus clustering schema that is minimally affected by the  outliers/noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In particular, we combine multiple experts’  opinions on data clustering using the robust ℓ1-loss func-  tion that aims to find the maximum consistency on the ex-  perts’ opinions with minimum conflict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Our work can be  viewed as an effective method of data clustering from het-  erogeneous information sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The proposed method is  practically feasible because it is independent of parameter  settings of each clustering algorithm before aggregating the  experts’ opinions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Driven by advertising applications, we ap-  ply the concensus clustering algorithm to cluster advertiser  profiles (Mahdian and Wang 2009) into different clusters so  as to accurately perform performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='. Click, Conver-  sions) forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  The main contribution of this paper is summarized as fol-  lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  To address the issue of consensus clustering performance  degradation in existence of noise and outliers, we rigor-  ously formulate the problem of robust consensus cluster-  ing as an optimization problem using the robust loss func-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  To find the best solution for robust consensus clustering,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='00717v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='LG]  27 Dec 2022  we develop an effective algorithm upon ADMM to de-  rive the optimal solution, whose convergence can be rig-  orously proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  We experimentally evaluate our proposed method on both  benchmarks and real-world advertiser segmentation and  forecasting tasks, and show that our method is effective  and efficient in producing the better clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' As  an application, the proposed algorithm is applied for ad-  vertising forecasting tasks with more stable and trustful  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Robust consensus clustering Model  Problem setting Assume we have n data points, each data  point can generate features from different views (total view  number=V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' For example, an image can generate features  using different descriptors, such as SIFT1, HOG2, CNN fea-  tures (Girshick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' More formally, let xv  i ∈ ℜpv  be v-th (1 ≤ v ≤ V ) view/modal feature of a data point i,  pv is the dimensionality of feature extracted from v-th view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Consider all data points,  Xv = [xv  1, xv  2, · · · , xv  n],  where each data column vector is xv  i ∈ ℜpv×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Each data  point has the ground-truth label yi ∈ ℜK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Simple concate-  nation of all data views gives  X =  �  ���  X1  X2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  XV  �  ��� = [x1, x2, · · · , xn],  xi =  �  �  �  �  x1  i  x2  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  xV  i  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Suppose we are given the clustering/partition results (P)  for data points {X1, X2, · · · , XV } from V different views,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', P = {G1, G2, · · · , GV }, where Gv ∈ {0, 1}n×K (for  K clusters) is the clustering result from view v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The clus-  ters/partitions assignment might be different for different  views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Let the connectivity matrix (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='a co-association ma-  trix) M ∈ {0, 1}n×n, where Mij(GV ) for view v between  data point i, j is:  Mij(Gv) =  �  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (i,j) belongs  to  the same  clustering  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  otherwise  Co-association consensus clustering  Standard consensus  clustering looks for a consensus clustering (based on major-  ity agreement) G∗ ∈ {0, 1}n×K, such that G∗ is closest to  the all the given partitions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  min  G∗ J(G∗) = 1  V  V  �  v=1  n  �  i=1  n  �  j=1  D(Mij(Gv) − Mij(G∗)),  (1)  where D(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=') is a distance function between the optimal solu-  tion and the solution from each view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Generally, least square  1https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='org/wiki/  Scale-invariant feature transform  2https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='org/wiki/  Histogram of oriented gradients  loss is used for function D(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' ):  min  G∗ J(G∗) = 1  V  V  �  v=1  n  �  i=1  n  �  j=1  �  Mij(Gv) − Mij(G∗)  �2  ,  (2)  Notice that Mij(Gv) = 0, 1, therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (2) can be  equivalently written as:  min  G∗ J2(G∗) = 1  V  V  �  v=1  n  �  i=1  n  �  j=1  |Mij(Gv) − Mij(G∗)|  = 1  V  V  �  v=1  ∥M(Gv) − M(G∗)∥1,  (3)  with ∥X∥1 := �  i,j |Xij|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Let  Gℓ2 = arg min  G∗  J(G∗)  (4)  Gℓ1 = arg min  G∗  J2(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (5)  Clearly,  Gℓ2 = Gℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (6)  Analysis Given little (or no) prior information of cluster-  ing result from each view, one natural way to compute the  associate between data points i and j is to get the expected  value of average association ˜  M = [ ˜  Mij], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  ˜  Mij = 1  V  V  �  v=1  Mij(Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (7)  Back to model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (3), the upper bound of J2 is given by:  J2 ≤ 1  V  V  �  v=1  ∥M(Gv) − ˜M∥1 + ∥ ˜M − M(G∗)∥1  = C + ∥ ˜M − M(G∗)∥1  = C + J3(G∗)  (8)  where  J3(G∗) = ∥ ˜M − M(G∗)∥1  (9)  C = 1  V  V  �  v=1  ∥M(Gv) − ˜M∥1  (10)  Note the first term is a constant C because it measures the  average difference between the final clustering assignment  and the average consensus association3  ˜M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The smaller  value of this term gives the closer distance between the final  clustering assignment and current partition from a specific  view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  On the other hand,  J3 ≤ 1  V  V  �  v=1  �  ∥M(Gv) − M(G∗)∥1 + ∥ ˜M − M(Gv)∥1  �  = J2(G∗) + C (11)  3The weighted average using attention can be adopted as well  with learned weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  This analysis indicates that the optimization of J2 can be  achieved by J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Clustering indicator One way to optimize J3 of ∥ ˜M −  M(G∗)∥1 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (9) is to assign the optimal clustering indi-  cator G ∈ {0, 1}n×K, with the constraint that �  k Gik = 1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', there is only one ‘1’ in each row of G, and the rest  are zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The connection between the averaged connectiv-  ity matrix M(G∗) and G is:  M(G∗) = G∗G∗⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (12)  Therefore the objective of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (9) becomes J4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  minG J4(G) = ∥ ˜M − GG⊺∥1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  G ∈ {0, 1}n×K, �  k Gik = 1  (13)  Relaxation in continuous domain  However, the major  difficulty of solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (13) is that is involves the dis-  crete clustering indicator G and the algorithm is NP-  complete (Filkov and Skiena 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Also the objective in-  volves non-smooth ℓ1 norm, which generally, is hard to han-  dle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Thus, as most of the spectral clustering solvers, we do  normalization on cluster indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In particular, we use  H ∈ ℜn×K to denote the new clustering indicator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  H = G(G⊺G)− 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Notice that  G⊺G =  �  �  �  �  �  n1  n2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  nk  �  �  �  �  � ,  (14)  where nk is the number of data points that falls in category  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Let diagonal matrix D ∈ ℜk×k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  D := G⊺G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  then  GG⊺ = HDH⊺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (15)  H⊺H = G(G⊺G)−1 = Ik,  (16)  where Ik is an identity matrix with size kxk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Therefore the  objective function to be solved becomes:  min  H,D J5(H, D) = ∥ ˜M − HDH⊺∥1  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  H⊺H = IK, D ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' D is diagonal, (17)  where G = HD  1  2 can be obtained in continous domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  In practice, the solution obtained from ˆM may not be ac-  curate due to noise or outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' For this reason, we term our  method as “robust” because we use ℓ1 distance to measure  the differences between the consensus optimal solution and  the solution computed from each view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The errors are not  squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Therefore, our model is capable of handling any  noises/outliers in some view/execution in data clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Further, we have:  Lemma 1 Set ˆM to be the normalized pairwise similarity  matrix ˆ  SS = D− 1  2 SSD− 1  2 (SS ∈ ℜn×n and Sij is the  similarity between data point i and j), using robust ℓ1-norm  as the loss function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  min  H J0(H) = ∥ ˆ  SS − HH⊺∥2  F  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  H⊺H = Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (18)  This is identical to normalized cut spectral clustering (Shi  and Malik 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Proof  Recall that in standard normalized cut spectral  clustering, the objective is:  min  H Tr(H⊺(I − ˆ  SS)H)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  H⊺H = Ik,  (19)  where ˆ  SS = D− 1  2 SSD− 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' This is equivalent to optimiz-  ing:  max  H Tr(H⊺ ˆ  SSH)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  H⊺H = Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (20)  Notice that  ∥ ˆ  SS − HH⊺∥2  F = Tr( ˆ  SS  ⊺ ˆ  SS + HH⊺H⊺H − 2 ˆ  SS  ⊺HH⊺),  Tr(HH⊺H⊺H)  =  const, and therefore minH ∥ ˆ  SS −  HH⊺∥2  F is equivalent to maxH Tr(H⊺ ˆ  SSH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' This com-  pletes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  In this paper next, we discuss how to solve the robust con-  sensus clustering model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Optimization Algorithm  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17) seems difficult to solve since it involves non-smooth  loss and high-order matrix optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We show how to  apply alternating direction method of multipliers (ADMM)  to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' ADMM method decomposes a large optimization  problem by breaking them into smaller pieces, each of which  are then easier to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' ADMM combines the benefits of  dual decomposition and augmented Lagrangian method for  constrained optimization problem (Bertsekas 1996) and has  been applied in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The problem solved by  ADMM method usually has the following general forms4,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  min  X,Z f(X) + g(Z),  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  AX + BZ = C(21)  After enforcing the Lagrangian multiplier while introducing  more variables, the problem can be solved alternatively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  ℓ(X, Z, µ) = f(X) + g(Z) + Ω⊤(AX + BZ − C)  +µ  2 ||AX + BZ − C||2  F ,  (22)  Xt+1 := arg min  X  ℓ(X, Zt, µt),  Zt+1 := arg min  Z  ℓ(Xt+1, Z, µt),  Ωt+1 := Ωt + µ(AXt+1 + BXt+1 − C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  and Ω is the augmented Lagrangian multiplier, and µ is the  non-negative step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  4Here X, Z can be vectors or matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Algorithm 1 Solving robust consensus clustering model of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17)  Input: clustering results from different views ˜  M, parameter ρ >  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Output: final clustering indicator H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Procedure:  1: Initialize E0, H0, D0, Ω0, µ0 > 0, t = 0  2: while Not converge do  3:  Update E via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (26)  4:  Update H via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (28)  5:  Update D via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (29)  6:  µt+1 := ρµt  7:  Ω := Ω + µ(E − ( ˜  M − HDH  ⊺))  8:  t := t + 1  9: end while  Optimization Algorithm to solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17)  According to ADMM algorithm, by imposing constraint  variable  E = ˜M − HDH  ⊺,  the problem of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17) is equivalent to solving,  min  E, H, D ∥E∥1,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  E − ( ˜M − HDH⊺) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', H  ⊺ ˜H = IK, D ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' D is diagonal (23)  In ADMM algorithm, to solve f(E) = ∥E∥1, under the con-  straint  h(E) = E − ( ˜M − HDHT ),  the ADMM function can be formulated as follows,  ℓ(E, H, D, Ω, µ) = f(E) + Ω⊤h(E) + µ  2 ||h(E)||2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  where Lagrange multiplier is Ω and µ is the penalty constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  We solve a sequence of subproblems  min  E,H,D ∥E∥1 + Ω  ⊤(E − ( ˜M − HDH  ⊺))  +µ  2 ∥E − ( ˜M − HDH  ⊺)∥2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (24)  with Ω and µ updated in a specified pattern:  Ω ← Ω + µ(E − ( ˜M − HDH  ⊺)),  µ ← ρµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  To solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (24), we search for optimal E, H, D itera-  tively until the algorithm converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Now we discuss how to  solve E, H, D in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='1 summarizes the complete  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Update E  To update the error matrix E, we derive Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (25)  with fixed H, D and obtain the following form:  min  E  µ  2 ||E − A||2  F + ||E||1  (25)  where  A = HDHT − ˜M + Ω  µ ,  It is well-known that the solution to the above LASSO type  problem (Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2009) is given by  Eij = sign(Aij) max(|Aij| − 1  µ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (26)  Table 1: dataset descriptions  datasets  #data points  # feature  #class  CSTR  475  1000  4  Glass  214  9  7  Ionosphere  351  34  2  Iris  150  4  3  Reuters  2900  1000  10  Soybean  47  35  4  Wine  178  13  3  Zoo  101  18  7  Update H, D  To update H, D while fixing E, we mini-  mize the relevant part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (24) which is  min  H,D  µ  2 ∥B − HDHT ∥2  F ,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', HT H = IK, D ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' D is diagonal (27)  where  B = ˜M − E + Ω  µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  It is easy to see the optimal solution H is given by the  k-largest eigenvectors of B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', H = [h1, h2, · · ·, hk],  Bhk = λkhk,  (28)  where λk is the associated eigen-value with respect to eigen-  vector hk, and Λ is a diagonal matrix, and  Λ = diag([λ1, λ2, · · ·, λk]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Then the optimal solution of D is given by  D = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (29)  The completes the algorithm for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Time Complexity Analysis Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17) is not a convex  problem, in each iteration, given µ, Σ, ρ, the algorithm will  find its local solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The convergence of ADMM algorithm  has been proved and widely discussed in (Bertsekas 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  The overall time cost for the algorithm depends on itera-  tions and time cost for each variable updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The compu-  tation of E takes O(nk2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The major burden is from the  updating of D and H, which requires computation of top k  eigen-vector of B,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', O(n3) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Overall, the cost of the  algorithm 1 is O(T(n3 + nk2)), where T is the iteration  number before convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' To make the solution scalable  for large-scale dataset, We can accelerate this via Hessen-  berg reduction and QR iteration with multi-thread solver us-  ing GPU acceleration (Gates, Haidar, and Dongarra 2014)  and also use divide-and-conquer (Talwalkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2013) to  accelerate the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
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+page_content='  In our experiment, the consensus co-association matrix  is obtained by running k-means algorithm for 40 times and  make an average of results using different initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' All  datasets have the ground-truth of clustering labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We also  compare against the following baseline methods:  k-means on original feature space  KC: k-means on the consensus similarity matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  NMFCC (Li, Ding, and Jordan 2007): NMF-based con-  sensus clustering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  CSPA (Domeniconi and Al-Razgan 2009): cluster based  similarity partition algorithm  HPGA (Strehl and Ghosh 2003): Hypergraph partitioning  algorithm  WCC (Li and Ding 2008): weighted consensus clustering  L2CC: using standard ℓ2 loss for solving the consensus  clustering on the consensus matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  RCC: proposed robust consensus clustering algorithm  Correlation Consensus (CorC) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2014): Corre-  lation based consensus clustering by exploiting the rank-  ing of different views;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Ensemble Consensus(ES) (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2015): ensemble  consensus via exploiting Kullback-Leibler divergence of  different aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Evaluation metrics As the other clustering tasks, we use  the clustering accuracy as the metric to evaluate the per-  formance of our algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Clustering accuracy(ACC) is  defined as,  ACC =  �n  i=1 δ(li, map(ci))  n  ,  (30)  where li is the true class label and ci is the obtained cluster  label of xi, map(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=') is the best mapping function, δ(x, y) is the  delta function where δ(x, y) = 1 if x = y, and δ(x, y) = 0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The mapping function map(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=') matches the true  class label and the obtained clustering label, where the best  mapping is solved by Hungarian algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The larger, the  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Experiment result analysis We report clustering results  in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Clearly, our method is very robust and outper-  forms the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The accuracy gain is very signif-  icant especially on dataset Soybean, Wine and zoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In  some cases, the clustering result from one execution might  be severely biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' This can be addressed by running clus-  tering multiple times to reduce the variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The Correla-  tion Consensus algorithm (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2014) is designed for  multi-modal data, which did not show very promising per-  formance for the multiple execution of datasets with differ-  ent initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Experiment results on multi-view datasets  Dataset Our algorithm can be easily extended for pro-  cessing multi-view data when  ˜M of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17) is com-  puted using multi-view features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We evaluate this over sev-  eral public datasets: Caltech101 (Fei-Fei, Fergus, and Per-  ona 2007) and Microsoft Research Cambridge Volume 1  (MSRCV1) (Winn and Jojic 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' MSRCV1 has images  from 7 classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', airplane, bicycle, building, car, cow,  face, and tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Each class has 30 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Caltech-101 is an  image dataset of 8677 images and 101 object classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We  follow the same experimental procedure as (Dueck and Frey  2007) and extract 7 and 20 classes for each experimental  setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We extract 1984-dimensional LBP (OJA 1996) fea-  tures, 4000-dimensional GIST (Oliva and Torralba 2001)  features, 768-dimensional HOG (Dalal and Triggs 2005)  features, 2659-dimensional classemes (Torresani, Szummer,  and Fitzgibbon 2010) features from each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Comparison results We compare our methods against  clustering using each view of features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', LBP, HOG, GIST  and Classemes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We also compare several multi-  view clustering methods: Spectral clustering using simple  multi-view feature concatenation (denoted as FC), multi-  Figure 1: (left) eCPM cost c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' eCPM bid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (right) win rate  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' eCPM bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  view k-means (denoted as MVKmean) (Cai, Nie, and Huang  2013), affinity propagation using multi-view features ( de-  noted as AP) (Dueck and Frey 2007), low-rank consen-  sus clustering (denoted as LCC) (Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 3  presents the clustering accuracy results, which demonstrates  the superiority of using our method for solving multi-view  clusering problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Moreover, our method is flexible for in-  corporating any view of features after properly feature ex-  traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Essentially, our method learns the low-rank sub-  space using eigen-decomposition using co-association ma-  trix from multi-view data, which plays the similar role as  those in (Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2017), (Cai, Nie, and Huang 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Applications in Advertising Segmentation  Computational Advertising refers to serving the most rele-  vant advertisement (ads for short) matching to the particular  context in internet webs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' One import problem is to segment  advertisers into different segmentations and provide person-  alized service for targeting selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', provide the best  target audiences) and forecasting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', predict clicks and  conversions for advertisers in winning auctions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  To cluster advertisers into different segmentations, we  have three views of information: (i) advertisers’ textual de-  scriptions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (ii) advertisers’ win-rate cumulative distribution  function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f for short);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (iii) advertisers’ eCPM cost  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In advertising, win rate is a percentage metric in  programmatic media marketing that measures the number  of impressions won over the number of impressions bid,  and win rate cumulative distribution function (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f) mea-  sures the percentages of impressions won blow certain bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  eCPM cost is the effective cost per thousand impressions  given the bid, and eCPM cost cumulative distribution func-  tion (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f) measures the cost of impressions below cer-  tain bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='1 give an example of advertisers’ win-rate  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f and eCPM cost c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='a bid landscape model  that allows one to estimate performance of advertising cam-  paigns under different bidding scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We extract adver-  tisers’ textual description for clustering purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Connection to the proposed robust clustering method  Our method consists of two key steps: (i) generating clus-  tering results using each view information and obtain the  co-association matrix of ˜  M of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (7) (ii) using the robust  consensus clustering model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (17) to generate the con-  sensus clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Similarity Computation using KS Statistics Given the  pairwise similarity in each view, a natural way is to segment  different them into different disjoint K subsets where each  subset denotes a clustering using graph cut type algorithm  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', Normalized Cut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In order to generate the clustering  results using win-rate c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f and eCPM cost c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f, we  refer to Kolmogorov- Smirnov (KS for short) statistics (Kol-  mogorov 1933), (Smirnov 1948), which offers a nonpara-  metric way to quantify the distances between the empiri-  cal distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The traditional way is to compare  the mean or median of two distributions with parametric  tests (using Z-test or t-test), or non-parametric tests (Mann-  Whitney or Kruskal-Wallis test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' However, these methods  only consider the domain differences of two distributions,  not the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Moreover, some parametric methods impose  strong normal distribution assumption, which is not realistic  for ad campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Suppose we have independent identical distributed  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d) samples X1, · · · , Xm of size m with cumulative  distribution function (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f) F(x) and the second i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d  sample Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=', Yn of size n with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f G(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' One wants  to test the null hypothesis (H0 : F = G) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' the alternative  (H1 : F ̸= G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Fm(x) and Gn(x) are the corresponding  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='fs, where  Fm(x) = 1  m  m  �  i=1  I(Xi ≤ x),  Gn(x) = 1  n  n  �  i=1  I(Yi ≤ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  The KS test statistic is defined as the maximum value of the  absolute difference between two c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=',  Dmn =  � mn  m + n sup  x |Fm(x) − Gn(x)|  (31)  The distribution of KS test statistic doesn’t depend on the  known distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In addition, it converge to KS distribu-  tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  lim  m,n→∞ P(Dmn ≤ z) = 1 − 2  ∞  �  i=1  (−1)i−1exp(−2i2z2)  (32)  This nice property enables the wide application of KS-  statistics for solving real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In the context of  advertiser clustering, let fm(i) be the winning rate function  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f for advertiser i generated from m observations, and  fn(j) be the win rate function c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f for advertiser j gen-  erated from n observations, then distance dist(i, j) between  i and j is defined as:  dist(i, j|m, n) =  � mn  m + n sup  x |fm(BFi) − fn(BFj)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  (33)  Therefore the similarity (denoted as Swin  ij ) of pairwise adver-  tisers i, j is computed based on  Swin  ij  =  �  1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  if  dist(i, j|m, n) ≤ Cα  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  otherwise,  (34)  where Cα is the confidence value set corresponding to p-  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In particular, if one chooses a critical value Cα such  eCPM cost vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' eCPM bid  wining rate vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' eCPM bio  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='6  eCPM cost  rate  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='5  wining I  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='0  0  5  10  15  20  0  5  10  15  20  eCPM bid  eCPM bidthat  Pr(dist(i, j|m, n) > Cα) = α,  then a band of width Cα around the distribution of  dist(i, j|m, n) will entirely contain the estimated value with  probability (1−α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' eCPM cost similarity is computed using  the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Experimental Results  The ad campaign data are collected from a major web por-  tal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In this study, we consider 1,268 advertisers over a week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  As is introduced in Kolmogorov-Smirnov statistics, we set  p-value = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='1} (p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='01 lower bound and  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='1 upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Table 4 shows the clustering number  using the win rate c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f and eCPM cost c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f features  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Clearly, when p-value increases, the critical  Cα-value for determination of similar campaigns decreases  because there are smaller chances for advertisers falling in  the range of P(dist(i, j|m, n)) > Cα using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Table 4: # of clusterings at different p  Category  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='01  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='05  p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='1  win rate  65  47  32  eCPM  73  51  40  Consistency of clustering For any pairwise advertisers,  if both win rate c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f and eCPM cost c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f cluster them  to the same group (or the different group), then they are  viewed as true consistency, otherwise, if one method clus-  ters them in the same group while the other clusters them in  different groups, then viewed as inconsistency (FPN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' More  mathematically, let Ci, Cj be the cluster label obtained us-  ing eCPM cost for i, j and let ℓi, ℓj be the cluster using win  rate for i and j , then  TPN =  �  i<j  ��  ICi=Cj  � � �  Iℓi=ℓj  �� � ��  ICi̸=Cj  � � �  Iℓi̸=ℓj  ��  FPN =  �  i<j  ��  ICi=Cj  � � �  Iℓi̸=ℓj  �� � ��  ICi̸=Cj  � � �  Iℓi=ℓj  ��  Table 5: Consistency of clustering using win rate c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f  and eCPM cost c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f  Category  Percentage %  TPN  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='21%  FPN  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='79%  Table 5 shows the consensus clustering results using these  two views of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Clearly, around three fourth advertis-  ers share very similar results even using different features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Clustering performance Given the fact that we do not  have ground-truth for advertiser clustering results, we can-  not directly evaluate the performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In-  stead, we view the robust consensus clustering result as the  advertiser segmentation result, and use it to re-generate the  Table 6: Forecast impression error (the smaller, the better)  Feature type  # forecasting error  win c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='10%  eCPM c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='31%  textual  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='98%  consensus clustering  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='87%  propose method  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='96%  Table 7: Forecast spend error (the smaller, the better)  Feature type  # forecasting error  win c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='34%  eCPM c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='56%  textual  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='78%  consensus clustering  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='78%  propose method  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='64%  eCPM cost and win-rate c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='ds using all the information  from the same advertiser clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' With the updated eCPM  cost (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' 2011) and win-rate c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='s, we forecast  the clicks and spend, and compute the relative percentage er-  rors to validate the performance of updated eCPM cost and  win-rate distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' In particular, if the clustering result  is better, then the re-generated eCPM and win-rate distribu-  tions are more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  # Impression = Total supply × win-rate  # Spend  = # Impression ×  eCPM cost  which is to say, the forecasted impression (# Impression) is  equal to the total number of supplies by applying the win-  rate, and the forecasted spend is equal to the forecasted im-  pressions multiplied by eCPM cost for per impression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The  accurate estimation of win-rate and eCPM will make the  forecasted impressions and spend more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Therefore, we re-generate the eCPM and win-rate distri-  butions using the following several consensus clustering re-  sults: (i) only ads textual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (ii) only win-rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (iii) only eCPM  cost;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (iv) proposed robust consensus clustering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' (v) consen-  sus clustering using ℓ2 distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' and compare their perfor-  mances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The mean of absolute percentage error is defined  as: MAPE = 1  n  �n  i=1  |yi−ˆyi|  yi  , where yi is the ground-truth  for advertiser i and ˆyi is the forecasted impression (or spend)  for advertiser i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The smaller of these values, the better per-  formance of clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Tables 6, 7 show the forecasted im-  pression and spend errors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' Thanks to the pro-  posed robust consensus clustering algorithm, the forecast-  ing error has been dropped significantly for both forecasted  clicks and spend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content='  Conclusion  This paper presents a novel approach for consensus cluster-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' The new clustering objective is more robust to noise  and outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We formulate the problem rigorously and show  that the optimal solution can be derived using ADMM algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
+page_content=' We apply our method to solve real-world advertiser  segmentation problem, where the consensus clustering pro-  duces better forecasting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdAyT4oBgHgl3EQf0fkK/content/2301.00717v1.pdf'}
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diff --git a/uNE4T4oBgHgl3EQfWQz4/content/tmp_files/2301.05032v1.pdf.txt b/uNE4T4oBgHgl3EQfWQz4/content/tmp_files/2301.05032v1.pdf.txt
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+Online Hyperparameter Optimization for Class-Incremental Learning
+Yaoyao Liu1
+Yingying Li2
+Bernt Schiele1
+Qianru Sun3
+1Max Planck Institute for Informatics, Saarland Informatics Campus
+2Computing and Mathematical Sciences, California Institute of Technology
+2School of Computing and Information Systems, Singapore Management University
+{yaoyao.liu, schiele}@mpi-inf.mpg.de
+yingli2@caltech.edu
+qianrusun@smu.edu.sg
+Abstract
+Class-incremental learning (CIL) aims to train a classi-
+fication model while the number of classes increases phase-
+by-phase.
+An inherent challenge of CIL is the stability-
+plasticity tradeoff, i.e., CIL models should keep stable to
+retain old knowledge and keep plastic to absorb new knowl-
+edge. However, none of the existing CIL models can achieve
+the optimal tradeoff in different data-receiving settings—
+where typically the training-from-half (TFH) setting needs
+more stability, but the training-from-scratch (TFS) needs
+more plasticity. To this end, we design an online learn-
+ing method that can adaptively optimize the tradeoff with-
+out knowing the setting as a priori. Specifically, we first
+introduce the key hyperparameters that influence the trade-
+off, e.g., knowledge distillation (KD) loss weights, learning
+rates, and classifier types. Then, we formulate the hyperpa-
+rameter optimization process as an online Markov Decision
+Process (MDP) problem and propose a specific algorithm
+to solve it. We apply local estimated rewards and a classic
+bandit algorithm Exp3 [4] to address the issues when apply-
+ing online MDP methods to the CIL protocol. Our method
+consistently improves top-performing CIL methods in both
+TFH and TFS settings, e.g., boosting the average accuracy
+of TFH and TFS by 2.2 percentage points on ImageNet-Full,
+compared to the state-of-the-art [23].1
+1. Introduction
+Real-world problems are ever-changing, with new con-
+cepts and new data being continuously observed.
+Ideal
+AI systems should have the ability to learn new concepts
+from the new data, also known as plasticity, while main-
+taining the ability to recognize old concepts, also known
+as stability. However, there is a fundamental tradeoff be-
+tween plasticity and stability: too much plasticity may re-
+1Code: https://class-il.mpi-inf.mpg.de/online/code/
+45
+50
+55
+60
+65
+70
+75
+Training from half (TFH)
+45
+50
+55
+60
+65
+Training from scratch (TFS)
+LwF
+iCaRL
+LUCIR
+AANets
+RMM
+Ours
+Figure 1. Average accuracy (%) on CIFAR-100 25-phase, using
+two data-receiving settings: 1) training-from-half (TFH): a large
+amount of data is available beforehand to pre-train the encoder; 2)
+training-from-scratch (TFS): classes come evenly in each phase.
+Dark blue and orange indicate the baselines and our method, re-
+spectively. Light-color circles are confidence intervals. Notice that
+methods with strong KD losses, e.g., LUCIR [14], AANets [22],
+and RMM [23], tend to provide worse performance in TFS than
+TFH, while methods with weak KD losses, e.g., iCaRL [30] and
+LwF [21], tend to provide worse performance in TFH than TFS.
+Our method uses an online learning algorithm to produce the key
+hyperparameters, e.g., the weights that control which KD losses
+are used. Thus, our method achieves the highest performance in
+both TFS and TFH.
+sult in the catastrophic forgetting of old concepts, while
+too much stability restricts the ability to adapt to new con-
+cepts [25, 26, 29]. To encourage related research, Rebuffi
+et al. [30] defined the class-incremental learning (CIL) pro-
+tocol, where the training samples of different classes come
+to the model phase-by-phase and most of the past data are
+removed from the memory.
+Recently, many methods have been proposed to balance
+the stability-plasticity tradeoff for different data-receiving
+settings of CIL. For example, the strong feature knowl-
+1
+arXiv:2301.05032v1  [cs.LG]  11 Jan 2023
+
+edge distillation (KD) loss function is usually adopted when
+a large amount of data is available beforehand (e.g., the
+training-from-half (TFH) setting) since it encourages sta-
+bility [14,22,23]; while the weak logit KD loss function is
+popular when data and classes are received evenly in each
+phase (e.g., the training-from-scratch (TFS) setting) since it
+provides more plasticity [7,21,30].
+Most CIL algorithms pre-fix the tradeoff balancing
+methods, usually according to which data-receiving setting
+will be used in the experiments. However, in real-world
+scenarios, it is difficult to anticipate how data will be re-
+ceived in the future. Hence, a pre-fixed method is no longer
+proper for balancing the stability and plasticity of the actual
+data stream, thus generating worse performance. This can
+also be validated in Figure 1. Notice that the methods with
+weak KD, e.g., iCaRL [30] and LwF [21], provide worse
+performance in TFH than in TFS since weak KD provides
+too much plasticity for TFH, while the methods with strong
+KD, e.g., LUCIR [14], AANets [22], and RMM [23], per-
+form worse in TFS than in TFH due to too much stability.
+Therefore, a natural question is:
+how to design an
+adaptive trade-off balancing method to achieve good per-
+formance without knowing how data will be received be-
+forehand? To tackle this, we propose an online-learning-
+inspired method to adaptively adjust key hyperparameters
+that affect the trade-off balancing performance in CIL. In
+our method, we introduce hyperparameters to control the
+choice of KD loss functions, learning rates, and classi-
+fier types, which are key algorithm choices that affect the
+tradeoff balancing performance.2 In this way, deciding this
+choice is transformed into a hyperparameter optimization
+(HO) problem.
+This HO problem cannot be directly solved because fu-
+ture data are not available. Thus, we borrow ideas from on-
+line learning, which is a widely adopted approach to adap-
+tively tune the decisions without knowing the future data a
+priori while still achieving good performance in hindsight.
+However, in CIL, our decisions affect not only the next
+phase but also all the future phases, which is different from
+the standard online learning setting [3]. To capture the de-
+pendence across phases, we formulate the HO problem in
+CIL as an online MDP, which is a generalized version of
+online learning. Further, we propose a new algorithm based
+on [10] to solve this online MDP problem. Our algorithm
+differs from the standard online MDP algorithm in [10] in
+two aspects:
+• In CIL, we cannot directly observe the reward (i.e., val-
+idation accuracy) because the validation data is not ac-
+2The impact of KD losses has been discussed before. Learning rates
+naturally affect how fast the model learns new concepts. We also adjust
+the classifier type because empirical results [14, 30] show that the near-
+est class mean (NCM) and fully-connected (FC) classifiers perform better
+under more plasticity and stability, respectively.
+cessible during training. To address this issue, we es-
+timate the reward by rebuilding local training and val-
+idation sets during policy learning in each phase, and
+computing the estimated reward on the local validation
+sets.
+• In CIL, we only have access to the model generated
+by the selected hyperparameters instead of the models
+generated by other hyperparameters. In other words,
+we only have bandit feedback instead of full feedback
+as assumed in [10]. To address this, we revise the al-
+gorithm in [10] by combining it with a classic bandit
+algorithm, Exp3 [4].
+Empirically, we find our method performs well consis-
+tently.
+We conduct extensive CIL experiments by plug-
+ging our method into three top-performing methods (LU-
+CIR [14], AANets [22], and RMM [23]) and testing them
+on three benchmarks (i.e., CIFAR-100, ImageNet-Subset,
+and ImageNet-Full). Our results show the consistent im-
+provements of the proposed method, e.g., boosting the av-
+erage accuracy of TFH and TFS by 2.2 percentage points
+on ImageNet-Full, compared to the state-of-the-art [23].
+Lastly, it is worth mentioning that our method can also
+be applied to optimize other key hyperparameters in CIL,
+e.g., memory allocation [23].
+Summary of our contributions. Our contributions are
+three-fold: 1) an online MDP formulation that allows on-
+line updates of hyperparameters that affect the balance of
+the stability and plasticity in CIL; 2) an Exp3-based online
+MDP algorithm to generate adaptive hyperparameters using
+bandit and estimated feedback; 3) extensive comparisons
+and visualizations for our method in three CIL benchmarks,
+taking top-performing methods as baselines.
+2. Related Work
+Class-incremental learning (CIL). There are three main
+lines of work to address the stability-plasticity trade-off in
+CIL. Distillation-based methods [8, 15, 16, 21, 27, 37, 41]
+introduce different knowledge distillation (KD) losses to
+consolidate previous knowledge when training the model
+on new data.
+The key idea is to enforce model predic-
+tion logits [21, 30], feature maps [8, 14], or topologies in
+the feature space [36] to be close to those of the pre-phase
+model. Memory-based methods [7,24,28,30,34] preserve a
+small number of old class data (called exemplars) and train
+the model on them together with new class data. Network-
+architecture-based methods [1,32,38–40] design incremen-
+tal network architectures by expanding the network capac-
+ity for new data or freezing partial network parameters to
+keep the knowledge of old classes. However, all the above
+methods are either too stable or too plastic to perform well
+in both TFS and TFH settings. Our method learns an on-
+line policy to generate hyperparameters that balance stabil-
+2
+
+Model Θ1
+Model
+New class 
+data D2
+Model
+New class 
+data D3
+Exemplars
+M2
+...
+...
+Phase 0
+Phase 1
+Phase 2
+...
+Exemplars
+New class 
+data D1
+(a)
+Eq. 3
+Eq. 4
+Model
+  Data   
+Data   
+Data sequence 
+from different classes
+Phase 1
+Model
+Phase i
+Policy
+Model
+Hyperparameters
+Train
+Train
+Train
+Initialize
+...
+Data   
+Policy
+Model
+Train
+Train
+...
+Initialize
+...
+...
+...
+Hyperparameters
+Figure 2. The computing flow of our method. We formulate the CIL task as an online MDP: each phase in CIL is a stage in the MDP, and
+the CIL models are the states. We train the policy to produce actions, which contain the hyperparameters we use in the CIL training. We
+illustrate the training process of each phase in Figure 3.
+ity and plasticity. Therefore, our method performs well in
+both TFS and TFH settings.
+Reinforcement learning (RL) aims to learn a policy in
+an environment, which is typically formulated as an MDP.
+Some CIL papers also deploy RL algorithms in their frame-
+works. Xu et al. [39] used RL to expand its backbone net-
+work when a new task arrives adaptively. Liu et al. [23]
+used RL to learn a policy to adjust the memory allocation
+between old and new class data dynamically along with the
+learning phases. Our method focuses on learning a pol-
+icy to produce key hyperparameters. Besides, the existing
+methods need to solve the complete MDP, which is time-
+consuming. Here, we formulate the CIL task as an online
+MDP. Thus, our method is more time-efficient.
+Online learning observes a stream of samples and makes a
+prediction for each element in the stream. There are mainly
+two settings in online learning: full feedback and bandit
+feedback. Full feedback means that the full reward function
+is given at each stage. It can be solved by Best-Expert algo-
+rithms [9]. Bandit feedback means that only the reward of
+the implemented decision is revealed. If the rewards are in-
+dependently drawn from a fixed and unknown distribution,
+we may use e.g., Thompson sampling [2] and UCB [5] to
+solve it. If the rewards are generated in a non-stochastic ver-
+sion, we can solve it by e.g., Exp3 [4]. Online MDP is an
+extension of online learning. Many studies [10,18–20] aim
+to solve it by converting it to online learning. In our case,
+we formulate the CIL as an online MDP and convert it into
+a classic online learning problem. The rewards in our MDP
+are non-stochastic because the training and validation data
+change in each phase. Therefore, we design our algorithm
+based on Exp3 [4].
+Hyperparameter optimization (HO). There are mainly
+two popular lines of HO methods: gradient-based and meta-
+learning-based. Gradient-based HO methods [6] make it
+possible to tune the entire weight vectors associated with
+a neural network layer as hyperparameters. Meta-learning-
+based HO methods [11] use a bilevel program to optimize
+the hyperparameters.
+However, all these methods only
+consider time-invariant environments. Our online method
+learns hyperparameters that adapt to the time-varying envi-
+ronments in CIL.
+3. Preliminaries
+Class-incremental learning (CIL). The general CIL
+pipeline is as follows. There are multiple phases during
+which the number of classes gradually increases to the max-
+imum [8,14,15,24]. In the 0-th phase, we observe data D0,
+and use it to learn an initial model Θ0. After this phase, we
+can only store a small subset of D0 (i.e., exemplars denoted
+as E0) in memory used as replay samples in later phases.
+In the i-th phase (i≥1), we get new class data Di and load
+exemplars E0:i−1=E0 ∪ · · · ∪ Ei−1 from the memory. Then,
+we initialize Θi with Θi−1, and train it using E0:i−1 ∪ Di.
+We evaluate the model Θi on a test set Q0:i for all classes
+observed so far. After that, we select exemplars E0:i from
+E0:i−1 ∪ Di and save them in the memory.
+Existing work mainly focus on two data-receiving set-
+tings: training-from-half (TFH) [14] and training-from-
+scratch (TFS) [30] settings. In TFH, we are given half of all
+classes in the 0-th phase, and observe the remaining classes
+evenly in the subsequent N phases. In TFS, we are given
+the same number of classes in all N phases.
+4. Methodology
+As illustrated in Figures 2 and 3, we formulate CIL as
+an online MDP and learn a policy to produce the hyperpa-
+rameters in each phase. In this section, we introduce the
+online MDP formulation, show optimizable hyperparame-
+ters, and provide an online learning algorithm to train the
+policy. Algorithm 1 summarizes the overall training steps
+of the proposed method. Algorithm 2 presents the training
+and evaluation process for a specific action.
+4.1. An Online MDP Formulation for CIL
+Optimizing hyperparameters in CIL should be online
+inherently: training and validation data changes in each
+phase, so the hyperparameters should be adjusted accord-
+ingly. Thus, it is intuitive to formulate the CIL as an online
+3
+
+8
+00
+2(β1, 1, 01, 入1)(Bi, Ti, Oi, i)Local validation
+Local training
+New data
+Old exemplars
+Temporary model
+Temporary model
+Train
+Test
+Policy
+CIL model
+Hyperparameters (action)
+Hyperparameters 
+Train
+Policy learning
+CIL training
+Update
+Validation acc. (reward)
+Figure 3. The training process of our online learning method in the i-th phase. It includes policy learning and CIL training. (a) Policy
+learning. 1) We construct a class-balanced subset from all training data as the local validation set and use the remaining data as the
+local training set. 2) We initialize the temporary model with Θi−1. 3) We sample an action using the current policy and deploy the
+hyperparameters on the temporary model according to the action. 4) We train it on the local training set for M1 epochs, and evaluate it on
+the local test set. 5) We use the validation accuracy as the reward and update the policy. We update the policy for T iterations by repeating
+Steps 2-5. (b) CIL training. We sample an action using the learned policy and deploy the hyperparameters on the CIL model. Then, we
+train the CIL model on all training data for M2 epochs. We set M1 = ⌈0.1M2⌉ to speed up the policy learning.
+MDP. In the following, we provide detailed formulations.
+Stages. Each phase in the CIL task can be viewed as a stage
+in the online MDP.
+States. The state should define the current situation of the
+intelligent agent. In CIL, we use the model Θi as the state
+of the i-th phase/stage. We use S to denote the state space.
+Actions. We use a vector consisting of the hyperparameters
+in the i-th phase as the action ai. When we take the ac-
+tion ai, we deploy the corresponding hyperparameters. We
+denote the action space as A. Please refer to the next sub-
+section, Optimizable Hyperparameters, for more details.
+Policy p={p(a|Θi)}a∈A is a probability distribution over
+the action space A, given the current state Θi.
+Environments. We define the training and validation data
+in each phase as the environment. In the i-th phase, the
+environment is Hi=(E0:i−1 ∪ Di, Q0:i), where E0:i−1 ∪ Di
+is the training data and Q0:i is the corresponding validation
+data. The environment is time-varying because we are given
+different training and validation data in each phase.
+Rewards.
+CIL aims to train a model that is efficient
+in recognizing all classes seen so far.
+Therefore, we
+use the validation accuracy as the reward in each phase.
+Our objective is to maximize a cumulative reward, i.e.,
+R = �N
+i=1 rHi(Θi, ai), where rHi(Θi, ai) denotes the i-
+th phase reward, i.e., the validation accuracy of Θi. The
+reward function rHi changes with Hi, so it is time-varying.
+4.2. Optimizable Hyperparameters
+In this part, we introduce the optimizable hyperparame-
+ters, and how to define the actions and action space based on
+the hyperparameters. We consider three kinds of hyperpa-
+rameters that significantly affect stability and plasticity: 1)
+KD loss weights, 2) learning rates, and 3) classified types.
+1) KD loss weights. We first introduce two KD losses (i.e.,
+logit and feature KD losses) and then show how to use the
+KD loss weights to balance them.
+Logit KD loss is proposed in [13] and widely applied in
+CIL methods [21,22,30]. Its motivation is to make the cur-
+rent model Θi mimic the prediction logits of the old model
+Θi−1:
+Llogi = −
+K
+�
+k=1
+ηk(µ(x; Θi−1)) log ηk(µ(x; Θi)),
+(1)
+where µ(x; Θ) is a function that maps the input mini-batch
+(x, y) to the prediction logits using the model Θ. ηk(v) =
+v1/τ
+k
+/ �
+j v1/τ
+j
+is a re-scaling function for the k-th class pre-
+diction logit and τ is a scalar set to be greater than 1.
+Feature KD loss [8,14] aims to enforce a stronger constraint
+on the previous knowledge by minimizing the cosine simi-
+larity between the features from the current model Θi and
+the old model Θi−1. It can be computed as follows,
+Lfeat = 1 − Sc(f(x; Θi), f(x; Θi−1)),
+(2)
+where f(x; Θ) denotes a function that maps the input image
+x to the features using the model Θ. Sc(v1, v2) denotes the
+cosine similarity between v1 and v2.
+The overall loss in the i-th phase is a weighted sum of the
+classification and different KD losses:
+Loverall = LCE + βiLlogi + γiLfeat,
+(3)
+4
+
+(Bi, Ti, Oi, i)where βi and γi are the weights of the logit and feature
+KD losses, respectively. LCE is the standard cross-entropy
+classification loss. Existing methods [14,21–23,30] can be
+viewed as using fixed heuristic KD loss weights, e.g., βi≡1
+and γi≡0 in iCaRL [21,30]. Instead, our method optimizes
+βi and γi online. Thus, we can balance the model’s stabil-
+ity and plasticity by adjusting βi and γi. We apply different
+weights for the logit and feature KD losses so that we can
+achieve fine-grained control over the intensity of knowledge
+distillation.
+2) Learning rate is another important hyperparameter that
+affects the model’s stability and plasticity. We empirically
+find that a lower learning rate makes the CIL model more
+stable, while a higher learning rate makes the CIL model
+more plastic. If we use λi to denote the learnable learning
+rate in the i-th phase, we update the CIL model as follows,
+Θi ← Θi − λi∇ΘiLoverall.
+(4)
+Another hyperparameter, the number of training epochs, has
+similar properties to the learning rate. We choose to opti-
+mize the learning rate and fix the number of epochs because
+it empirically works better with our online learning algo-
+rithm.
+3) Classifier type. Motivated by the empirical analysis, we
+consider two classifier types in our study: nearest class
+mean (NCM) [30, 35] and fully-connected (FC) [14, 24]
+classifiers. For the NCM classifier, we first compute the
+mean feature for each class using the new data and old ex-
+emplars. Then we perform a nearest neighbor search using
+the Euclidean distance on the L2 normalized mean features
+to get the final predictions. It is observed empirically that
+the NCM classifier tends to work better on the models with
+high plasticity, while the FC classifier performs better on
+the models with high stability [14,30]. Thus, we propose to
+use a hyperparameter, classifier type indicator δi, to control
+the final predictions during the evaluation:
+µ(x; Θi) = µncm(x; Θi)[δi = 1] + µfc(x; Θi)[δi = 0], (5)
+where δi ∈ {0, 1}, µncm and µfc are the predictions on the
+input image x using the NCM and FC classifiers, respec-
+tively.
+Summary: actions and action space. In summary, we
+define the action as ai=(βi, γi, λi, δi, ), which consists of
+the following hyperparameters: KD loss weights βi and γi,
+learning rate λi, and classifier type indicator δi. For the
+hyperparameters that may vary in a continuous range, we
+discretize them to define a finite action space.3 In the next
+subsection, we show how to learn the policy in each phase.
+3Though discretization suffers the curse of dimensionality, our experi-
+ments show that with a coarse grid, we already have significant improve-
+ments over pre-fixed hyperparameters.
+4.3. Policy Learning
+A common approach to solving an online MDP is to ap-
+proximate it as an online learning problem and solve it us-
+ing online learning algorithms [2,4,9]. We also take this ap-
+proach, and our approximation follows [10], which achieves
+the optimal regret. In their paper, Even-Dar et al. [10] relax
+the Markovian assumption of the MDP by decoupling the
+cumulative reward function and letting it be time-dependent
+so that they can solve the online MDP by standard online
+learning algorithms. Such a decoupling requires the fol-
+lowing assumptions. 1) Fast mixing: in CIL, the hyperpa-
+rameters in an early phase do not have much impact on the
+test accuracy of the classes observed in the current phase.
+2) The algorithm changes the hyperparameters slowly (this
+can be observed in Experiments & Figure 5). Thus, these
+assumptions fit our CIL problem.
+However, we cannot directly apply the algorithms pro-
+posed in [10] to our problem. It is because their paper as-
+sumes full feedback, i.e., we can observe the rewards of all
+actions in each phase. Therefore, its online learning prob-
+lem could be solved by Best Expert algorithms [9]. In CIL,
+we cannot observe any reward (i.e., validation accuracy) be-
+cause the validation data is not accessible during training.
+To address this issue, we rebuild the local training and val-
+idation sets in each phase. In this way, our problem has
+bandit feedback: we can compute the reward of the imple-
+mented action. Therefore, we can solve our online learning
+problem based on Exp3 [4], a famous bandit algorithm.
+In the following, we show how to rebuild the local train-
+ing and validation sets, compute the decoupled cumulative
+reward, and learn the policy with Exp3.
+Rebuilding local datasets. In the i-th phase, we need to ac-
+cess the validation set Q0:i to compute the reward (i.e., the
+validation accuracy). However, we are not allowed to use
+Q0:i during training because it violates the CIL benchmark
+protocol. Therefore, we replace Q0:i with a class-balanced
+subset B0:i sampled from the training data E0:i−1 ∪Di. B0:i
+contains the same number of samples for both the old and
+new classes. In this way, we can rebuild the local train-
+ing and validation sets, and obtain the local environment
+hi = ((E0:i−1 ∪ Di) \ B0:i, B0:i).
+Decoupled cumulative reward. We create the decoupled
+cumulative reward function ˆR based on the original cumu-
+lative reward function R = �N
+j=1 rHj(Θj, aj). In the i-th
+phase, we compute ˆR as follows,
+ˆR(ai, hi) =
+i−1
+�
+j=1
+rHj(Θj, aj)
+�
+��
+�
+Part I
++
+i+n
+�
+j=i
+rhi(Θj, ai)
+�
+��
+�
+Part II
+,
+(6)
+where Part I is the historical rewards from the 1-st phase
+to the (i-1)-th phase. It is a constant and doesn’t influence
+5
+
+policy optimization. Part II is the long-term reward of a
+time-invariant local MDP based on the local environment
+hi. We use Part II as an estimation of the future rewards,
+following [10]. Because we don’t know the total number of
+phases N during training, we assume there are n phases in
+the future. Furthermore, we fix the action ai in Part II to
+simplify the training process. Thus, ˆR can be reviewed as a
+function of ai and hi.
+Training policy with Exp3. Exp3 [4] introduces an auxil-
+iary variable w = {w(a)}a∈A. After updating w, we can
+determine the policy p={p(a|Θi)}a∈A by
+p = w/||w||.
+(7)
+The updating rule of w is provided below.
+In the 1-st phase, we initialize w as {1, . . . , 1}. In each
+phase i (i≥1), we update w for T iterations. In the t-th
+iteration, we sample an action at∼p, apply the action at
+to the CIL system, and compute the decoupled cumulative
+reward ˆR(at, hi) using Eq. 6. After that, we update w(at)
+in w as,
+w(at) ← w(at) exp(ξ ˆR(at, hi)/p(at|Θi)),
+(8)
+where ξ is a constant, which can be regarded as the learning
+rate in Exp3.
+5. Experiments
+We evaluate the proposed method on three CIL bench-
+marks, incorporate our method into three top-performing
+baseline methods, and boost their performances consis-
+tently in all settings. Below we describe the datasets and
+implementation details, followed by the results and analy-
+ses.
+5.1. Datasets and Implementation Details
+Datasets.
+We employ CIFAR-100 [17],
+ImageNet-
+Subset [30] (100 classes), and ImageNet-Full [31] (1000
+classes) as the benchmarks. We use the same data splits
+and class orders as the related work [22,23,30].
+Network architectures.
+We use a modified 32-layer
+ResNet [30] for CIFAR-100 and an 18-layer ResNet [12]
+for ImageNet, following [14, 22, 30].
+We deploy the
+AANets [22] for the experiments based on AANets and
+RMM [23]. Further, we use a cosine normalized classifier
+without bias terms as the FC classifier, following [14,24].
+Configurations. We discretize the hyperparameter search
+space into 50 actions, i.e., card(A)=50. We update the pol-
+icy for 25 iterations in each phase, i.e., T=25. For other
+configurations, we follow the corresponding baselines.
+5.2. Results and Analyses
+Tables 1 and 2 present the results of top-performing
+baselines w/ and w/o our method and some recent related
+Algorithm 1: Our CIL algorithm in Phase i (i≥1)
+Input : Old model Θi−1, training data E0:i−1 ∪ Di,
+validation data Q0:i,
+learnable parameters w, numbers of
+epochs M1 and M2.
+Output: New model Θi, new exemplars E0:i,
+learnable parameters w.
+1 // Policy learning
+2 if i=1 then
+3
+Initialize w = {1, . . . , 1};
+4 for t in i, ..., T do
+5
+Randomly sample a class-balanced subset B0:i
+from E0:i−1 ∪ Di,;
+6
+Create the local environment
+hi = ((E0:i−1 ∪ Di) \ B0:i, B0:i);
+7
+Set the policy p = w/||w||;
+8
+Sample an action at ∼ p;
+9
+for j in i, ..., i + n do
+10
+Train Θj for M1 epochs by Algorithm 2
+with inputs Θj−1, at, hi;
+11
+Collect the reward rhi(Θj, at);
+12
+Compute the cumulative reward ˆR(at, hi) by
+Eq. 6;
+13
+Update w by Eq. 8;
+14 // CIL training
+15 Sample an action ai ∼ p;
+16 Train Θi for M2 epochs by Algorithm 2 with inputs
+Θi−1, ai, Hi = (E0:i−1 ∪ Di, Q0:i);
+17 Select new exemplars E0:i from E0:i−1 ∪ Di using
+herding [30].
+Algorithm 2: Training and evaluation for action a
+Input : Old model Θold, action a = {β, γ, δ, λ},
+environment h = {T , Q}.
+Output: New model Θ, reward rh(Θ, a) (i.e., the
+validation accuracy).
+1 Initialize Θ with Θold;
+2 for epochs do
+3
+for mini-batch (x, y)∈ T do
+4
+Compute the loss Loverall(x, y; Θold, β, γ) by
+Eq. 3;
+5
+Update Θ by Eq. 4 with λ;
+6 for mini-batch (xval, yval)∈ Q do
+7
+Compute predictions µ(xval; Θ) by Eq. 5 with δ;
+8 Compute the reward rh(Θ, a) using
+{(µ(xval), yval)}(xval,yval)∈Q.
+work. Table 3 summarizes the results in seven ablative set-
+tings. Figure 4 compares the activation maps (using Grad-
+CAM [33]) produced by diffident methods in TFH and TFS.
+6
+
+Methods
+CIFAR-100, N=5
+CIFAR-100, N=25
+ImageNet-Subset, N=5
+ImageNet-Subset, N=25
+TFH
+TFS
+Avg.
+TFH
+TFS
+Avg.
+TFH
+TFS
+Avg.
+TFH
+TFS
+Avg.
+iCaRL [30]
+58.1
+64.0
+61.0
+48.1
+53.2
+50.7
+65.3
+70.4
+67.9
+53.0
+53.5
+53.3
+PODNet [8]
+64.7
+63.6
+64.2
+60.3
+45.3
+52.8
+64.3
+58.9
+61.6
+68.3
+39.1
+53.7
+DER [40]
+67.6
+72.3
+70.0
+65.5
+67.3
+66.4
+78.4
+76.9
+77.7
+75.4
+71.0
+73.2
+FOSTER [38]
+70.4
+72.5
+71.5
+63.8
+70.7
+67.3
+80.2
+78.3
+79.3
+69.3
+72.9
+71.1
+LUCIR [14]
+63.1±0.7 63.0±0.6 63.1±0.7
+57.5±0.4 49.2±0.5 53.4±0.5
+65.3±0.6 66.7±0.5 66.0±0.6
+61.4±0.7 46.2±0.8 53.8±0.8
+w/ ours
+63.9±0.6 64.9±0.5 64.4±0.6
+59.3±0.5 52.4±0.5 55.9±0.5
+70.6±0.7 68.4±0.6 69.5±0.7
+62.9±0.6 54.1±0.6 58.5±0.6
+↑0.8
+↑1.9
+↑1.3
+↑1.8
+↑3.2
+↑2.5
+↑5.3
+↑1.7
+↑3.5
+↑1.5
+↑7.9
+↑4.7
+AANets [22]
+65.3±0.4 63.1±0.3 64.2±0.4
+63.2±0.3 44.4±0.4 53.8±0.4
+77.0±0.7 68.9±0.6 73.0±0.7
+72.2±0.6 60.7±0.5 66.5±0.6
+w/ ours
+67.0±0.3 65.1±0.3 66.1±0.3
+64.1±0.4 50.3±0.5 57.2±0.5
+77.3±0.6 70.6±0.5 74.0±0.6
+72.9±0.5 64.8±0.5 68.9±0.5
+↑1.7
+↑2.0
+↑1.9
+↑0.9
+↑5.9
+↑3.4
+↑0.3
+↑1.7
+↑1.0
+↑0.7
+↑4.1
+↑2.4
+RMM [23]
+67.6±0.7 70.4±0.8 69.0±0.8
+65.6±0.6 58.4±0.6 62.0±0.6
+79.5±0.2 80.5±0.3 80.0±0.3
+75.0±0.3 71.6±0.3 73.3±0.3
+w/ ours
+70.8±0.7 72.7±0.6 71.8±0.7
+69.5±0.8 65.9±0.7 67.7±0.8
+81.0±0.3 82.2±0.4 81.6±0.4
+76.1±0.2 73.2±0.4 74.7±0.3
+↑3.2
+↑2.3
+↑2.8
+↑3.9
+↑7.5
+↑5.7
+↑1.5
+↑1.7
+↑1.6
+↑1.1
+↑1.6
+↑1.4
+Table 1. Average accuracy (%) across all phases on CIFAR-100 and ImageNet-Subset. The first block shows some recent CIL methods. The
+second block shows three top-performing baselines [14,22,23] w/ and w/o our method plugged in. “TFH” and “TFS” denote the training-
+from-half and training-from-scratch settings, respectively. “Avg.” shows the average of the “TFH” and “TFS” results. For “AANets” [22],
+we use its version based on PODNet [8]. We rerun the baselines using their open-source code in a unified setting for a fair comparison.
+Methods
+ImageNet-Full, N=5
+TFH
+TFS
+Avg.
+LUCIR [14]
+64.5±0.3
+62.7±0.4
+62.0±0.4
+w/ ours
+65.8±0.3 ↑1.3
+66.1±0.3 ↑3.4
+66.0±0.3 ↑2.4
+RMM [23]
+69.0±0.5
+66.1±0.4
+67.6±0.5
+w/ ours
+70.7±0.5 ↑1.7
+68.9±0.5 ↑2.8
+69.8±0.5 ↑2.2
+Table 2. Average accuracy (%) on ImageNet-Full.
+Figure 5 shows the values of hyperparameters produced by
+our method.
+Comparison with the state-of-the-art.
+Tables 1 and 2
+show that taking our method as a plug-in module for the
+state-of-the-art [23] and other baselines [14,22] consistently
+improves their performance. For example, RMM [23] w/
+ours gains 4.3 and 2.2 percentage points on CIFAR-100
+and ImageNet-Full, respectively. Interestingly, we find that
+we can surpass the baselines more when the number of
+phases N is larger. E.g., on CIFAR-100, our method im-
+proves RMM by 5.7 percentage points when N=25, while
+this number is 2.8 percentage points when N=5. Our expla-
+nation is that the forgetting problem is more serious when
+the number of phases is larger. Thus, we need better hyper-
+parameters to balance stability and plasticity.
+Ablation study. Table 3 concerns eight ablation settings,
+and shows the results in both TFH and TFS for different
+numbers of phases (N=5/25). The detailed analyses are as
+follows.
+1) First block. Row 1 shows the baseline [14].
+2) Second block: optimizable hyperparameters.
+In our
+study, we optimize three kinds of hyperparameters that af-
+fect the model’s stability and plasticity: KD loss weights
+(β, γ), learning rate λ, and classifier type indicator δ. Com-
+No.
+Optimizing
+N=5
+N=25
+(β, γ)
+(aδa,)
+(aλa)
+TFH
+TFS
+TFH
+TFS
+1
+Baseline
+63.11
+62.96
+57.47
+49.16
+2
+✓
+63.20
+63.60
+58.27
+50.91
+3
+✓
+✓
+63.23
+64.08
+58.20
+51.94
+4
+✓
+✓
+✓
+63.88
+64.92
+59.27
+52.44
+5
+Cross-val fixed
+63.33
+64.02
+57.50
+51.64
+6
+Offline RL [23]
+63.42
+63.88
+58.12
+51.53
+7
+Bilevel HO [11]
+63.20
+63.02
+57.56
+49.42
+Table 3. Ablation results (average accuracy %) on CIFAR-100.
+(β, γ) are KD loss weights. λ and δ denote learning rates and
+classifier types, respectively. The baseline is LUCIR [14]. Row 4
+shows our best result.
+paring Row 2 to Row 1, we can observe that optimizing
+the KD loss weights boosts the TFS accuracy more sig-
+nificantly. It is because the baseline, LUCIR [14], applies
+a strong regularization term (i.e., feature KD loss) which
+harms the TFH performance. Our method changes the reg-
+ularization term by adjusting the KD loss weights, so it
+achieves better performance. Comparing Row 3 to Row 2,
+we can see that optimizing the classifier type indicator δ
+performs further improvements, especially in TFS. It is be-
+cause the baseline deploys an FC classifier by default while
+our method learns to switch between different classifiers in
+different settings. Comparing Row 4 to Row 3, we can see
+the effectiveness of optimizing the learning rates in CIL. In
+summary, it is impressive that optimizing three kinds of hy-
+perparameters together achieves the best results.
+3) Third block: hyperparameter learning methods. For
+Row 5, we use cross-validation (i.e., all past, future, and
+7
+
+Training from half (TFH)
+Images iCaRL
+Ours
+Phase 0
+Phase 3
+Phase 5
+Training from scratch (TFS)
+Images iCaRL
+Ours
+Phase 0
+Phase 3
+Phase 5
+LUCIR
+LUCIR
+Figure 4. The activation maps using Grad-CAM [33] for the last
+phase model on ImageNet-Subset 5-phase. Samples are selected
+from the classes coming in the 0-th, 3-rd, and 5-th phases. Green
+ticks mean successful activation of discriminative features on ob-
+ject regions, while red crosses mean unsuccessful.
+validation data are accessible) to find a set of fixed hyperpa-
+rameters and apply them to all phases. We can see that Row
+5 results are consistently lower than ours in Row 4, although
+it can access more data. It shows that we need to update
+the hyperparameters online in different phases. For Row 6,
+we use the policy pre-trained in the 0-th phase of the target
+CIL task using the framework proposed in [23] (compared
+to ours, it is offline). Comparing Row 6 with Row 4, we
+are happy to see that our online learning algorithm achieves
+better performance than the offline RL while we use much
+less training time (see the analysis in the supplementary).
+For Row 7, we use the bilevel hyperparameter optimiza-
+tion method [11]. Comparing Row 7 with Row 4, we can
+observe that our method achieves more significant perfor-
+mance improvements. The reason is that [11] is designed
+for time-invariant environments, while our online algorithm
+can adapt to the time-varying environments in CIL..
+Visualizing activation maps. Figure 4 demonstrates the
+activation maps visualized by Grad-CAM [33] for the final
+model (obtained after five phases) on ImageNet-Subset 5-
+phase. The left and right sub-figures show the results for
+training-from-half (TFH) and training-from-scratch (TFS),
+respectively. We can observe: 1) LUCIR [14] makes predic-
+tions according to foreground (correct) and background (in-
+correct) regions in the TFH and TFS settings, respectively;
+2) iCaRL [30] behaves opposite to LUCIR in the two set-
+tings; 3) our method always makes predictions according to
+foreground (correct) regions in both TFH and TFS. The rea-
+sons are as follows. LUCIR applies a strong (feature) KD
+by default, so it performs better in TFH. iCaRL applies a
+weak (logit) KD by default, so it performs better in TFS.
+Our method can adjust the hyperparameters to change be-
+tween different KD losses, so it performs well in both set-
+tings.
+Hyperparameter values. Figure 5 shows the hyperparam-
+eter values produced by our policy on CIFAR-100 25-phase.
+1) KD loss weights β and γ. From the figure, we have two
+observations. a) The policy learns to produce a larger ini-
+5
+10
+15
+20
+#phases (TFH)
+0.2
+0.4
+0.6
+0.8
+1.0
+Logit KD ( )
+Feature KD ( )
+5
+10
+15
+20
+#phases (TFS)
+0
+0.20
+0.40
+0.60
+0.80
+1.00
+5
+10
+15
+20
+#phases (TFH)
+0
+0.05
+0.10
+0.15
+Learning rate ( )
+5
+10
+15
+20
+#phases (TFS)
+0
+0.05
+0.10
+0.15
+Figure 5. The hyperparameter values produced by our policy on
+CIFAR-100 25-phase for LUCIR [14] w/ ours. We smooth all
+curves with a rate of 0.8 for better visualization.
+tial value for γ and β in TFH and TFS, respectively. Our
+explanation is as follows. In TFH, we already have a pre-
+trained model, so we need a strong regularization term (i.e.,
+feature KD loss) to make the model more stable and avoid
+forgetting. In TFS, we start from random initialization, so
+we need a weak regularization term (i.e., logit KD loss)
+to improve the model’s plasticity.
+b) Both β and γ in-
+crease in TFH, while β decreases and γ increases in TFS.
+It is because we need stronger regularization to maintain
+the knowledge when more data is observed. The policy
+achieves that in different ways: it assigns higher weights
+for both KD losses in TFH, while it transfers from logit KD
+to feature KD in TFS.
+2) Learning rates λ. In Figure 5, we can observe that the
+learning rate in TFS is much higher than in TFH. It can be
+explained that 1) we need a higher learning rate in TFS as
+the model is trained from scratch and needs to capture more
+new knowledge; 2) we need a lower learning rate in TFH
+because we need to avoid forgetting the pre-trained model.
+3) Classifier type indicator δ. On CIFAR-100 25-phase, our
+policy learned to choose the NCM and FC classifiers in TFS
+and TFH, respectively.
+6. Conclusions
+In this study, we introduce a novel framework that allows
+us to optimize hyperparameters online to balance the stabil-
+ity and plasticity in CIL. To achieve this, we formulate the
+CIL task as an online MDP and learn a policy to produce
+the hyperparameters. Our approach is generic, and it can be
+easily applied to existing methods to achieve large-margin
+improvements in both TFS and TFH settings. It is worth
+mentioning that our method can also be applied to optimize
+other key hyperparameters in CIL.
+8
+
+ORAcknowledgments
+This research was supported by A*STAR under its AME
+YIRG Grant (Project No. A20E6c0101).
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+page_content='Online Hyperparameter Optimization for Class-Incremental Learning  Yaoyao Liu1  Yingying Li2  Bernt Schiele1  Qianru Sun3  1Max Planck Institute for Informatics, Saarland Informatics Campus  2Computing and Mathematical Sciences, California Institute of Technology  2School of Computing and Information Systems, Singapore Management University  {yaoyao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
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+page_content='sg  Abstract  Class-incremental learning (CIL) aims to train a classi-  fication model while the number of classes increases phase-  by-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  An inherent challenge of CIL is the stability-  plasticity tradeoff, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', CIL models should keep stable to  retain old knowledge and keep plastic to absorb new knowl-  edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' However, none of the existing CIL models can achieve  the optimal tradeoff in different data-receiving settings—  where typically the training-from-half (TFH) setting needs  more stability, but the training-from-scratch (TFS) needs  more plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' To this end, we design an online learn-  ing method that can adaptively optimize the tradeoff with-  out knowing the setting as a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Specifically, we first  introduce the key hyperparameters that influence the trade-  off, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', knowledge distillation (KD) loss weights, learning  rates, and classifier types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Then, we formulate the hyperpa-  rameter optimization process as an online Markov Decision  Process (MDP) problem and propose a specific algorithm  to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We apply local estimated rewards and a classic  bandit algorithm Exp3 [4] to address the issues when apply-  ing online MDP methods to the CIL protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our method  consistently improves top-performing CIL methods in both  TFH and TFS settings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', boosting the average accuracy  of TFH and TFS by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2 percentage points on ImageNet-Full,  compared to the state-of-the-art [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Introduction  Real-world problems are ever-changing, with new con-  cepts and new data being continuously observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Ideal  AI systems should have the ability to learn new concepts  from the new data, also known as plasticity, while main-  taining the ability to recognize old concepts, also known  as stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' However, there is a fundamental tradeoff be-  tween plasticity and stability: too much plasticity may re-  1Code: https://class-il.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='mpi-inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='de/online/code/  45  50  55  60  65  70  75  Training from half (TFH)  45  50  55  60  65  Training from scratch (TFS)  LwF  iCaRL  LUCIR  AANets  RMM  Ours  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Average accuracy (%) on CIFAR-100 25-phase, using  two data-receiving settings: 1) training-from-half (TFH): a large  amount of data is available beforehand to pre-train the encoder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 2)  training-from-scratch (TFS): classes come evenly in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Dark blue and orange indicate the baselines and our method, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Light-color circles are confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Notice that  methods with strong KD losses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', LUCIR [14], AANets [22],  and RMM [23], tend to provide worse performance in TFS than  TFH, while methods with weak KD losses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', iCaRL [30] and  LwF [21], tend to provide worse performance in TFH than TFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Our method uses an online learning algorithm to produce the key  hyperparameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', the weights that control which KD losses  are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, our method achieves the highest performance in  both TFS and TFH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  sult in the catastrophic forgetting of old concepts, while  too much stability restricts the ability to adapt to new con-  cepts [25, 26, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' To encourage related research, Rebuffi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' [30] defined the class-incremental learning (CIL) pro-  tocol, where the training samples of different classes come  to the model phase-by-phase and most of the past data are  removed from the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Recently, many methods have been proposed to balance  the stability-plasticity tradeoff for different data-receiving  settings of CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' For example, the strong feature knowl-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='05032v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='LG]  11 Jan 2023  edge distillation (KD) loss function is usually adopted when  a large amount of data is available beforehand (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', the  training-from-half (TFH) setting) since it encourages sta-  bility [14,22,23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' while the weak logit KD loss function is  popular when data and classes are received evenly in each  phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', the training-from-scratch (TFS) setting) since it  provides more plasticity [7,21,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Most CIL algorithms pre-fix the tradeoff balancing  methods, usually according to which data-receiving setting  will be used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' However, in real-world  scenarios, it is difficult to anticipate how data will be re-  ceived in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Hence, a pre-fixed method is no longer  proper for balancing the stability and plasticity of the actual  data stream, thus generating worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' This can  also be validated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Notice that the methods with  weak KD, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', iCaRL [30] and LwF [21], provide worse  performance in TFH than in TFS since weak KD provides  too much plasticity for TFH, while the methods with strong  KD, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', LUCIR [14], AANets [22], and RMM [23], per-  form worse in TFS than in TFH due to too much stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Therefore, a natural question is:  how to design an  adaptive trade-off balancing method to achieve good per-  formance without knowing how data will be received be-  forehand?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' To tackle this, we propose an online-learning-  inspired method to adaptively adjust key hyperparameters  that affect the trade-off balancing performance in CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In  our method, we introduce hyperparameters to control the  choice of KD loss functions, learning rates, and classi-  fier types, which are key algorithm choices that affect the  tradeoff balancing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2 In this way, deciding this  choice is transformed into a hyperparameter optimization  (HO) problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  This HO problem cannot be directly solved because fu-  ture data are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, we borrow ideas from on-  line learning, which is a widely adopted approach to adap-  tively tune the decisions without knowing the future data a  priori while still achieving good performance in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  However, in CIL, our decisions affect not only the next  phase but also all the future phases, which is different from  the standard online learning setting [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' To capture the de-  pendence across phases, we formulate the HO problem in  CIL as an online MDP, which is a generalized version of  online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Further, we propose a new algorithm based  on [10] to solve this online MDP problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our algorithm  differs from the standard online MDP algorithm in [10] in  two aspects:  In CIL, we cannot directly observe the reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', val-  idation accuracy) because the validation data is not ac-  2The impact of KD losses has been discussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Learning rates  naturally affect how fast the model learns new concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We also adjust  the classifier type because empirical results [14, 30] show that the near-  est class mean (NCM) and fully-connected (FC) classifiers perform better  under more plasticity and stability, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  cessible during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' To address this issue, we es-  timate the reward by rebuilding local training and val-  idation sets during policy learning in each phase, and  computing the estimated reward on the local validation  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  In CIL, we only have access to the model generated  by the selected hyperparameters instead of the models  generated by other hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In other words,  we only have bandit feedback instead of full feedback  as assumed in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' To address this, we revise the al-  gorithm in [10] by combining it with a classic bandit  algorithm, Exp3 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Empirically, we find our method performs well consis-  tently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  We conduct extensive CIL experiments by plug-  ging our method into three top-performing methods (LU-  CIR [14], AANets [22], and RMM [23]) and testing them  on three benchmarks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', CIFAR-100, ImageNet-Subset,  and ImageNet-Full).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our results show the consistent im-  provements of the proposed method, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', boosting the av-  erage accuracy of TFH and TFS by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2 percentage points  on ImageNet-Full, compared to the state-of-the-art [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Lastly, it is worth mentioning that our method can also  be applied to optimize other key hyperparameters in CIL,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', memory allocation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Summary of our contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our contributions are  three-fold: 1) an online MDP formulation that allows on-  line updates of hyperparameters that affect the balance of  the stability and plasticity in CIL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 2) an Exp3-based online  MDP algorithm to generate adaptive hyperparameters using  bandit and estimated feedback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 3) extensive comparisons  and visualizations for our method in three CIL benchmarks,  taking top-performing methods as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Related Work  Class-incremental learning (CIL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' There are three main  lines of work to address the stability-plasticity trade-off in  CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Distillation-based methods [8, 15, 16, 21, 27, 37, 41]  introduce different knowledge distillation (KD) losses to  consolidate previous knowledge when training the model  on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  The key idea is to enforce model predic-  tion logits [21, 30], feature maps [8, 14], or topologies in  the feature space [36] to be close to those of the pre-phase  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Memory-based methods [7,24,28,30,34] preserve a  small number of old class data (called exemplars) and train  the model on them together with new class data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Network-  architecture-based methods [1,32,38–40] design incremen-  tal network architectures by expanding the network capac-  ity for new data or freezing partial network parameters to  keep the knowledge of old classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' However, all the above  methods are either too stable or too plastic to perform well  in both TFS and TFH settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our method learns an on-  line policy to generate hyperparameters that balance stabil-  2  Model Θ1  Model  New class  data D2  Model  New class  data D3  Exemplars  M2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Phase 0  Phase 1  Phase 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Exemplars  New class  data D1  (a)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 3  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 4  Model  Data  Data  Data sequence  from different classes  Phase 1  Model  Phase i  Policy  Model  Hyperparameters  Train  Train  Train  Initialize  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Data  Policy  Model  Train  Train  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Initialize  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Hyperparameters  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The computing flow of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We formulate the CIL task as an online MDP: each phase in CIL is a stage in the MDP, and  the CIL models are the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We train the policy to produce actions, which contain the hyperparameters we use in the CIL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We  illustrate the training process of each phase in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  ity and plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Therefore, our method performs well in  both TFS and TFH settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Reinforcement learning (RL) aims to learn a policy in  an environment, which is typically formulated as an MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Some CIL papers also deploy RL algorithms in their frame-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' [39] used RL to expand its backbone net-  work when a new task arrives adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' [23]  used RL to learn a policy to adjust the memory allocation  between old and new class data dynamically along with the  learning phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our method focuses on learning a pol-  icy to produce key hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Besides, the existing  methods need to solve the complete MDP, which is time-  consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Here, we formulate the CIL task as an online  MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, our method is more time-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Online learning observes a stream of samples and makes a  prediction for each element in the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' There are mainly  two settings in online learning: full feedback and bandit  feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Full feedback means that the full reward function  is given at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It can be solved by Best-Expert algo-  rithms [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Bandit feedback means that only the reward of  the implemented decision is revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' If the rewards are in-  dependently drawn from a fixed and unknown distribution,  we may use e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', Thompson sampling [2] and UCB [5] to  solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' If the rewards are generated in a non-stochastic ver-  sion, we can solve it by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', Exp3 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Online MDP is an  extension of online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Many studies [10,18–20] aim  to solve it by converting it to online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In our case,  we formulate the CIL as an online MDP and convert it into  a classic online learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The rewards in our MDP  are non-stochastic because the training and validation data  change in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Therefore, we design our algorithm  based on Exp3 [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Hyperparameter optimization (HO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' There are mainly  two popular lines of HO methods: gradient-based and meta-  learning-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Gradient-based HO methods [6] make it  possible to tune the entire weight vectors associated with  a neural network layer as hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Meta-learning-  based HO methods [11] use a bilevel program to optimize  the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  However, all these methods only  consider time-invariant environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our online method  learns hyperparameters that adapt to the time-varying envi-  ronments in CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Preliminaries  Class-incremental learning (CIL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The general CIL  pipeline is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' There are multiple phases during  which the number of classes gradually increases to the max-  imum [8,14,15,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In the 0-th phase, we observe data D0,  and use it to learn an initial model Θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' After this phase, we  can only store a small subset of D0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', exemplars denoted  as E0) in memory used as replay samples in later phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  In the i-th phase (i≥1), we get new class data Di and load  exemplars E0:i−1=E0 ∪ · · · ∪ Ei−1 from the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Then,  we initialize Θi with Θi−1, and train it using E0:i−1 ∪ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  We evaluate the model Θi on a test set Q0:i for all classes  observed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' After that, we select exemplars E0:i from  E0:i−1 ∪ Di and save them in the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Existing work mainly focus on two data-receiving set-  tings: training-from-half (TFH) [14] and training-from-  scratch (TFS) [30] settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In TFH, we are given half of all  classes in the 0-th phase, and observe the remaining classes  evenly in the subsequent N phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In TFS, we are given  the same number of classes in all N phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Methodology  As illustrated in Figures 2 and 3, we formulate CIL as  an online MDP and learn a policy to produce the hyperpa-  rameters in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In this section, we introduce the  online MDP formulation, show optimizable hyperparame-  ters, and provide an online learning algorithm to train the  policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Algorithm 1 summarizes the overall training steps  of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Algorithm 2 presents the training  and evaluation process for a specific action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' An Online MDP Formulation for CIL  Optimizing hyperparameters in CIL should be online  inherently: training and validation data changes in each  phase, so the hyperparameters should be adjusted accord-  ingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, it is intuitive to formulate the CIL as an online  3  8  00  2(β1, 1, 01, 入1)(Bi, Ti, Oi, i)Local validation  Local training  New data  Old exemplars  Temporary model  Temporary model  Train  Test  Policy  CIL model  Hyperparameters (action)  Hyperparameters  Train  Policy learning  CIL training  Update  Validation acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' (reward)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The training process of our online learning method in the i-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It includes policy learning and CIL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' (a) Policy  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 1) We construct a class-balanced subset from all training data as the local validation set and use the remaining data as the  local training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 2) We initialize the temporary model with Θi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 3) We sample an action using the current policy and deploy the  hyperparameters on the temporary model according to the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 4) We train it on the local training set for M1 epochs, and evaluate it on  the local test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 5) We use the validation accuracy as the reward and update the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We update the policy for T iterations by repeating  Steps 2-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' (b) CIL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We sample an action using the learned policy and deploy the hyperparameters on the CIL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Then, we  train the CIL model on all training data for M2 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We set M1 = ⌈0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='1M2⌉ to speed up the policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In the following, we provide detailed formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Each phase in the CIL task can be viewed as a stage  in the online MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The state should define the current situation of the  intelligent agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In CIL, we use the model Θi as the state  of the i-th phase/stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We use S to denote the state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We use a vector consisting of the hyperparameters  in the i-th phase as the action ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' When we take the ac-  tion ai, we deploy the corresponding hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We  denote the action space as A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Please refer to the next sub-  section, Optimizable Hyperparameters, for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Policy p={p(a|Θi)}a∈A is a probability distribution over  the action space A, given the current state Θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We define the training and validation data  in each phase as the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In the i-th phase, the  environment is Hi=(E0:i−1 ∪ Di, Q0:i), where E0:i−1 ∪ Di  is the training data and Q0:i is the corresponding validation  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The environment is time-varying because we are given  different training and validation data in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  CIL aims to train a model that is efficient  in recognizing all classes seen so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Therefore, we  use the validation accuracy as the reward in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Our objective is to maximize a cumulative reward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=',  R = �N  i=1 rHi(Θi, ai), where rHi(Θi, ai) denotes the i-  th phase reward, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', the validation accuracy of Θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The  reward function rHi changes with Hi, so it is time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Optimizable Hyperparameters  In this part, we introduce the optimizable hyperparame-  ters, and how to define the actions and action space based on  the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We consider three kinds of hyperpa-  rameters that significantly affect stability and plasticity: 1)  KD loss weights, 2) learning rates, and 3) classified types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  1) KD loss weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We first introduce two KD losses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=',  logit and feature KD losses) and then show how to use the  KD loss weights to balance them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Logit KD loss is proposed in [13] and widely applied in  CIL methods [21,22,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Its motivation is to make the cur-  rent model Θi mimic the prediction logits of the old model  Θi−1:  Llogi = −  K  �  k=1  ηk(µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θi−1)) log ηk(µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θi)),  (1)  where µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θ) is a function that maps the input mini-batch  (x, y) to the prediction logits using the model Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' ηk(v) =  v1/τ  k  / �  j v1/τ  j  is a re-scaling function for the k-th class pre-  diction logit and τ is a scalar set to be greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Feature KD loss [8,14] aims to enforce a stronger constraint  on the previous knowledge by minimizing the cosine simi-  larity between the features from the current model Θi and  the old model Θi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It can be computed as follows,  Lfeat = 1 − Sc(f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θi), f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θi−1)),  (2)  where f(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θ) denotes a function that maps the input image  x to the features using the model Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Sc(v1, v2) denotes the  cosine similarity between v1 and v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  The overall loss in the i-th phase is a weighted sum of the  classification and different KD losses:  Loverall = LCE + βiLlogi + γiLfeat,  (3)  4  (Bi, Ti, Oi, i)where βi and γi are the weights of the logit and feature  KD losses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' LCE is the standard cross-entropy  classification loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Existing methods [14,21–23,30] can be  viewed as using fixed heuristic KD loss weights, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', βi≡1  and γi≡0 in iCaRL [21,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Instead, our method optimizes  βi and γi online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, we can balance the model’s stabil-  ity and plasticity by adjusting βi and γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We apply different  weights for the logit and feature KD losses so that we can  achieve fine-grained control over the intensity of knowledge  distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  2) Learning rate is another important hyperparameter that  affects the model’s stability and plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We empirically  find that a lower learning rate makes the CIL model more  stable, while a higher learning rate makes the CIL model  more plastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' If we use λi to denote the learnable learning  rate in the i-th phase, we update the CIL model as follows,  Θi ← Θi − λi∇ΘiLoverall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  (4)  Another hyperparameter, the number of training epochs, has  similar properties to the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We choose to opti-  mize the learning rate and fix the number of epochs because  it empirically works better with our online learning algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  3) Classifier type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Motivated by the empirical analysis, we  consider two classifier types in our study: nearest class  mean (NCM) [30, 35] and fully-connected (FC) [14, 24]  classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' For the NCM classifier, we first compute the  mean feature for each class using the new data and old ex-  emplars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Then we perform a nearest neighbor search using  the Euclidean distance on the L2 normalized mean features  to get the final predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It is observed empirically that  the NCM classifier tends to work better on the models with  high plasticity, while the FC classifier performs better on  the models with high stability [14,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, we propose to  use a hyperparameter, classifier type indicator δi, to control  the final predictions during the evaluation:  µ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θi) = µncm(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θi)[δi = 1] + µfc(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θi)[δi = 0], (5)  where δi ∈ {0, 1}, µncm and µfc are the predictions on the  input image x using the NCM and FC classifiers, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Summary: actions and action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In summary, we  define the action as ai=(βi, γi, λi, δi, ), which consists of  the following hyperparameters: KD loss weights βi and γi,  learning rate λi, and classifier type indicator δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' For the  hyperparameters that may vary in a continuous range, we  discretize them to define a finite action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='3 In the next  subsection, we show how to learn the policy in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  3Though discretization suffers the curse of dimensionality, our experi-  ments show that with a coarse grid, we already have significant improve-  ments over pre-fixed hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Policy Learning  A common approach to solving an online MDP is to ap-  proximate it as an online learning problem and solve it us-  ing online learning algorithms [2,4,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We also take this ap-  proach, and our approximation follows [10], which achieves  the optimal regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In their paper, Even-Dar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' [10] relax  the Markovian assumption of the MDP by decoupling the  cumulative reward function and letting it be time-dependent  so that they can solve the online MDP by standard online  learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Such a decoupling requires the fol-  lowing assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 1) Fast mixing: in CIL, the hyperpa-  rameters in an early phase do not have much impact on the  test accuracy of the classes observed in the current phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  2) The algorithm changes the hyperparameters slowly (this  can be observed in Experiments & Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, these  assumptions fit our CIL problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  However, we cannot directly apply the algorithms pro-  posed in [10] to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It is because their paper as-  sumes full feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', we can observe the rewards of all  actions in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Therefore, its online learning prob-  lem could be solved by Best Expert algorithms [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In CIL,  we cannot observe any reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', validation accuracy) be-  cause the validation data is not accessible during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  To address this issue, we rebuild the local training and val-  idation sets in each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In this way, our problem has  bandit feedback: we can compute the reward of the imple-  mented action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Therefore, we can solve our online learning  problem based on Exp3 [4], a famous bandit algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  In the following, we show how to rebuild the local train-  ing and validation sets, compute the decoupled cumulative  reward, and learn the policy with Exp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Rebuilding local datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In the i-th phase, we need to ac-  cess the validation set Q0:i to compute the reward (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', the  validation accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' However, we are not allowed to use  Q0:i during training because it violates the CIL benchmark  protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Therefore, we replace Q0:i with a class-balanced  subset B0:i sampled from the training data E0:i−1 ∪Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' B0:i  contains the same number of samples for both the old and  new classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In this way, we can rebuild the local train-  ing and validation sets, and obtain the local environment  hi = ((E0:i−1 ∪ Di) \\ B0:i, B0:i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Decoupled cumulative reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We create the decoupled  cumulative reward function ˆR based on the original cumu-  lative reward function R = �N  j=1 rHj(Θj, aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In the i-th  phase, we compute ˆR as follows,  ˆR(ai, hi) =  i−1  �  j=1  rHj(Θj, aj)  �  ��  �  Part I  +  i+n  �  j=i  rhi(Θj, ai)  �  ��  �  Part II  ,  (6)  where Part I is the historical rewards from the 1-st phase  to the (i-1)-th phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It is a constant and doesn’t influence  5  policy optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Part II is the long-term reward of a  time-invariant local MDP based on the local environment  hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We use Part II as an estimation of the future rewards,  following [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Because we don’t know the total number of  phases N during training, we assume there are n phases in  the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Furthermore, we fix the action ai in Part II to  simplify the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, ˆR can be reviewed as a  function of ai and hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Training policy with Exp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Exp3 [4] introduces an auxil-  iary variable w = {w(a)}a∈A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' After updating w, we can  determine the policy p={p(a|Θi)}a∈A by  p = w/||w||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  (7)  The updating rule of w is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  In the 1-st phase, we initialize w as {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' , 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In each  phase i (i≥1), we update w for T iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In the t-th  iteration, we sample an action at∼p, apply the action at  to the CIL system, and compute the decoupled cumulative  reward ˆR(at, hi) using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' After that, we update w(at)  in w as,  w(at) ← w(at) exp(ξ ˆR(at, hi)/p(at|Θi)),  (8)  where ξ is a constant, which can be regarded as the learning  rate in Exp3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Experiments  We evaluate the proposed method on three CIL bench-  marks, incorporate our method into three top-performing  baseline methods, and boost their performances consis-  tently in all settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Below we describe the datasets and  implementation details, followed by the results and analy-  ses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Datasets and Implementation Details  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  We employ CIFAR-100 [17],  ImageNet-  Subset [30] (100 classes), and ImageNet-Full [31] (1000  classes) as the benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We use the same data splits  and class orders as the related work [22,23,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Network architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  We use a modified 32-layer  ResNet [30] for CIFAR-100 and an 18-layer ResNet [12]  for ImageNet, following [14, 22, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  We deploy the  AANets [22] for the experiments based on AANets and  RMM [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Further, we use a cosine normalized classifier  without bias terms as the FC classifier, following [14,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We discretize the hyperparameter search  space into 50 actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', card(A)=50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We update the pol-  icy for 25 iterations in each phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', T=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' For other  configurations, we follow the corresponding baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Results and Analyses  Tables 1 and 2 present the results of top-performing  baselines w/ and w/o our method and some recent related  Algorithm 1: Our CIL algorithm in Phase i (i≥1)  Input : Old model Θi−1, training data E0:i−1 ∪ Di,  validation data Q0:i,  learnable parameters w, numbers of  epochs M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Output: New model Θi, new exemplars E0:i,  learnable parameters w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  1 // Policy learning  2 if i=1 then  3  Initialize w = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' , 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  4 for t in i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', T do  5  Randomly sample a class-balanced subset B0:i  from E0:i−1 ∪ Di,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  6  Create the local environment  hi = ((E0:i−1 ∪ Di) \\ B0:i, B0:i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  7  Set the policy p = w/||w||;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  8  Sample an action at ∼ p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  9  for j in i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', i + n do  10  Train Θj for M1 epochs by Algorithm 2  with inputs Θj−1, at, hi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  11  Collect the reward rhi(Θj, at);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  12  Compute the cumulative reward ˆR(at, hi) by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  13  Update w by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  14 // CIL training  15 Sample an action ai ∼ p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  16 Train Θi for M2 epochs by Algorithm 2 with inputs  Θi−1, ai, Hi = (E0:i−1 ∪ Di, Q0:i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  17 Select new exemplars E0:i from E0:i−1 ∪ Di using  herding [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Algorithm 2: Training and evaluation for action a  Input : Old model Θold, action a = {β, γ, δ, λ},  environment h = {T , Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Output: New model Θ, reward rh(Θ, a) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', the  validation accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  1 Initialize Θ with Θold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  2 for epochs do  3  for mini-batch (x, y)∈ T do  4  Compute the loss Loverall(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θold, β, γ) by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  5  Update Θ by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 4 with λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  6 for mini-batch (xval, yval)∈ Q do  7  Compute predictions µ(xval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Θ) by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 5 with δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  8 Compute the reward rh(Θ, a) using  {(µ(xval), yval)}(xval,yval)∈Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Table 3 summarizes the results in seven ablative set-  tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Figure 4 compares the activation maps (using Grad-  CAM [33]) produced by diffident methods in TFH and TFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  6  Methods  CIFAR-100, N=5  CIFAR-100, N=25  ImageNet-Subset, N=5  ImageNet-Subset, N=25  TFH  TFS  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  TFH  TFS  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  TFH  TFS  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  TFH  TFS  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  iCaRL [30]  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='1  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='0  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='0  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='1  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='7  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='3  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='4  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='9  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
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+page_content=' Average accuracy (%) across all phases on CIFAR-100 and ImageNet-Subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
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+page_content=' “TFH” and “TFS” denote the training-  from-half and training-from-scratch settings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
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+page_content='  Methods  ImageNet-Full, N=5  TFH  TFS  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
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+page_content='5  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='4  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='5  w/ ours  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='5 ↑1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='7  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='5 ↑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='5 ↑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Average accuracy (%) on ImageNet-Full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Figure 5 shows the values of hyperparameters produced by  our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Comparison with the state-of-the-art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Tables 1 and 2  show that taking our method as a plug-in module for the  state-of-the-art [23] and other baselines [14,22] consistently  improves their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' For example, RMM [23] w/  ours gains 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2 percentage points on CIFAR-100  and ImageNet-Full, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Interestingly, we find that  we can surpass the baselines more when the number of  phases N is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', on CIFAR-100, our method im-  proves RMM by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='7 percentage points when N=25, while  this number is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='8 percentage points when N=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our expla-  nation is that the forgetting problem is more serious when  the number of phases is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Thus, we need better hyper-  parameters to balance stability and plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Table 3 concerns eight ablation settings,  and shows the results in both TFH and TFS for different  numbers of phases (N=5/25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The detailed analyses are as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  1) First block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Row 1 shows the baseline [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  2) Second block: optimizable hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  In our  study, we optimize three kinds of hyperparameters that af-  fect the model’s stability and plasticity: KD loss weights  (β, γ), learning rate λ, and classifier type indicator δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Com-  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Optimizing  N=5  N=25  (β, γ)  (aδa,)  (aλa)  TFH  TFS  TFH  TFS  1  Baseline  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='11  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='96  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='47  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='16  2  ✓  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='20  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='60  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='27  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='91  3  ✓  ✓  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='23  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='08  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='20  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='94  4  ✓  ✓  ✓  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='88  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='92  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='27  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='44  5  Cross-val fixed  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='33  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='02  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='50  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='64  6  Offline RL [23]  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='42  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='88  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='12  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='53  7  Bilevel HO [11]  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='20  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='02  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='56  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='42  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Ablation results (average accuracy %) on CIFAR-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  (β, γ) are KD loss weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' λ and δ denote learning rates and  classifier types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The baseline is LUCIR [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Row 4  shows our best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  paring Row 2 to Row 1, we can observe that optimizing  the KD loss weights boosts the TFS accuracy more sig-  nificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It is because the baseline, LUCIR [14], applies  a strong regularization term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', feature KD loss) which  harms the TFH performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our method changes the reg-  ularization term by adjusting the KD loss weights, so it  achieves better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Comparing Row 3 to Row 2,  we can see that optimizing the classifier type indicator δ  performs further improvements, especially in TFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It is be-  cause the baseline deploys an FC classifier by default while  our method learns to switch between different classifiers in  different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Comparing Row 4 to Row 3, we can see  the effectiveness of optimizing the learning rates in CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In  summary, it is impressive that optimizing three kinds of hy-  perparameters together achieves the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  3) Third block: hyperparameter learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' For  Row 5, we use cross-validation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', all past, future, and  7  Training from half (TFH)  Images iCaRL  Ours  Phase 0  Phase 3  Phase 5  Training from scratch (TFS)  Images iCaRL  Ours  Phase 0  Phase 3  Phase 5  LUCIR  LUCIR  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The activation maps using Grad-CAM [33] for the last  phase model on ImageNet-Subset 5-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Samples are selected  from the classes coming in the 0-th, 3-rd, and 5-th phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Green  ticks mean successful activation of discriminative features on ob-  ject regions, while red crosses mean unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  validation data are accessible) to find a set of fixed hyperpa-  rameters and apply them to all phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We can see that Row  5 results are consistently lower than ours in Row 4, although  it can access more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It shows that we need to update  the hyperparameters online in different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' For Row 6,  we use the policy pre-trained in the 0-th phase of the target  CIL task using the framework proposed in [23] (compared  to ours, it is offline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Comparing Row 6 with Row 4, we  are happy to see that our online learning algorithm achieves  better performance than the offline RL while we use much  less training time (see the analysis in the supplementary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  For Row 7, we use the bilevel hyperparameter optimiza-  tion method [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Comparing Row 7 with Row 4, we can  observe that our method achieves more significant perfor-  mance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The reason is that [11] is designed  for time-invariant environments, while our online algorithm  can adapt to the time-varying environments in CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='.  Visualizing activation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Figure 4 demonstrates the  activation maps visualized by Grad-CAM [33] for the final  model (obtained after five phases) on ImageNet-Subset 5-  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The left and right sub-figures show the results for  training-from-half (TFH) and training-from-scratch (TFS),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We can observe: 1) LUCIR [14] makes predic-  tions according to foreground (correct) and background (in-  correct) regions in the TFH and TFS settings, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  2) iCaRL [30] behaves opposite to LUCIR in the two set-  tings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 3) our method always makes predictions according to  foreground (correct) regions in both TFH and TFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The rea-  sons are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' LUCIR applies a strong (feature) KD  by default, so it performs better in TFH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' iCaRL applies a  weak (logit) KD by default, so it performs better in TFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Our method can adjust the hyperparameters to change be-  tween different KD losses, so it performs well in both set-  tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  Hyperparameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Figure 5 shows the hyperparam-  eter values produced by our policy on CIFAR-100 25-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  1) KD loss weights β and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' From the figure, we have two  observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' a) The policy learns to produce a larger ini-  5  10  15  20  #phases (TFH)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='0  Logit KD ( )  Feature KD ( )  5  10  15  20  #phases (TFS)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='00  5  10  15  20  #phases (TFH)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='15  Learning rate ( )  5  10  15  20  #phases (TFS)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='15  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The hyperparameter values produced by our policy on  CIFAR-100 25-phase for LUCIR [14] w/ ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' We smooth all  curves with a rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='8 for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  tial value for γ and β in TFH and TFS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our  explanation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In TFH, we already have a pre-  trained model, so we need a strong regularization term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=',  feature KD loss) to make the model more stable and avoid  forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In TFS, we start from random initialization, so  we need a weak regularization term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=', logit KD loss)  to improve the model’s plasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  b) Both β and γ in-  crease in TFH, while β decreases and γ increases in TFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  It is because we need stronger regularization to maintain  the knowledge when more data is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' The policy  achieves that in different ways: it assigns higher weights  for both KD losses in TFH, while it transfers from logit KD  to feature KD in TFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  2) Learning rates λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' In Figure 5, we can observe that the  learning rate in TFS is much higher than in TFH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It can be  explained that 1) we need a higher learning rate in TFS as  the model is trained from scratch and needs to capture more  new knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' 2) we need a lower learning rate in TFH  because we need to avoid forgetting the pre-trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  3) Classifier type indicator δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' On CIFAR-100 25-phase, our  policy learned to choose the NCM and FC classifiers in TFS  and TFH, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Conclusions  In this study, we introduce a novel framework that allows  us to optimize hyperparameters online to balance the stabil-  ity and plasticity in CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' To achieve this, we formulate the  CIL task as an online MDP and learn a policy to produce  the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' Our approach is generic, and it can be  easily applied to existing methods to achieve large-margin  improvements in both TFS and TFH settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' It is worth  mentioning that our method can also be applied to optimize  other key hyperparameters in CIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content='  8  ORAcknowledgments  This research was supported by A*STAR under its AME  YIRG Grant (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
+page_content=' A20E6c0101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/uNE4T4oBgHgl3EQfWQz4/content/2301.05032v1.pdf'}
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diff --git a/v9E3T4oBgHgl3EQflApa/content/tmp_files/2301.04602v1.pdf.txt b/v9E3T4oBgHgl3EQflApa/content/tmp_files/2301.04602v1.pdf.txt
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+arXiv:2301.04602v1  [astro-ph.HE]  8 Dec 2022
+MNRAS 000, 1–4 (0000)
+Preprint 12 January 2023
+Compiled using MNRAS LATEX style file v3.0
+On the origin of magnetar glitch-associated activities
+Shuang Du1⋆
+1College of Physics and Astronomy, Jialidun University, Guangde, Anhui, 242214, China
+12 January 2023
+ABSTRACT
+Magnetars are the most strongly magnetized subclass of pulsars. Observed magnetar
+activities, such as glitches and bursts, have not yet been systematically explained.
+Inspired by the dynamo theory, in this letter, we suggest a uniform picture for illus-
+trating these observations. We speculate that the perturbation of velocity fields in
+these magnetars due to glitches acts as the springhead for glitch-associated activities.
+We argue that comparing the spark frequencies of pulsed radio pulses before and after
+glitches may examine the imaginary picture.
+Key words: pulsars: general - stars: magnetars
+1
+INTRODUCTION
+Magnetars are a subclass of pulsars with surface magnetic
+fields as large as ∼ 1014−1015G (Thompson & Duncan 1993;
+Kouveliotou et al. 1998, see Kaspi & Beloborodov 2017 for
+a review). A feature that can distinguish magnetars from
+normal pulsars is that the former maintains a large X-ray
+luminosity beyond the spin-down luminosity, while the lat-
+ter does the opposite. Strong magnetic fields are thought to
+be responsible for the discrepancies between magnetars and
+normal pulsars (see, e.g., Thompson & Duncan 1995, 1996;
+Beloborodov 2009). Of course, strong magnetic fields do not
+make magnetars completely different from normal pulsars,
+for example, both subclasses occasionally exhibit sudden
+spin-up events (i.e., glitches, Radhakrishnan & Manchester
+1969; Reichley & Downs 1969; see Haskell & Melatos 2015
+for a review) and similar post-glitch activities (e.g., spin-
+down lags, Lyne et al. 1993; Ge et al. 2022). The exception
+is that glitches in magnetars (including few high-magnetic-
+field
+and
+low-magnetic-field
+pulsars,
+e.g.,
+Gavriil et al.
+2008; Rea et al. 2010) are often accompanied by X-ray
+bursts and even radio bursts (e.g., Bochenek et al. 2020;
+CHIME/FRB Collaboration et al. 2020; Mereghetti et al.
+2020; Ridnaia et al. 2021; Li et al. 2021; Tavani et al. 2021;
+Ge et al. 2022). It is wildly believed that the unpinning of
+neutron superfluid vortices in pulsar crusts triggers these
+glitches since the repinning of the vortices to the crusts
+seems to be the only reasonable way to explain post-glitch
+recovery processes (Anderson & Itoh 1975). The question
+then arises, why would magnetars behave differently from
+normal pulsars in X-ray and even radio bands when same-
+origin glitches happen? To answer this question more clearly,
+maybe we should revise a deeper question, namely, how these
+⋆ E-mail: dushuang@pku.edu.cn
+strong magnetic fields arise. The popular model for the gen-
+eration of pulsar magnetic fields, at least for magnetars, is
+based on dynamo theory (see, e.g., Moffatt 1978 for a review
+of the general theory and see Thompson & Duncan 1993 for
+the magnetar case). In this scenario, convection (α effect)
+and differential rotation (ω effect) convert the thermal en-
+ergy and kinetic energy of the fluid in a magnetar into the
+magnetic energy. Thus, the starting point of understanding
+these glitch-associated activities should focus on the effect of
+the variation of velocity fields in these magnetars. Inspired
+by the dynamo theory, in the next section, we present a
+uniform picture for illustrating these similar or different ob-
+servations between magnetars and normal pulsars.
+2
+THE UNIFORM PICTURE
+For a magnetar, the evolution of the magnetic field, B, is sec-
+ular. During a short duration prior to a glitch, the magnetic
+field can be treated as a constant. Thus, in this steady state,
+the magnetic field is governed by the induction equation,
+0 ≈ ∂B
+∂t ≈ ∇ × (v0 × B),
+(1)
+the Navier-Stokes equations in the presence of a strong mag-
+netic field,
+∂v0
+∂t ≈ −ρg − 1
+ρ∇P + 1
+cρJ0 × B,
+(2)
+and a constraint, for example, the Taylor’s constraint,
+�
+L
+(J0 × B)dl = 0,
+(3)
+where v0 is the velocity field of charge particles before the
+glitch, ρ is mass density of the pulsar, g is the gravitational
+acceleration, c is the speed of light, J0 is the current density
+© 0000 The Authors
+
+2
+DS
+before the glitch and L is a circle perpendicular to the rota-
+tion axis. For convenience, we still call these three equations
+as dynamo even though the convective motion has ceased in
+this letter.
+When the glitch is generated, a series of effects will ap-
+pear. (i) The sudden unpinning of superfluid vortices per-
+turbs the velocity field in the magnetar in a moment that
+v0 → v0 +δv, as well as the current field that J0 → J0 +δJ.
+As a result, equations (1) and (3) are no longer satisfied and
+the dynamo in the star starts to rearrange the distribution of
+the velocity field and magnetic field. Moreover, as expected
+for a magnetar, if there is a strong toroidal component of
+the magnetic field, the breaking of mechanical equilibrium
+(e.g., equation 3) would result in a net torque acting on the
+crust. Under this unsteady state, the torque tends to slow
+the differential rotation (Spruit 1999), that is, some of energy
+stored in the toroidal magnetic field can be converted into
+kinetic energy of the crust. (ii) After the glitch, superfluid
+vortices will continuously repin to the crust which can lead
+to a sustaining variation of the velocity field. Therefore, dur-
+ing the recovery stage, the velocity field and magnetic field
+can not be transformed into a stationary state, but instead,
+their distributions are continuously adjusted by the dynamo.
+Note that equations (1)-(3) describe a self-sustaining
+process, hence a slight perturbation to the magnetic field
+may be amplified. All of the above effects could be related
+to the observed glitch-associated activities. (a) If the accel-
+eration of the crust due to the torque of the twisty magnetic
+field is more effective than the deceleration due to the su-
+perfluid vortices repinning, the spin up would be lengthened
+after the glitch. Only when the torque decays (e.g., due to
+the release of the magnetic energy stored in the toroidal
+component and rearrangement of magnetic field by the dy-
+namo) to a value where the deceleration effect dominates
+the rotation, the rotation period begins to increase. There-
+fore, spin-down lags are more likely to be observed in young
+pulsars, especially magnetars. If this conjecture is true, it im-
+plies that there exists an evolution of the dynamo, that is,
+differential-rotation-effect dynamo (hereafter ω dynamo; for
+younger pulsars, e.g., Crab pulsar) gradually transforms into
+the dynamo without differential rotation (hereafter no-ω dy-
+namo; for older pulsars, e.g., Vela pulsar). (b) Since magnetic
+field lines are pined to the crust, when the dynamo rear-
+ranges the magnetic field, stress in the crust is also changed.
+This could lead to crust breaking and trigger an X-ray or soft
+gamma-ray burst. Under this conjecture, X-ray bursts must
+come after glitches. (c) The ω dynamo persistently changes
+the configuration of the magnetic field, so the profile (mainly
+effected by poloidal component) and polarization (mainly ef-
+fected by toroidal component) of the pulsed radiation may
+be also changed, especially for the pulsed radio radiation
+since it is emitted in the light cylinder. After the recovery
+process, the configuration of the magnetic field gradually
+becomes stationary and should be different from its origi-
+nal shape before the glitch. Correspondingly, there may be
+a persistent shift (see, e.g., Lyne et al. 1993). Due to the ab-
+sence of strong toroidal magnetic fields, the no-ω dynamo is
+insensitive to the vortex-unpinning-induced toroidal pertur-
+bations of velocity fields (the energy density of the toroidal
+field should be large enough so that a small toroidal pertur-
+bation can not be restrained by the tension of the magnetic
+field) so that spin-down lags and persistent shifts can not
+appear in older pulsars. This may explain the differences
+between the Crab and Vela pulsars after glitches. (d) As the
+dipole magnetic field is the leading order of the actual mag-
+netic field, when the ω dynamo works in an unsteady state,
+multipole component of the magnetic field should be gen-
+erated. As a result, a radio-quiet magnetar could become a
+radio-loud one after some glitches (see the next section for
+more details).
+3
+THE CHECK METHOD
+It is difficult to formulate a mathematical formalism for the
+above conjectures, especially in the presence of the Navier-
+Stokes equations. Simulations are needed, but this is beyond
+my scope. However, conjecture (d) provides a clue to test
+above imaginations.
+Some magnetars have a unique property that their ex-
+tinct pulsed radio emission may brighten again after/during
+some outbursts (Camilo et al. 2006, 2007; Levin et al. 2010;
+Shannon & Johnston 2013; Zhang et al. 2020). Models in-
+voking gap-like accelerators (Ruderman & Sutherland 1975;
+Arons & Scharlemann 1979) may reasonably account for
+the cessation of pulsed radio emission. These models are
+able to correctly predict the death line shown in the two-
+dimensional pulsar parameter phase space (e.g., P − ˙P di-
+agram, where P is the pulsar period and
+˙P is the pul-
+sar spin-down rate; Chen & Ruderman 1993; Zhang et al.
+2000). According to these models, the cessation of pulsed
+radio emission is the consequence of the potential across the
+accelerator no longer being able to sustain the avalanche of
+discharges (i.e., the spark). Thus, the rebrightening of the
+radio pulses implies an increased potential. An operational
+way to improve the potential is to provide a multipole com-
+ponent of the magnetic field, since the multipole component
+of the magnetic field can enhance the potential across the
+gap (Chen & Ruderman 1993; Zhang et al. 2000)1. As men-
+tioned at the end of Section 2, the generation of multipole
+component is not groundless, but induced by the perturba-
+tion of velocity field.
+The multipole component not only raises the poten-
+tial but also lowers the height of the spark (because curva-
+ture radii of magnetic field lines are decreased). Therefore,
+when the newly generated multipole component is strong
+enough, the frequency of sparks will rise significantly. Here,
+we present an equivalent RC circuit of the spark under the
+inner gap model (Ruderman & Sutherland 1975) for illus-
+trating this issue.
+Let us first briefly review the inner gap model. For a
+pulsar with net positive charges being in its open field line
+region, when these positive charges move outwards (Sturrock
+1971; Holloway 1973), the lost charges can not be replen-
+ished by the stellar surface due to large binding energies
+of positive charges, Whereafter, a gap grows on the polar
+cap. The potential across the gap, U, increases with the gap
+height, h. When the voltage is increased to a critical value,
+positrons (e.g., originally produced by the thermal photons
+from the pulsar surface through γ + B → e− + e+ + B)
+1 Other scenarios can be found in Morozova et al. (2010) and
+Lin et al. (2015).
+MNRAS 000, 1–4 (0000)
+
+Origin of magnetar activities after glitches
+3
+in the gap can be accelerated to high energies that the
+photons emitted by these positrons via curvature radiation
+and even inverse Compton scattering (Qiao & Lin 1998) will
+again be converted into electron-positron pairs by the reac-
+tion γ + B → e− + e+ + B. The avalanche of discharges,
+e+e− → γ → e+e− → · · · , leads to the eventual reduction
+of the potential, after which the spark ceases. The growth
+and reduction of the gap happens back and forth, generating
+a large number of secondary charges to sustain the pulsed
+radio emission. However, the maximum value of the poten-
+tial across the gap does not grow unconstrainedly with the
+gap height, but is enslaved by the pulsar period (the po-
+tential is inversely proportional to the period). There is a
+moment when the maximum value of the potential is not
+able to sustain a spark due to the spin-down of the pulsar,
+and then the pulsed radio emission is extinguished.
+We note that the electric field across the gap behaves
+like that of a parallel-plate capacitor since the gap height is
+much smaller than the pulsar radius, r∗. For example, the
+electric field of the parallel-plate capacitor is
+E′ = U ′
+d′ = 4πkQ′
+S′
+(4)
+where U ′ and d′ are the voltage and interval between the
+two plates, Q′ and S′ are the charge and area of each plate,
+and k is the electrostatic constant. As a contrast, the electric
+field across the gap is (Ruderman & Sutherland 1975)
+E ≈ 2U
+h = 2ΩBs
+c
+h
+(5)
+where Ω is the spin velocity, and Bs is the magnetic field
+strength on the polar cap. Therefore, we equivalently treat
+the inner gap as a parallel-plate capacitor with the voltage
+being U and the area of the plate, S, being the area of the
+polar cap. Then, the equivalent interval is h/2, the equiva-
+lent charge is
+Q = ΩBshS
+2πck ,
+(6)
+and the equivalent capacitance is
+C = Q
+U = Sε0
+2πkh,
+(7)
+where ε0 is the permittivity of vacuum. The increase rate
+of gap height is exactly the velocity of the charge particles
+which are flowing outwards.
+The charge and discharge in the inner gap can be equiv-
+alent to the charge and break down of the capacitor. How-
+ever, such an equality is nonlinear in the sense that both of
+the capacitance and charge increase with the gap height un-
+til the capacitor breaks down. Moreover, as with the charge
+of a realistic RC circuit, we assume that the charge in the
+inner gap is also a transient process (so that equation 9, i.e.,
+the “ dynamic version” of equation 5, holds) and have
+Rd(CU)
+dt
++ U = Umax
+(8)
+where R is the equivalent constant resistance of the circuit
+(the magnetosphere should be a semi-static state during a
+short duration),
+U(t) ≈ ΩBs
+c
+h(t)2
+(9)
+is the potential across the gap at time t, and
+Umax ≈ ΩBs
+c
+h2
+max
+(10)
+is the supply voltage (e.g., due to unipolar induction) with
+hmax being the maximum thickness of the inner gap. It is
+worth reminding that U may not be able to reach Umax since
+before that the capacitor may be broken down. For example,
+as illustrated in the inner gap model, the maximum possible
+potential drop along any magnetic field line within the polar
+cap is
+∆Vmax ≈ ΩBs
+c
+r2
+p
+2 ,
+(11)
+(i.e., hmax = 0.7rp). However, this potential drop is usually
+larger than the voltage, Usp, that a spark needed (see, e.g.,
+equation (23) in Ruderman & Sutherland 1975).
+Note that S ≈ 4πr2
+p with rp being the radius of the
+polar cap and
+rp = r∗
+�Ωr∗
+c
+�1/2
+,
+(12)
+equations (7)-(10) can be solved as
+h(t) = 1
+A
+c1
+√
+ABe
+√
+ABt − c2
+√
+ABe−
+√
+ABt
+c1e
+√
+ABt + c2e−
+√
+ABt
+,
+(13)
+where
+A =
+kc
+2Ωr3∗Rε0 ,
+B =
+kc
+2Ωr3∗Rε0 h2
+max,
+(14)
+and c1, c2 are integration constants. We set the initial con-
+dition as
+h(t = 0) = 0,
+(15)
+and have c1 = c2 through equation (13), so that
+h(t) = hmax tanh(
+√
+ABt).
+(16)
+Through equation (16), the velocity of charge outflow is
+vof = dh(t)
+dt
+= hmax
+√
+AB[1 − tanh2(
+√
+ABt)].
+(17)
+According to equations (9) and (16), we get
+ΩBs
+c
+[hmax tanh(
+√
+ABτ)]2 = Usp,
+(18)
+where τ is the time that the potential across the gap in-
+creases to trigger a spark. The spark frequency is ∼ 1/τ.
+Since the multipole component can increase curvature radii
+of magnetic field lines, the necessary spatial scale for the
+reaction γ + B → e− + e+ + B will be reduced, as well as
+the height and voltage (i.e., Usp) of the gap. According to
+equation (18), it is easy to see that the value of τ will be
+also reduced (i.e., the spark frequency becomes faster). It is
+worth noting that the values of R and vof can be roughly
+estimated via equations (17) and (18) if the current strength
+of the charge outflow and the value of the spark frequency
+can be constrained.
+The increase of spark frequency may be checked by com-
+paring the fine structures of the radio pulses before and after
+the glitch. If conjecture (d) is supported by observations, this
+suggests that, at the very least, the dynamo mechanism is
+at work, then the other conjectures are not any more insub-
+stantial.
+MNRAS 000, 1–4 (0000)
+
+4
+DS
+4
+SUMMARY AND DISCUSSION
+In this letter, we suggest a uniform picture for illustrating
+glitch-associated activities of magnetars, arguing that per-
+turbations of the velocity fields in magnetars due to su-
+perfluid vortex unpinning induce these activities via dy-
+namo mechanism. We believe that these conjectures can be
+tested by comparing the spark frequencies of the pulsed ra-
+dio pulses before and after glitches.
+Although the imaginary picture is not presented in a
+rigorous manner, we believe that the logic is reasonable.
+If the result of the method shown in Section 3 is positive,
+the idea shown in Section 2 can be extended. For exam-
+ple, the nascent multiple field may bridge the positive and
+negative charge regions of the magnetar surface, then the
+primordial force-free condition (Goldreich & Julian 1969) is
+violated and currents moved along these multiple magnetic
+field lines will appear before the force-free condition building
+again. As long as the curvature radiation of the current can
+emit coherent radio emission (maybe pinch instability is a
+good mechanism since it can cut the current into bunches),
+this could be a potential scenario to explain fast-radio-burst-
+like bursts.
+We note that three fast radio burst-like bursts are ob-
+served following a spin-up glitch (Younes et al. 2022). Our
+picture may be still suitable for the spin-up glitch situation,
+because, empirically, the appearance of the spin-up glitch de-
+mands that some extra superfluid vortices pin to the crust
+suddenly. No matter pinning or unpinning, the perturbation
+of the velocity field should be induced. Therefore, the “tran-
+sition” of the dynamo from the previous steady state to an
+unsteady state could still generate the multipole field.
+5
+ACKNOWLEDGMENTS
+I thank Dr. Weihua Wang for sharing information about
+glitches with me.
+6
+DATA AVAILABILITY
+No new data is produced.
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+
diff --git a/v9E3T4oBgHgl3EQflApa/content/tmp_files/load_file.txt b/v9E3T4oBgHgl3EQflApa/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf,len=360
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='04602v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='HE]  8 Dec 2022  MNRAS 000, 1–4 (0000)  Preprint 12 January 2023  Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='0  On the origin of magnetar glitch-associated activities  Shuang Du1⋆  1College of Physics and Astronomy, Jialidun University, Guangde, Anhui, 242214, China  12 January 2023  ABSTRACT  Magnetars are the most strongly magnetized subclass of pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Observed magnetar  activities, such as glitches and bursts, have not yet been systematically explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Inspired by the dynamo theory, in this letter, we suggest a uniform picture for illus-  trating these observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' We speculate that the perturbation of velocity fields in  these magnetars due to glitches acts as the springhead for glitch-associated activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  We argue that comparing the spark frequencies of pulsed radio pulses before and after  glitches may examine the imaginary picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Key words: pulsars: general - stars: magnetars  1  INTRODUCTION  Magnetars are a subclass of pulsars with surface magnetic  fields as large as ∼ 1014−1015G (Thompson & Duncan 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Kouveliotou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 1998, see Kaspi & Beloborodov 2017 for  a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' A feature that can distinguish magnetars from  normal pulsars is that the former maintains a large X-ray  luminosity beyond the spin-down luminosity, while the lat-  ter does the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Strong magnetic fields are thought to  be responsible for the discrepancies between magnetars and  normal pulsars (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Thompson & Duncan 1995, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Beloborodov 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Of course, strong magnetic fields do not  make magnetars completely different from normal pulsars,  for example, both subclasses occasionally exhibit sudden  spin-up events (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', glitches, Radhakrishnan & Manchester  1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Reichley & Downs 1969;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' see Haskell & Melatos 2015  for a review) and similar post-glitch activities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', spin-  down lags, Lyne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The exception  is that glitches in magnetars (including few high-magnetic-  field  and  low-magnetic-field  pulsars,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=',  Gavriil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Rea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2010) are often accompanied by X-ray  bursts and even radio bursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Bochenek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  CHIME/FRB Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Mereghetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Ridnaia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Tavani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' It is wildly believed that the unpinning of  neutron superfluid vortices in pulsar crusts triggers these  glitches since the repinning of the vortices to the crusts  seems to be the only reasonable way to explain post-glitch  recovery processes (Anderson & Itoh 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The question  then arises, why would magnetars behave differently from  normal pulsars in X-ray and even radio bands when same-  origin glitches happen?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' To answer this question more clearly,  maybe we should revise a deeper question, namely, how these  ⋆ E-mail: dushuang@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='cn  strong magnetic fields arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The popular model for the gen-  eration of pulsar magnetic fields, at least for magnetars, is  based on dynamo theory (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Moffatt 1978 for a review  of the general theory and see Thompson & Duncan 1993 for  the magnetar case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' In this scenario, convection (α effect)  and differential rotation (ω effect) convert the thermal en-  ergy and kinetic energy of the fluid in a magnetar into the  magnetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Thus, the starting point of understanding  these glitch-associated activities should focus on the effect of  the variation of velocity fields in these magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Inspired  by the dynamo theory, in the next section, we present a  uniform picture for illustrating these similar or different ob-  servations between magnetars and normal pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  2  THE UNIFORM PICTURE  For a magnetar, the evolution of the magnetic field, B, is sec-  ular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' During a short duration prior to a glitch, the magnetic  field can be treated as a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' in this steady state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  the magnetic field is governed by the induction equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  0 ≈ ∂B  ∂t ≈ ∇ × (v0 × B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  (1)  the Navier-Stokes equations in the presence of a strong mag-  netic field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  ∂v0  ∂t ≈ −ρg − 1  ρ∇P + 1  cρJ0 × B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  (2)  and a constraint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' for example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' the Taylor’s constraint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  �  L  (J0 × B)dl = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  (3)  where v0 is the velocity field of charge particles before the  glitch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' ρ is mass density of the pulsar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' g is the gravitational  acceleration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' c is the speed of light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' J0 is the current density  © 0000 The Authors  2  DS  before the glitch and L is a circle perpendicular to the rota-  tion axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' For convenience, we still call these three equations  as dynamo even though the convective motion has ceased in  this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  When the glitch is generated, a series of effects will ap-  pear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (i) The sudden unpinning of superfluid vortices per-  turbs the velocity field in the magnetar in a moment that  v0 → v0 +δv, as well as the current field that J0 → J0 +δJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  As a result, equations (1) and (3) are no longer satisfied and  the dynamo in the star starts to rearrange the distribution of  the velocity field and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Moreover, as expected  for a magnetar, if there is a strong toroidal component of  the magnetic field, the breaking of mechanical equilibrium  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', equation 3) would result in a net torque acting on the  crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Under this unsteady state, the torque tends to slow  the differential rotation (Spruit 1999), that is, some of energy  stored in the toroidal magnetic field can be converted into  kinetic energy of the crust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (ii) After the glitch, superfluid  vortices will continuously repin to the crust which can lead  to a sustaining variation of the velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Therefore, dur-  ing the recovery stage, the velocity field and magnetic field  can not be transformed into a stationary state, but instead,  their distributions are continuously adjusted by the dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Note that equations (1)-(3) describe a self-sustaining  process, hence a slight perturbation to the magnetic field  may be amplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' All of the above effects could be related  to the observed glitch-associated activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (a) If the accel-  eration of the crust due to the torque of the twisty magnetic  field is more effective than the deceleration due to the su-  perfluid vortices repinning, the spin up would be lengthened  after the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Only when the torque decays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', due to  the release of the magnetic energy stored in the toroidal  component and rearrangement of magnetic field by the dy-  namo) to a value where the deceleration effect dominates  the rotation, the rotation period begins to increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' There-  fore, spin-down lags are more likely to be observed in young  pulsars, especially magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' If this conjecture is true, it im-  plies that there exists an evolution of the dynamo, that is,  differential-rotation-effect dynamo (hereafter ω dynamo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' for  younger pulsars, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Crab pulsar) gradually transforms into  the dynamo without differential rotation (hereafter no-ω dy-  namo;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' for older pulsars, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Vela pulsar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (b) Since magnetic  field lines are pined to the crust, when the dynamo rear-  ranges the magnetic field, stress in the crust is also changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  This could lead to crust breaking and trigger an X-ray or soft  gamma-ray burst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Under this conjecture, X-ray bursts must  come after glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (c) The ω dynamo persistently changes  the configuration of the magnetic field, so the profile (mainly  effected by poloidal component) and polarization (mainly ef-  fected by toroidal component) of the pulsed radiation may  be also changed, especially for the pulsed radio radiation  since it is emitted in the light cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' After the recovery  process, the configuration of the magnetic field gradually  becomes stationary and should be different from its origi-  nal shape before the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Correspondingly, there may be  a persistent shift (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Lyne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Due to the ab-  sence of strong toroidal magnetic fields, the no-ω dynamo is  insensitive to the vortex-unpinning-induced toroidal pertur-  bations of velocity fields (the energy density of the toroidal  field should be large enough so that a small toroidal pertur-  bation can not be restrained by the tension of the magnetic  field) so that spin-down lags and persistent shifts can not  appear in older pulsars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' This may explain the differences  between the Crab and Vela pulsars after glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (d) As the  dipole magnetic field is the leading order of the actual mag-  netic field, when the ω dynamo works in an unsteady state,  multipole component of the magnetic field should be gen-  erated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' As a result, a radio-quiet magnetar could become a  radio-loud one after some glitches (see the next section for  more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  3  THE CHECK METHOD  It is difficult to formulate a mathematical formalism for the  above conjectures, especially in the presence of the Navier-  Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Simulations are needed, but this is beyond  my scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' However, conjecture (d) provides a clue to test  above imaginations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Some magnetars have a unique property that their ex-  tinct pulsed radio emission may brighten again after/during  some outbursts (Camilo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2006, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Levin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Shannon & Johnston 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Models in-  voking gap-like accelerators (Ruderman & Sutherland 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Arons & Scharlemann 1979) may reasonably account for  the cessation of pulsed radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' These models are  able to correctly predict the death line shown in the two-  dimensional pulsar parameter phase space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', P − ˙P di-  agram, where P is the pulsar period and  ˙P is the pul-  sar spin-down rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Chen & Ruderman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' According to these models, the cessation of pulsed  radio emission is the consequence of the potential across the  accelerator no longer being able to sustain the avalanche of  discharges (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', the spark).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Thus, the rebrightening of the  radio pulses implies an increased potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' An operational  way to improve the potential is to provide a multipole com-  ponent of the magnetic field, since the multipole component  of the magnetic field can enhance the potential across the  gap (Chen & Ruderman 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2000)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' As men-  tioned at the end of Section 2, the generation of multipole  component is not groundless, but induced by the perturba-  tion of velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  The multipole component not only raises the poten-  tial but also lowers the height of the spark (because curva-  ture radii of magnetic field lines are decreased).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Therefore,  when the newly generated multipole component is strong  enough, the frequency of sparks will rise significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Here,  we present an equivalent RC circuit of the spark under the  inner gap model (Ruderman & Sutherland 1975) for illus-  trating this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Let us first briefly review the inner gap model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' For a  pulsar with net positive charges being in its open field line  region, when these positive charges move outwards (Sturrock  1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Holloway 1973), the lost charges can not be replen-  ished by the stellar surface due to large binding energies  of positive charges, Whereafter, a gap grows on the polar  cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The potential across the gap, U, increases with the gap  height, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' When the voltage is increased to a critical value,  positrons (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', originally produced by the thermal photons  from the pulsar surface through γ + B → e− + e+ + B)  1 Other scenarios can be found in Morozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (2010) and  Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  MNRAS 000, 1–4 (0000)  Origin of magnetar activities after glitches  3  in the gap can be accelerated to high energies that the  photons emitted by these positrons via curvature radiation  and even inverse Compton scattering (Qiao & Lin 1998) will  again be converted into electron-positron pairs by the reac-  tion γ + B → e− + e+ + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The avalanche of discharges,  e+e− → γ → e+e− → · · · , leads to the eventual reduction  of the potential, after which the spark ceases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The growth  and reduction of the gap happens back and forth, generating  a large number of secondary charges to sustain the pulsed  radio emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' However, the maximum value of the poten-  tial across the gap does not grow unconstrainedly with the  gap height, but is enslaved by the pulsar period (the po-  tential is inversely proportional to the period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' There is a  moment when the maximum value of the potential is not  able to sustain a spark due to the spin-down of the pulsar,  and then the pulsed radio emission is extinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  We note that the electric field across the gap behaves  like that of a parallel-plate capacitor since the gap height is  much smaller than the pulsar radius, r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' For example, the  electric field of the parallel-plate capacitor is  E′ = U ′  d′ = 4πkQ′  S′  (4)  where U ′ and d′ are the voltage and interval between the  two plates, Q′ and S′ are the charge and area of each plate,  and k is the electrostatic constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' As a contrast, the electric  field across the gap is (Ruderman & Sutherland 1975)  E ≈ 2U  h = 2ΩBs  c  h  (5)  where Ω is the spin velocity, and Bs is the magnetic field  strength on the polar cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Therefore, we equivalently treat  the inner gap as a parallel-plate capacitor with the voltage  being U and the area of the plate, S, being the area of the  polar cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Then, the equivalent interval is h/2, the equiva-  lent charge is  Q = ΩBshS  2πck ,  (6)  and the equivalent capacitance is  C = Q  U = Sε0  2πkh,  (7)  where ε0 is the permittivity of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The increase rate  of gap height is exactly the velocity of the charge particles  which are flowing outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  The charge and discharge in the inner gap can be equiv-  alent to the charge and break down of the capacitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' How-  ever, such an equality is nonlinear in the sense that both of  the capacitance and charge increase with the gap height un-  til the capacitor breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Moreover, as with the charge  of a realistic RC circuit, we assume that the charge in the  inner gap is also a transient process (so that equation 9, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=',  the “ dynamic version” of equation 5, holds) and have  Rd(CU)  dt  + U = Umax  (8)  where R is the equivalent constant resistance of the circuit  (the magnetosphere should be a semi-static state during a  short duration),  U(t) ≈ ΩBs  c  h(t)2  (9)  is the potential across the gap at time t, and  Umax ≈ ΩBs  c  h2  max  (10)  is the supply voltage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', due to unipolar induction) with  hmax being the maximum thickness of the inner gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' It is  worth reminding that U may not be able to reach Umax since  before that the capacitor may be broken down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' For example,  as illustrated in the inner gap model, the maximum possible  potential drop along any magnetic field line within the polar  cap is  ∆Vmax ≈ ΩBs  c  r2  p  2 ,  (11)  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', hmax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='7rp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' However, this potential drop is usually  larger than the voltage, Usp, that a spark needed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=',  equation (23) in Ruderman & Sutherland 1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Note that S ≈ 4πr2  p with rp being the radius of the  polar cap and  rp = r∗  �Ωr∗  c  �1/2  ,  (12)  equations (7)-(10) can be solved as  h(t) = 1  A  c1  √  ABe  √  ABt − c2  √  ABe−  √  ABt  c1e  √  ABt + c2e−  √  ABt  ,  (13)  where  A =  kc  2Ωr3∗Rε0 ,  B =  kc  2Ωr3∗Rε0 h2  max,  (14)  and c1, c2 are integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' We set the initial con-  dition as  h(t = 0) = 0,  (15)  and have c1 = c2 through equation (13), so that  h(t) = hmax tanh(  √  ABt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  (16)  Through equation (16), the velocity of charge outflow is  vof = dh(t)  dt  = hmax  √  AB[1 − tanh2(  √  ABt)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  (17)  According to equations (9) and (16), we get  ΩBs  c  [hmax tanh(  √  ABτ)]2 = Usp,  (18)  where τ is the time that the potential across the gap in-  creases to trigger a spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' The spark frequency is ∼ 1/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Since the multipole component can increase curvature radii  of magnetic field lines, the necessary spatial scale for the  reaction γ + B → e− + e+ + B will be reduced, as well as  the height and voltage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Usp) of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' According to  equation (18), it is easy to see that the value of τ will be  also reduced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', the spark frequency becomes faster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' It is  worth noting that the values of R and vof can be roughly  estimated via equations (17) and (18) if the current strength  of the charge outflow and the value of the spark frequency  can be constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  The increase of spark frequency may be checked by com-  paring the fine structures of the radio pulses before and after  the glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' If conjecture (d) is supported by observations, this  suggests that, at the very least, the dynamo mechanism is  at work, then the other conjectures are not any more insub-  stantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  MNRAS 000, 1–4 (0000)  4  DS  4  SUMMARY AND DISCUSSION  In this letter, we suggest a uniform picture for illustrating  glitch-associated activities of magnetars, arguing that per-  turbations of the velocity fields in magnetars due to su-  perfluid vortex unpinning induce these activities via dy-  namo mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' We believe that these conjectures can be  tested by comparing the spark frequencies of the pulsed ra-  dio pulses before and after glitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  Although the imaginary picture is not presented in a  rigorous manner, we believe that the logic is reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  If the result of the method shown in Section 3 is positive,  the idea shown in Section 2 can be extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' For exam-  ple, the nascent multiple field may bridge the positive and  negative charge regions of the magnetar surface, then the  primordial force-free condition (Goldreich & Julian 1969) is  violated and currents moved along these multiple magnetic  field lines will appear before the force-free condition building  again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' As long as the curvature radiation of the current can  emit coherent radio emission (maybe pinch instability is a  good mechanism since it can cut the current into bunches),  this could be a potential scenario to explain fast-radio-burst-  like bursts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  We note that three fast radio burst-like bursts are ob-  served following a spin-up glitch (Younes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Our  picture may be still suitable for the spin-up glitch situation,  because, empirically, the appearance of the spin-up glitch de-  mands that some extra superfluid vortices pin to the crust  suddenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' No matter pinning or unpinning, the perturbation  of the velocity field should be induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Therefore, the “tran-  sition” of the dynamo from the previous steady state to an  unsteady state could still generate the multipole field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  5  ACKNOWLEDGMENTS  I thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' Weihua Wang for sharing information about  glitches with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  6  DATA AVAILABILITY  No new data is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
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+page_content=', Harding, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2000, ApJ, 531,  L135  Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', Men, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content=' 2020, The As-  tronomer’s Telegram, 13699  This paper has been typeset from a TEX/LATEX file prepared by  the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
+page_content='  MNRAS 000, 1–4 (0000)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9E3T4oBgHgl3EQflApa/content/2301.04602v1.pdf'}
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+Vacuum stability, fixed points, and phases of QED3 at large Nf
+Lorenzo Di Pietro,1 Edoardo Lauria,2 and Pierluigi Niro3
+1Dipartimento di Fisica, Universit`a di Trieste, Strada Costiera 11, I-34151 Trieste, Italy,
+& INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy
+2CPHT, CNRS, Institut Polytechnique de Paris, France
+3Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy,
+University of California, Los Angeles, CA 90095, USA
+We consider three-dimensional Quantum Electrodynamics in the presence of a Chern-Simons term
+at level k and Nf flavors, in the limit of large Nf and k with k/Nf fixed. We consider either bosonic
+or fermionic matter fields, with and without quartic terms at criticality: the resulting theories are
+critical and tricritical bosonic QED3, Gross-Neveu and fermionic QED3. For all such theories we
+compute the effective potentials and the β functions of classically marginal couplings, at the leading
+order in the large Nf limit and to all orders in k/Nf and in the couplings. We determine the RG
+fixed points and discuss the quantum stability of the corresponding vacua. While critical bosonic
+and fermionic QED3 are always stable CFTs, we find that tricritical bosonic and Gross-Neveu QED3
+exist as stable CFTs only for specific values of k/Nf. Finally, we discuss the phase diagrams of these
+theories as a function of their relevant deformations.
+I.
+INTRODUCTION
+Three-dimensional Quantum Electrodynamics (QEDs)
+with either bosonic or fermionic degrees of freedom are
+among the simplest and yet very rich examples of gauge
+theories. At high energies, these theories are defined in
+terms of a three-dimensional U(1) Maxwell field a with
+gauge coupling e2 (of mass dimension one) and a Chern-
+Simons term at level k, coupled to Nf flavors of charged
+bosons or fermions. The Lagrangian is (we will always
+work in Euclidean signature)
+L =
+1
+2e2 da ∧ ⋆da + ik
+4π a ∧ da + Lmatter .
+(1)
+In absence of charged matter, if k = 0 the theory is
+dual to a compact scalar and it confines in the sense of
+[1, 2], whereas if k ̸= 0 it flows to a pure U(1)k Chern-
+Simons theory. In presence of charged matter, the low-
+energy behavior of QED theories for small Nf has not
+been rigorously established, and this question remains
+an open problem which has been studied with a vari-
+ety of approaches (see e.g. refs. [3–24]), including lattice
+simulations (see e.g. refs. [25–31]) and conformal boot-
+strap techniques (see e.g. [32–41] and references therein).
+When Nf is large enough, these theories are generically
+expected to flow to interacting Conformal Field Theories
+(CFTs) at low energies. This expectation is corroborated
+by the existence of fixed points of the Renormalization
+Group (RG) flow, which can be found in the limit of
+large Nf. However, as we will discuss in this work, some
+of these fixed points happen to lie in a region of instabil-
+ity of the theory, and so not all of them define a CFT at
+the quantum level.
+In order to clarify this issue, we will compute the ef-
+fective potentials Veff for a few instances of such Quan-
+tum Electrodynamics (to be defined below): two bosonic,
+called ‘tricritical’ and ‘critical’ QED3, and two fermionic,
+called ‘Gross-Neveu’ and ‘fermionic’ QED3. While criti-
+cal and fermionic QED3 do not admit any marginal defor-
+mations, tricritical and Gross-Neveu QED3 admit clas-
+sically marginal couplings. Imposing the stability of the
+vacuum gives rise to constraints for such couplings, which
+have to be satisfied at the fixed points of the RG flow.
+We will perform these computations by working at the
+leading order in the 1/Nf expansion, with Λ = e2Nf and
+κ = k/Nf held fixed. We study the low-energy limit by
+taking Λ → ∞ while retaining the dimensionless param-
+eter κ which does not run. The gauge interactions do
+not enter Veff at the leading order in the large-Nf limit,
+nevertheless the parameter κ will play an important role
+in the following analysis.1
+Next, we will compute the β functions of classically
+marginal couplings of tricritical and Gross-Neveu QEDs,
+at leading order in the 1/Nf expansion and to all orders in
+κ and in the couplings. Details of the RG computations
+will be presented elsewhere [43]. The zeroes of the β func-
+tions determine families of RG fixed points parametrized
+by κ. We finally impose on those fixed points the con-
+straint of vacuum stability from Veff.2 The result of this
+analysis is summarized in figures 1 and 2. Along the way,
+we will also present the expressions for the anomalous di-
+mensions of the lowest-lying singlet operators. In a final
+appendix, we derive the large-Nf phase diagrams of all
+four QEDs in presence of relevant deformations.
+1 The Maxwell (Yang-Mills) kinetic term in general does affect the
+vacuum structure of QED (QCD) theories with finite Nf. This
+term distinguishes between type I and type II superconductivity
+in critical bosonic QED3 with Nf = 1 and k = 0. Also, it affects
+the existence of a conformal critical point in SU(N)k QCD3 at
+large N and finite Nf, as a function of k/N [42]. However, for
+large-Nf QEDs, the whole dependence on Λ is encoded in the
+exact photon propagator in such a way that the limit Λ → ∞
+can be taken reliably.
+2 To avoid possible confusion, let us stress that here by “stability”
+we refer to the condition that the potential has a stable minimum,
+not to the RG stability of the fixed point under deformations by
+some coupling. In this letter we always use the term “stability”
+in the former sense.
+arXiv:2301.04611v1  [hep-th]  11 Jan 2023
+
+2
+II.
+BOSONIC THEORIES
+Tricritical QED
+The first theory we consider is tricritical QED3,
+the theory of Nf massless complex scalars φm (m =
+1, . . . , Nf) coupled to U(1)k and with the quartic cou-
+pling tuned to zero. The matter Lagrangian is
+Lmatter = (Dµφm)†(Dµφm) + h
+N 2
+f
+(φ†mφm)3 ,
+(2)
+where Dµ = ∂µ + iaµ denotes the covariant derivative
+and h is held fixed in the large-Nf limit. The continuous
+part of the global symmetry is SU(Nf) × U(1)m, where
+the first factor is a flavor symmetry and the second one is
+the magnetic symmetry of the gauge field. For κ = 0 the
+theory further enjoys parity symmetry. In the κ → ∞
+limit the gauge field decouples and the theory describes
+2Nf real scalars with a sextic interaction, restricted to
+the U(1)-invariant sector.
+Our first task will be to study the vacuum stability
+of tricritical QED3. To this end, we shall compute the
+effective potential of the theory, at the leading order in
+the 1/Nf expansion and exactly in κ and h. Following
+the standard strategy [44], we therefore rewrite the sextic
+interaction in (2) in terms of two auxiliary fields σ and ρ
+σ
+�
+φ†mφm
+�
+Nf
+− ρ
+�
++
+h
+�
+Nf
+ρ3 .
+(3)
+Here σ is a Lagrange multiplier field, while the field ρ
+is identified with the composite operator φ†mφm/
+�
+Nf.
+Then, we let φm =
+�
+Nfvm + δφm, ρ =
+�
+Nfη + δρ, and
+σ =
+�
+NfΣ+δσ, being vm, η, and Σ vacuum expectation
+values (v.e.v.) that scale as O(N 0
+f ).
+Finally we path-
+integrate over the fluctuations δφm, δρ, and δσ to get (in
+dimensional regularization) the following result
+Veff(v2, Σ, η) = Nf
+�
+Σv2 − Ση + hη3 − 1
+6π Σ
+3
+2
+�
+,
+(4)
+where v2 = vmvm ≥ 0 (without loss of generality we take
+vm = vδm1), and we require Σ ≥ 0. The derivatives of
+the potential w.r.t. v, Σ, and η are
+2Σv = 0 ,
+v2 − η −
+√
+Σ
+4π = 0 ,
+−Σ + 3hη2 = 0 .
+(5)
+We refer to these as “gap equations” for v, Σ, and η,
+respectively. As expected from the absence of scales in
+the problem, the only stationary point is at (Σ, v, η) =
+P ∗ ≡ (0, 0, 0).
+For later reference, it is also useful to
+consider only the gap equations for v and Σ, in which
+case we see that there are two classes of solutions:
+v ̸= 0 ,
+Σ = 0 ,
+η = v2 > 0 ,
+(Higgsed),
+v = 0 ,
+Σ ≥ 0 ,
+η = −
+√
+Σ
+4π ≤ 0 ,
+(unHiggsed).
+(6)
+Let us now discuss quantum stability. Unlike ρ, the
+auxiliary field σ does not correspond on-shell to a phys-
+ical operator of the theory. As a result we do not study
+the stability as a function of the expectation value Σ, but
+rather we “integrate it out” by plugging its gap equation
+back in the effective potential. Note that the existence
+of a solution Σ ≥ 0 imposes the constraint η ≤ v2 on the
+remaining variables. In this way we express Veff as
+Veff(v2, η) = Nf
+�16π2
+3
+�
+v2 − η
+�3 + hη3
+�
+, η ≤ v2 . (7)
+This potential has a global minimum in P ∗ if3
+0 < h < 16π2
+3
+.
+(8)
+Note that, while classically η ≥ 0 and so h ≥ 0, quan-
+tum mechanically η can be negative – the only constraint
+being η ≤ v2 – and the stability region is restricted to
+eq. (8). In other words, bosonic self-interactions are re-
+pulsive and tend to destabilize the vacuum. In the ap-
+pendix, we discuss the effective potential and the phases
+of the theory in presence of a massive and a quartic de-
+formation, showing how the same bound on h holds.
+As a consistency check, note that the determinant of
+the Hessian H of the effective potential in eq. (4)
+det H = −
+�√
+Σ
+4π + 4v2
+�
+6hη − 2Σ ,
+(9)
+when restricted to the Higgsed and unHiggsed directions
+of eq. (6) reads, respectively
+det HH = −24hη2 ,
+det HuH = (6h − 32π2)η2 .
+(10)
+These quantities are both negative everywhere (except
+P ∗) precisely when the bound in eq. (8) is satisfied. This
+has to be the case since a stable quantum vacuum P ∗
+should be at the same time a minimum of Veff in two di-
+rections and a maximum in the third direction. Indeed,
+while in the path integrals for δφm and δρ the integration
+contours run along the real axis, in the one for δσ it runs
+along the imaginary axis. Correspondingly, the Hessian
+of Veff must have two positive and one negative eigenval-
+ues in the neighborhood of P ∗ and its determinant should
+be negative.
+Our next task will be to verify whether the condition
+shown in eq. (8) is satisfied at the RG fixed points of
+tricritical QED3. To this end we shall compute the β
+3 This result is consistent with ref. [45], where the κ → ∞ limit
+of the stability bound can be recovered as a particular case of
+their eq. (1.10), namely in the limit λB → 0. We find perfect
+agreement using the dictionary 4π2λ2
+Bx6 = h.
+In addition, a
+discussion on the stability of the multicritical point in the large
+N bosonic vector model can also be found in section 10.2 of
+ref. [44]. Our stability bound agrees with eq. (10.22) therein.
+
+3
+function for h. At the leading order in the 1/Nf expan-
+sion and to all orders in κ and h, the result is [43, 46]
+βh(h, κ) =
+1
+π2Nf
+�
+− 9
+256h3 + 9
+4h2
++ 128π2(π2 − 128κ2)
+(π2 + 64κ2)2
+h −16384π4(π2 − 192κ2)
+3(π2 + 64κ2)3
+�
+.
+(11)
+This β function is a cubic polynomial in h and gauge
+interactions only affect the linear and constant terms.
+Furthermore, it is an even function of κ, so we can assume
+κ > 0 while solving for its zeros.
+Let us first inspect the zeros of βh for κ → ∞.
+In
+this limit eq. (11) reproduces the known results for the
+β function of the sextic coupling in the free ungauged
+O(N) model from ref. [47], i.e.4
+βh(h, ∞) =
+1
+π2Nf
+�
+− 9
+256h3 + 9
+4h2
+�
+.
+(12)
+Hence, we find a double zero at h∗ = 0 and a single zero
+at h∗ = 64. The sextic operator (φ†φ)3 has dimension
+exactly 3 in the free CFT at h∗ = 0, and is marginally
+irrelevant for h > 0 and marginally relevant for h < 0.
+On the other hand we have ∂hβh(h∗ = 64, ∞) < 0 making
+the operator relevant there. Note that h∗ = 64 is outside
+the window (8) of vacuum stability, while h∗ = 0 is at the
+boundary of the window, and the corresponding theory
+has a stable vacuum since it is a free CFT.
+As a next step, we shall inspect the zeros of βh for
+κ = 0. In this limit eq. (11) becomes
+βh(h, 0) =
+1
+π2Nf
+�
+− 9
+256h3
++9
+4h2 + 128 h − 16384
+3
+�
+,
+(13)
+and therefore we find one fixed point with irrelevant
+(φ†φ)3 at h∗ = 32(
+√
+17 − 1)/3 and two fixed points
+with relevant (φ†φ)3 at h∗
+= 256/3 and at h∗
+=
+−32(
+√
+17 + 1)/3.
+Note that only the fixed point at
+h∗ = 32(
+√
+17 − 1)/3 satisfies the vacuum stability bound
+(8). Having two singlet relevant deformations, φ†φ and
+(φ†φ)2, this CFT can be identified with tricritical QED3
+at large Nf and k = 0.
+Besides these special values of κ, we can solve numeri-
+cally for the zeros of βh and plot them as functions of κ,
+as shown in figure 1. We see that there are three fami-
+lies of solutions. A family of zeros with relevant (φ†φ)3
+(depicted in red in the figure) that exists for any value
+of κ ≥ 0, and two families of zeros with (φ†φ)3 irrele-
+vant/relevant (depicted in blue/green, respectively) that
+exist for κ ≤ κ0 ≃ 0.229, above which they annihilate
+4 In particular, our result (12) matches eq. (2a) of [47], using the
+dictionary N = 2Nf and 8π2λ = 3hN2
+f .
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+- 60
+- 40
+- 20
+0
+20
+40
+60
+80
+FIG. 1. Families of fixed points of βh as a function of κ for
+tricritical QED3.
+The fixed points corresponding to stable
+vacua must lie inside the gray region given by 0 < h < 16π2
+3
+.
+and move to the complex plane, until they reappear at
+(κ, h) = (∞, 0). In the figure, the region highlighted in
+gray corresponds to the values of h satisfying the vac-
+uum stability bound in eq. (8).
+The blue curve is in-
+side the stability region for κ between κ = 0 (where
+h∗ = 32(
+√
+17 − 1)/3) and κ = π/8
+√
+3 ≃ 0.227 (where
+h∗ = 0). The red curve is inside the stability region in
+the interval 0.518 ≲ κ ≲ 1.082.
+Outside these values
+of κ there is no CFT with a stable vacuum. Note that
+only the family of fixed points corresponding to the blue
+curve properly deserves the name of “tricritical QED”
+since the sextic operator is irrelevant there, so the sec-
+ond order transition is reached by tuning the mass and
+the quartic coupling, while also the sextic coupling re-
+quires tuning to reach the family of fixed points on the
+red curve.
+From the results of refs. [43, 46] we can also extract the
+anomalous dimension of the lowest-lying singlet φ†mφm,
+at the leading order in 1/Nf and to all orders in κ
+γφ†φ = 128
+3Nf
+π2 − 128κ2
+(π2 + 64κ2)2 ,
+(14)
+which does not depend on h at this order of the large Nf
+expansion. As a consistency check, note that for κ = 0
+we recover the results of ref. [21] for tricritical QED3,
+while for κ → ∞ the anomalous dimension vanishes, as
+we expect since the theory is simply the one of 2Nf free
+real scalars.
+
+4
+Critical QED
+The second theory we consider is critical QED3, where
+the mass of the Nf complex scalars is still tuned to zero,
+but the quartic coupling λ is at its non-trivial critical
+point, i.e. λ = ∞. The matter Lagrangian is
+Lmatter = (Dµφm)†(Dµφm) +
+σ
+�
+Nf
+φ†mφm ,
+(15)
+where Dµ = ∂µ + iaµ is the covariant derivative and σ
+is the Hubbard-Stratonovich field with scaling dimension
+2 + O(1/Nf). We did not include a sextic term in the
+Lagrangian, as from the equation of motion of σ we get
+that φ†mφm = 0. As in the case of tricritical QED3, the
+continuous global symmetry is SU(Nf) × U(1)m and the
+theory further enjoys parity symmetry when κ = 0. In
+the limit κ → ∞ the gauge field decouples and theory
+becomes the critical O(2Nf) vector model restricted to
+the U(1)-singlet sector. As it turns out, at the leading
+order in the 1/Nf expansion, critical QED3 exists as a
+stable CFT for any value of κ. One can verify this by
+computing the effective potential Veff of the theory at
+large Nf. Following a similar strategy as in the case of
+tricritical QED3, we write φm =
+�
+Nfvm +δφm and σ =
+�
+NfΣ + δσ, and we path-integrate over the fluctuations
+δφm and δσ to get
+Veff(v2, Σ) = Nf
+�
+Σv2 − 1
+6π Σ
+3
+2
+�
+,
+(16)
+with again the condition Σ ≥ 0. The gap equations for v
+and Σ read, respectively
+2Σv = 0 ,
+v2 −
+√
+Σ
+4π = 0 ,
+(17)
+which imply that the vacuum is (v, Σ) = P ∗ ≡ (0, 0). In
+order for P ∗ to be a stable vacuum, we require that, in
+the neighborhood of P ∗, one eigenvalue of the Hessian is
+positive and one is negative. This is equivalent to
+det H = −
+√
+Σ
+4π − 4v2 < 0 ,
+(18)
+which is satisfied everywhere (except at most P ∗). Since
+the eigenvalues never change sign, this condition turns
+out to be also sufficient and we conclude that the vacuum
+of critical QED3 is always stable for any value of κ.
+From the results of refs. [43, 46] we can also extract
+the anomalous dimension of the lowest-lying singlet σ, at
+the leading order in 1/Nf and to all orders in κ
+γσ = −
+16
+3π2Nf
+9π4 − 896π2κ2 + 4096κ4
+(π2 + 64κ2)2
+.
+(19)
+As a consistency check, note that for κ = 0 we recover
+the results of ref. [22], while for κ → ∞ we recover the
+results of refs. [48, 49] for the critical O(2Nf) model.
+III.
+FERMIONIC THEORIES
+Gross-Neveu QED
+The third theory we consider is Gross-Neveu QED3,
+the theory of Nf massless Dirac fermions ψm (m =
+1, . . . , Nf), coupled to U(1)k and with the quartic cou-
+pling g at the UV fixed point, i.e. g = ∞. The matter
+Lagrangian is
+Lmatter = ¯ψm /Dψm +
+σ
+�
+Nf
+¯ψmψm +
+y
+�
+Nf
+σ3 ,
+(20)
+where Dµ = ∂µ + iaµ is the covariant derivative and σ
+is the Hubbard-Stratonovich field with scaling dimen-
+sion 1 + O(1/Nf), whose equation of motion imposes
+¯ψmψm = 0.
+We have included in the Lagrangian the
+classically marginal coupling y (held fixed in the large
+Nf limit), which is generated for any finite κ ̸= 0. Simi-
+larly to the bosonic case, the continuous global symmetry
+is SU(Nf) × U(1)m, and theory also enjoys parity sym-
+metry if κ = y = 0 and Nf is even. In the limit κ → ∞
+we recover the O(2Nf) Gross-Neveu model restricted to
+the U(1)-singlet sector.
+We would like to study the vacuum stability of Gross-
+Neveu QED3, and to this end we shall compute the effec-
+tive potential Veff. Similarly to what we did for bosonic
+QEDs, we will perform the computation at the leading
+order in the 1/Nf expansion and to all orders in κ and y.
+Therefore, we write σ =
+�
+NfΣ+δσ, being Σ a v.e.v. that
+scales as O(N 0
+f ), and we path-integrate over ψm and δσ.
+Importantly, in contrast to the bosonic case, for fermionic
+vector models the Hubbard-Stratonovich field has to be
+path-integrated along the real axis, in order to get a con-
+vergent path integral. All in all we get
+Veff(Σ) = Nf
+�
+yΣ3 + 1
+6π |Σ|3
+�
+.
+(21)
+In eq. (21) the first term is the classical contribution,
+whereas the second term comes from quantum fluctua-
+tions. Due to the real path-integration contour of σ, the
+stability of the vacuum at Σ = 0 now requires that
+det H = 6
+�
+y sign(Σ) + 1
+6π
+�
+|Σ| > 0 ,
+(22)
+and therefore we get the stability bound5
+|y| < 1
+6π .
+(23)
+Again, this condition implies that the local minimum
+is also a global one, since the second derivative never
+5 This result is consistent with ref. [45], where the κ → ∞ limit of
+the stability bound can be recovered as a particular case of their
+eq. (1.10), namely in the limit |λF | = 1 − |λB| → 0. We find
+perfect agreement using the dictionary x6/λF = −16πy.
+
+5
+changes its sign. As is turns out, a cubic scalar potential
+for Σ – which would be classically unbounded from below
+since Σ can have both signs, corresponding to the sign
+of the effective fermion mass – is allowed by quantum
+corrections as long as y lies within the region of eq. (23).
+This is due to the fact that fermionic self-interactions are
+attractive and tend to stabilize the vacuum, as opposite
+to what happens in the bosonic case. In the appendix, we
+discuss the effective potential and the phases of the the-
+ory in presence of a massive and a quartic deformation,
+showing how the same bound on y holds.
+Next, we shall check whether the condition shown in
+eq. (23) is satisfied at the RG fixed points of Gross-Neveu
+QED3. To this end we shall compute the β function for
+y. At the leading order in the 1/Nf expansion and to all
+orders in κ and y we find [43]
+βy(y, κ) =
+32
+3π2Nf
+�
+−864y3 + 3(4096κ4 + 640π2κ2 − 3π4)
+(π2 + 64κ2)2
+y
++ 4π3(320κ2 − 3π2)κ
+(π2 + 64κ2)3
+�
+.
+(24)
+This β function is a cubic polynomial in y with no
+quadratic term, and gauge interactions only affect the
+linear and constant terms. Since the zeros of βy are odd
+functions of κ, we can assume κ > 0 without loss of gen-
+erality. Before discussing the zeros of βy for generic κ,
+let us first study the special cases for κ = 0 and κ → ∞.
+When κ = 0 from eq. (24) we get
+βy(y, 0) =
+96
+π2Nf
+�
+−96 y3 − 3y
+�
+,
+(25)
+and so we find only one fixed point corresponding to the
+real zero at y∗ = 0, which lies in the middle of the sta-
+bility region of eq. (23). Since ∂yβy(y∗ = 0, 0) < 0, the
+operator σ3 is relevant here, meaning that tuning of y is
+needed to reach this fixed point.
+In the limit κ → ∞ the β function agrees with the
+findings of ref. [45] for the ungauged Gross-Neveu model6
+βy(y, ∞) =
+32
+π2Nf
+�
+−288 y3 + y
+�
+.
+(26)
+This β function has a fixed point at y∗ = 0 with σ3 ir-
+relevant, and two (parity-related) zeros at y∗ = ±
+√
+2/24
+with σ3 relevant.
+Only the fixed point at y∗ = 0 lies
+within the vacuum stability bound, and corresponds to
+the usual Gross-Neveu CFT.
+In general, we can solve numerically for the zeros of
+βy as functions of κ. The results are presented in fig-
+ure 2, which shows that there are three families of ze-
+ros. A first family of fixed points, depicted by the red
+6 In particular, our result (26) matches eq. (3.65) of ref. [45], using
+the dictionary NF = Nf and λF
+6 = −16πy.
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+- 0.05
+0.00
+0.05
+0.10
+FIG. 2. Families of fixed points of βy as a function of κ for
+Gross-Neveu QED3. The fixed points corresponding to stable
+vacua must lie inside the gray region given by |y| <
+1
+6π .
+curve in the figure, has a relevant σ3 operator and exists
+for any value of κ, interpolating from (κ, y) = (0, 0) to
+(κ, y) = (∞, −
+√
+2/24). Two additional families of fixed
+points are depicted by the blue and green curve in the fig-
+ure. The operator σ3 is irrelevant/relevant there, respec-
+tively. They exist for every κ ≥ κ1 ≃ 0.273, below which
+they annihilate and become complex conjugate, and they
+reach the values y = 0 and y = +
+√
+2/24, respectively, for
+κ → ∞. The fixed points on the red curve instead are
+stable only for κ ≲ 0.162. The fixed points on the blue
+curve always lie within the region of vacuum stability.
+Finally, the fixed points on the green curve have a sta-
+ble vacuum only if κ1 ≲ κ ≲ 0.283. For 0.162 ≲ κ ≲ κ1
+there is a ‘blind spot’ where there is no CFT with a stable
+vacuum.
+The anomalous dimension of the lowest-lying singlet σ
+at the leading order in 1/Nf and to all orders in κ is [43]
+γσ = −
+16
+3π2Nf
+9π4 − 896π2κ2 + 4096κ4
+(π2 + 64κ2)2
+,
+(27)
+which does not depend on y.
+As already observed in
+ref. [46] for the special case of κ = 0, we find that we find
+that this result is identical to that for the anomalous
+dimension of σ in critical bosonic QED3, see eq. (19), up
+to O(1/N 2
+f ). As a consistency check, in the limit κ →
+∞ we indeed recover the anomalous dimension for the
+Hubbard-Stratonovich field of the O(2Nf) Gross-Neveu
+model [50], while for κ = 0 we recover the results of
+refs. [21, 51].
+
+6
+Fermionic QED
+The fourth theory we consider is fermionic QED3,
+where the mass of the Nf Dirac fermions is still tuned
+to zero, but there is no quartic fermionic self-interaction.
+The matter Lagrangian is
+Lmatter = ¯ψm /Dψm ,
+(28)
+where Dµ = ∂µ+iaµ is the covariant derivative. Similarly
+to the Gross-Neveu case, the continuous global symme-
+try is SU(Nf) × U(1)m, and theory also enjoys parity
+symmetry if κ = 0 and Nf is even. In the limit where
+κ → ∞, it describes 2Nf free Majorana fermions re-
+stricted to the U(1)-singlet sector. Fermionic QED3 has
+no marginal deformations and it leads to a stable CFT
+for any κ, as expected. Indeed, since the large-Nf effec-
+tive potential does not depend on gauge interactions, the
+stability analysis is the same as the one for a free fermion
+theory, which is clearly always stable.
+The anomalous dimension of the lowest-lying singlet
+¯ψmψm, at the leading order in 1/Nf and to all orders in
+κ reads [43]
+γ ¯
+ψψ = 128
+3Nf
+π2 − 128κ2
+(π2 + 64κ2)2 .
+(29)
+As a consistency check, in the limit κ → ∞ we get γ ¯
+ψψ =
+0, as it should be for a theory of free fermions, while for
+κ = 0 we recover the result of refs. [52, 53]. Again, we
+have that at this order of the large-Nf expansion, the
+fermionic result agrees with the bosonic one, γ ¯
+ψψ = γφ†φ.
+IV.
+DISCUSSION
+In this work, we have studied the fixed points of either
+bosonic or fermionic three-dimensional large-Nf QEDs
+coupled to a Chern-Simons term at level k, with fixed
+κ = k/Nf. For each of these theories we computed the
+effective potential, as well as the β functions of the clas-
+sically marginal couplings, at the leading order in the
+1/Nf expansion and to all orders in κ and in the cou-
+plings.
+For tricritical bosonic QED3 and Gross-Neveu
+QED3 our analysis shows that a CFT with a stable vac-
+uum can only exist when the parameter κ is restricted to
+lie in certain intervals, while there is no such restriction
+for critical bosonic QED3 and fermionic QED3.
+In a forthcoming paper [43] we provide details about
+the derivation of the results presented here, in particular
+regarding the computation of the β functions, and we
+couple the 3d large Nf models studied in this work to
+a 4d real scalar field via a bulk/boundary interaction.
+This allows us to give an explicit Lagrangian construction
+of interacting conformal boundary conditions for the 4d
+CFT of a free scalar.
+ACKNOWLEDGEMENTS
+We are grateful to Ofer Aharony and Marco Serone for
+discussions. LD acknowledges support from the program
+“Rita Levi Montalcini” for young researchers and from
+the INFN “Iniziativa Specifica ST&FI”. EL would like
+to thank Nikolay Bobev and Balt van Rees for their sup-
+port. PN would like to thank Thomas Dumitrescu for
+discussions, and Lorenzo Maffi for a useful conversation
+on global stability. PN is grateful to ICTP and University
+of Trieste for the kind hospitality during the preparation
+of this work. The work of PN is supported by the Mani
+L. Bhaumik Institute for Theoretical Physics and by a
+DOE Early Career Award under DE-SC0020421.
+Appendix: Deformations and phase diagrams
+In this appendix, we discuss the phase diagrams of the
+QED theories described above upon turning on relevant
+deformations. Since the effective potential does not de-
+pend on the gauge interactions at leading order in the
+large Nf expansion, our discussion is the same as for the
+case of ungauged vector models, which constitute a par-
+ticular case of the analysis performed in refs. [45, 54, 55].
+Let us first consider the bosonic cases.
+Tricritical
+QED3 admits two relevant deformations, the mass m2
+and the quartic coupling λ. The Lagrangian in the pres-
+ence of such terms reads
+Lmatter = (Dµφm)†(Dµφm) + σ
+�
+φ†mφm
+�
+Nf
+− ρ
+�
++
+h
+�
+Nf
+ρ3 + λ ρ2 +
+�
+Nfm2ρ .
+(A.1)
+The effective potential is easily found to be
+Veff = Nf
+�
+Σv2 − Ση − 1
+6π Σ
+3
+2 + h η3 + λ η2 + m2 η
+�
+.
+(A.2)
+The gap equations for v2 and Σ
+2Σv = 0 ,
+v2 − η −
+√
+Σ
+4π = 0 ,
+(A.3)
+do not depend on the scalar potential and they can be
+used to express the effective potential as a function of η
+only:
+Veff
+Nf
+=
+�
+h η3 + λ η2 + m2 η
+if η > 0 ,
+�
+h − 16π2
+3
+�
+η3 + λ η2 + m2 η
+if η < 0 . (A.4)
+Note that the condition for this potential to be bounded
+from below gives the same stability bound as in eq. (8)
+0 < h < 16π2
+3
+.
+(A.5)
+
+7
+FIG. 3. Phase diagram of tricritical QED3 as a function of
+λ and m2, in the range 0 < h < 16π2/3. We have depicted
+in blue the second-order line and the tricritical point. Above
+the dashed gray lines, there is a unique minimum (an unHig-
+gsed vacuum in green, a Higgsed vacuum in yellow), whereas
+below such lines the two distinct minima coexist and become
+degenerate on the first-order line, depicted in red. Its loca-
+tion is specified by f(h), here we have chosen for definiteness
+a value h < 8π2/3, so that the transition happens for m2 > 0.
+As we have previously shown, tricritical QED3 admits
+an IR fixed point h∗ satisfying the stability bound above
+only if 0 ≤ κ < π/8
+√
+3 and if the flow starts from a UV
+value of h in the basin of attraction of h∗, hgreen < h <
+hred (see figure 1). From this new Veff, the vacuum is
+determined by the gap equations for η, i.e.
+�
+3h η2 + 2λ η + m2 = 0
+if η > 0 ,
+(3h − 16π2)η2 + 2λ η + m2 = 0
+if η < 0 .
+(A.6)
+From these results we can determine the phase diagram
+of the theory, which is depicted in figure 3. First, let us
+introduce two useful auxiliary functions of the couplings
+η+ = −λ +
+√
+λ2 − 3hm2
+3h
+,
+η− = λ −
+�
+λ2 + (16π2 − 3h)m2
+16π2 − 3h
+.
+(A.7)
+• If λ ≥ 0 and m2 > 0, there is a unique minimum
+at η = η− < 0 (unHiggsed).
+• If λ ≥ 0 and m2 < 0 there is a unique minimum at
+η = η+ > 0 (Higgsed).
+• If λ ≥ 0 and m2 = 0 there is a second-order phase
+transition between the two vacua, which coalesce
+to η = 0.
+• If λ < 0 and m2 > λ2
+3h, there is a unique minimum
+at η = η− < 0 (unHiggsed).
+• If λ < 0 and m2 < −
+λ2
+16π2−3h, there is a unique
+minimum at η = η+ > 0 (Higgsed).
+• If λ < 0 and −
+λ2
+16π2−3h ≤ m2 ≤ λ2
+3h (dashed lines in
+the figure), there are two competing local minima,
+one at η = η+ > 0 and one at η = η− < 0. The
+two distinct minima become exactly degenerate at
+m2 = λ2f(h), where a first-order phase transition
+happens.
+The function f(h) is found to be
+f(h) =
+8π2 − 3h + 8π2 sin
+�
+1
+3 arctan
+8π2−3h
+√
+3h(16π2−3h)
+�
+3h(16π2 − 3h)
+.
+(A.8)
+It is straightforward to show that
+f
+�16π2
+3
+− h
+�
+= −f(h) ,
+1
+16π2 − 3h ≤ f(h) ≤ 1
+3h ,
+(A.9)
+so that f(h) is an odd function with respect to h = 8π2/3
+(where it vanishes), and it is monotonically decreasing
+from f(0) = +∞ to f(16π2/3) = −∞. Thus, the first-
+order line (λ < 0, m2 = λ2f(h)) – depicted in red in
+the figure – is at m2 > 0 (m2 < 0) if 0 < h < 8π2/3
+(8π2/3 < h < 16π2/3), and at m2 = 0 if h = 8π2/3. It
+terminates at the tricritical point (λ, m2) = (0, 0), where
+it merges with the second-order line (λ > 0, m2 = 0) –
+depicted in blue in the figure.
+Both transitions are between a phase (at η = η−, green
+in the figure) described by a U(1)k Chern-Simons the-
+ory with an unbroken SU(Nf) global symmetry, and a
+phase (at η = η+, yellow in the figure) where U(1)k is
+Higgsed and the SU(Nf) part of the global symmetry is
+broken down to SU(Nf−1). The tricritical point is where
+the tricritical QED3 CFT sits, whereas any point on the
+second-order line is described by the critical QED3 CFT.
+Indeed, at low energies, any λ > 0 flows to the critical
+value λ = ∞. Note that the location of the phase tran-
+sitions depends on the regularization scheme (we have
+used dimensional regularization in this work), but the
+existence of the various phases does not.
+Let us now discuss the phase diagram for critical
+QED3. The only relevant deformation is the mass term
+Mσ (whereas the quartic term σ2/λ is obviously irrele-
+vant), and so we consider
+Lmatter = (Dµφm)†(Dµφm) +
+σ
+�
+Nf
+φ†mφm +
+�
+NfM σ .
+(A.10)
+The effective potential reads
+Veff = Nf
+�
+Σv2 − 1
+6π Σ
+3
+2 + M Σ
+�
+,
+(A.11)
+
+8
+which is always stable, as expected. The gap equations
+for v2 and Σ
+2Σv = 0 ,
+v2 −
+√
+Σ
+4π + M = 0 ,
+(A.12)
+imply that there are two classes of vacua: if M > 0,
+then v2 = 0 and Σ = 16π2M, so that the theory is in
+the unHiggsed phase; if M < 0, then v2 = −M and
+Σ = 0, so that the theory is in the Higgsed phase. For
+M = 0 there is a second-order phase transition where
+the vacuum is (v2, Σ) = (0, 0), and this is described by
+the critical QED3 CFT. We see that any horizontal line
+at λ > 0 intersecting the phase diagram of figure 3 re-
+produces this one-dimensional phase diagram of critical
+QED3, as it should be. An alternative way to arrive to
+the same conclusion is to allow ourselves to use the gap
+equation for Σ to express Veff as a function of v2 only
+Veff
+Nf
+= 16π2
+3
+(v2 + M)3 ,
+(A.13)
+with the condition v2 ≥ max(−M, 0).
+Note that this
+potential is always bounded from below, confirming that
+the vacuum is always stable. Interestingly, the upper sta-
+bility bound that we found for tricritical QED3 amounts
+to require that the sextic coupling h does not exceed the
+value it has, effectively, in critical QED3.
+Finally, let us discuss the phase diagram when we are
+outside the stability bound.
+This scenario is depicted
+in figure 4.
+By investigating the gap equations, one
+can show that if h is outside the stability interval, then
+there is a region of the phase diagram where the gap
+equations (A.6) do not have any solution. This region
+stays in the half-plane of m2 < 0 (m2 > 0) for h < 0
+(h > 16π2/3) and it is delimited by the curves m2 = λ2
+3h
+and m2 =
+λ2
+3h−16π2 (yellow and green lines in the figure,
+respectively). Moreover, when the gap equations have a
+solution, this always corresponds to a metastable mini-
+mum, since the effective potential is unbounded from be-
+low. Curiously, while the first-order transition for λ < 0
+disappears, the second-order transition for λ > 0 is still
+present. We interpret this as the fact that the sextic term
+in critical QED3 is always (strongly) irrelevant. Hence,
+starting from tricritical QED3 with any sextic coupling
+h, as soon as we switch on the relevant quartic deforma-
+tion with positive λ, at low energies the theory always
+flows to the critical theory, irrespectively of the initial
+value of h. The metastable vacua of figure 4 become sta-
+ble in the limit λ → +∞, when the runaway behavior is
+suppressed.
+Let us now consider the fermionic case. Gross-Neveu
+QED3 admits two relevant deformations, the mass ˆm2
+and the quartic coupling ˆλ. The Lagrangian reads
+Lmatter = ¯ψm /Dψm +
+σ
+�
+Nf
+¯ψmψm
++
+y
+�
+Nf
+σ3 + ˆλ σ2 +
+�
+Nf ˆm2 σ ,
+(A.14)
+FIG. 4. Phase diagram of tricritical QED3 as a function of λ
+and m2, in the range h < 0 (top) and h > 16π2/3 (bottom).
+We have depicted in blue the second-order line and the multi-
+critical point, in green the unHiggsed vacuum, and in yellow
+the Higgsed vacuum. In the dark gray regions, no solutions
+to the gap equations exist.
+and leads to the following effective potential
+Veff
+Nf
+=
+��
+y +
+1
+6π
+�
+Σ3 + ˆλ Σ2 + ˆm2 Σ
+if Σ > 0 ,
+�
+y −
+1
+6π
+�
+Σ3 + ˆλ Σ2 + ˆm2 Σ
+if Σ < 0 . (A.15)
+Note that the condition for this potential to be bounded
+from below gives the same stability bound as in eq. (23)
+|y| < 1
+6π .
+(A.16)
+As we have previously shown, Gross-Neveu QED3 admits
+an IR fixed point y∗ satisfying the stability bound above
+only if κ ≳ 0.273 and the flow starts from a UV value
+
+9
+of y in the basin of attraction of y∗, yred < y < ygreen
+(see figure 2). The discussion of the phase diagram of
+Gross-Neveu QED3 is now exactly the same as the one
+of tricritical QED3, using the dictionary
+η ↔
+Σ
+24/3π ,
+h ↔ 16π3
+�
+y + 1
+6π
+�
+,
+λ ↔ 28/3π2 ˆλ ,
+m2 ↔ 24/3π ˆm2 .
+(A.17)
+Obviously, now the phases have a different low-energy
+description.
+Since the sign of Σ gives the sign of the
+effective fermion mass, phases with positive (negative)
+Σ are characterized by a low-energy U(1) Chern-Simons
+theory at level k ± Nf/2, respectively, both with an un-
+broken SU(Nf) global symmetry. The multicritical point
+at the origin is described by the Gross-Neveu QED3 CFT,
+whereas each point on the second-order line (ˆλ > 0,
+ˆm2 = 0) is described at low energies by the fermionic
+QED3 CFT. Indeed, any positive ˆλ flows to ˆλ = ∞ or,
+equivalently, the quartic fermion coupling g ∼ ˆλ−1 flows
+to g = 0.
+Finally, fermionic QED3 has only one relevant defor-
+mation, given by the mass term m ¯ψmψm to be added
+to the Lagrangian (28). Clearly, positive and negative
+m give rise exactly to the same phases we got in Gross-
+Neveu QED3 for positive and negative Σ, respectively.
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diff --git a/yNE3T4oBgHgl3EQflwrt/content/tmp_files/load_file.txt b/yNE3T4oBgHgl3EQflwrt/content/tmp_files/load_file.txt
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@@ -0,0 +1,756 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf,len=755
+page_content='Vacuum stability, fixed points, and phases of QED3 at large Nf  Lorenzo Di Pietro,1 Edoardo Lauria,2 and Pierluigi Niro3  1Dipartimento di Fisica, Universit`a di Trieste, Strada Costiera 11, I-34151 Trieste, Italy,  & INFN, Sezione di Trieste, Via Valerio 2, I-34127 Trieste, Italy  2CPHT, CNRS, Institut Polytechnique de Paris, France  3Mani L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy,  University of California, Los Angeles, CA 90095, USA  We consider three-dimensional Quantum Electrodynamics in the presence of a Chern-Simons term  at level k and Nf flavors, in the limit of large Nf and k with k/Nf fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We consider either bosonic  or fermionic matter fields, with and without quartic terms at criticality: the resulting theories are  critical and tricritical bosonic QED3, Gross-Neveu and fermionic QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' For all such theories we  compute the effective potentials and the β functions of classically marginal couplings, at the leading  order in the large Nf limit and to all orders in k/Nf and in the couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We determine the RG  fixed points and discuss the quantum stability of the corresponding vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' While critical bosonic  and fermionic QED3 are always stable CFTs, we find that tricritical bosonic and Gross-Neveu QED3  exist as stable CFTs only for specific values of k/Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Finally, we discuss the phase diagrams of these  theories as a function of their relevant deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  INTRODUCTION  Three-dimensional Quantum Electrodynamics (QEDs)  with either bosonic or fermionic degrees of freedom are  among the simplest and yet very rich examples of gauge  theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' At high energies, these theories are defined in  terms of a three-dimensional U(1) Maxwell field a with  gauge coupling e2 (of mass dimension one) and a Chern-  Simons term at level k, coupled to Nf flavors of charged  bosons or fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The Lagrangian is (we will always  work in Euclidean signature)  L =  1  2e2 da ∧ ⋆da + ik  4π a ∧ da + Lmatter .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (1)  In absence of charged matter, if k = 0 the theory is  dual to a compact scalar and it confines in the sense of  [1, 2], whereas if k ̸= 0 it flows to a pure U(1)k Chern-  Simons theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In presence of charged matter, the low-  energy behavior of QED theories for small Nf has not  been rigorously established, and this question remains  an open problem which has been studied with a vari-  ety of approaches (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [3–24]), including lattice  simulations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [25–31]) and conformal boot-  strap techniques (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [32–41] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  When Nf is large enough, these theories are generically  expected to flow to interacting Conformal Field Theories  (CFTs) at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' This expectation is corroborated  by the existence of fixed points of the Renormalization  Group (RG) flow, which can be found in the limit of  large Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' However, as we will discuss in this work, some  of these fixed points happen to lie in a region of instabil-  ity of the theory, and so not all of them define a CFT at  the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  In order to clarify this issue, we will compute the ef-  fective potentials Veff for a few instances of such Quan-  tum Electrodynamics (to be defined below): two bosonic,  called ‘tricritical’ and ‘critical’ QED3, and two fermionic,  called ‘Gross-Neveu’ and ‘fermionic’ QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' While criti-  cal and fermionic QED3 do not admit any marginal defor-  mations, tricritical and Gross-Neveu QED3 admit clas-  sically marginal couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Imposing the stability of the  vacuum gives rise to constraints for such couplings, which  have to be satisfied at the fixed points of the RG flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  We will perform these computations by working at the  leading order in the 1/Nf expansion, with Λ = e2Nf and  κ = k/Nf held fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We study the low-energy limit by  taking Λ → ∞ while retaining the dimensionless param-  eter κ which does not run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The gauge interactions do  not enter Veff at the leading order in the large-Nf limit,  nevertheless the parameter κ will play an important role  in the following analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='1  Next, we will compute the β functions of classically  marginal couplings of tricritical and Gross-Neveu QEDs,  at leading order in the 1/Nf expansion and to all orders in  κ and in the couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Details of the RG computations  will be presented elsewhere [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The zeroes of the β func-  tions determine families of RG fixed points parametrized  by κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We finally impose on those fixed points the con-  straint of vacuum stability from Veff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='2 The result of this  analysis is summarized in figures 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Along the way,  we will also present the expressions for the anomalous di-  mensions of the lowest-lying singlet operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In a final  appendix, we derive the large-Nf phase diagrams of all  four QEDs in presence of relevant deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  1 The Maxwell (Yang-Mills) kinetic term in general does affect the  vacuum structure of QED (QCD) theories with finite Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' This  term distinguishes between type I and type II superconductivity  in critical bosonic QED3 with Nf = 1 and k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Also, it affects  the existence of a conformal critical point in SU(N)k QCD3 at  large N and finite Nf, as a function of k/N [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' However, for  large-Nf QEDs, the whole dependence on Λ is encoded in the  exact photon propagator in such a way that the limit Λ → ∞  can be taken reliably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  2 To avoid possible confusion, let us stress that here by “stability”  we refer to the condition that the potential has a stable minimum,  not to the RG stability of the fixed point under deformations by  some coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In this letter we always use the term “stability”  in the former sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='04611v1  [hep-th]  11 Jan 2023  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  BOSONIC THEORIES  Tricritical QED  The first theory we consider is tricritical QED3,  the theory of Nf massless complex scalars φm (m =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' , Nf) coupled to U(1)k and with the quartic cou-  pling tuned to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The matter Lagrangian is  Lmatter = (Dµφm)†(Dµφm) + h  N 2  f  (φ†mφm)3 ,  (2)  where Dµ = ∂µ + iaµ denotes the covariant derivative  and h is held fixed in the large-Nf limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The continuous  part of the global symmetry is SU(Nf) × U(1)m, where  the first factor is a flavor symmetry and the second one is  the magnetic symmetry of the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' For κ = 0 the  theory further enjoys parity symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In the κ → ∞  limit the gauge field decouples and the theory describes  2Nf real scalars with a sextic interaction, restricted to  the U(1)-invariant sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Our first task will be to study the vacuum stability  of tricritical QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' To this end, we shall compute the  effective potential of the theory, at the leading order in  the 1/Nf expansion and exactly in κ and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Following  the standard strategy [44], we therefore rewrite the sextic  interaction in (2) in terms of two auxiliary fields σ and ρ  σ  �  φ†mφm  �  Nf  − ρ  �  +  h  �  Nf  ρ3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (3)  Here σ is a Lagrange multiplier field, while the field ρ  is identified with the composite operator φ†mφm/  �  Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Then, we let φm =  �  Nfvm + δφm, ρ =  �  Nfη + δρ, and  σ =  �  NfΣ+δσ, being vm, η, and Σ vacuum expectation  values (v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=') that scale as O(N 0  f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Finally we path-  integrate over the fluctuations δφm, δρ, and δσ to get (in  dimensional regularization) the following result  Veff(v2, Σ, η) = Nf  �  Σv2 − Ση + hη3 − 1  6π Σ  3  2  �  ,  (4)  where v2 = vmvm ≥ 0 (without loss of generality we take  vm = vδm1), and we require Σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The derivatives of  the potential w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' v, Σ, and η are  2Σv = 0 ,  v2 − η −  √  Σ  4π = 0 ,  −Σ + 3hη2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (5)  We refer to these as “gap equations” for v, Σ, and η,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' As expected from the absence of scales in  the problem, the only stationary point is at (Σ, v, η) =  P ∗ ≡ (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  For later reference, it is also useful to  consider only the gap equations for v and Σ, in which  case we see that there are two classes of solutions:  v ̸= 0 ,  Σ = 0 ,  η = v2 > 0 ,  (Higgsed),  v = 0 ,  Σ ≥ 0 ,  η = −  √  Σ  4π ≤ 0 ,  (unHiggsed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (6)  Let us now discuss quantum stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Unlike ρ, the  auxiliary field σ does not correspond on-shell to a phys-  ical operator of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' As a result we do not study  the stability as a function of the expectation value Σ, but  rather we “integrate it out” by plugging its gap equation  back in the effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Note that the existence  of a solution Σ ≥ 0 imposes the constraint η ≤ v2 on the  remaining variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In this way we express Veff as  Veff(v2, η) = Nf  �16π2  3  �  v2 − η  �3 + hη3  �  , η ≤ v2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (7)  This potential has a global minimum in P ∗ if3  0 < h < 16π2  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (8)  Note that, while classically η ≥ 0 and so h ≥ 0, quan-  tum mechanically η can be negative – the only constraint  being η ≤ v2 – and the stability region is restricted to  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In other words, bosonic self-interactions are re-  pulsive and tend to destabilize the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In the ap-  pendix, we discuss the effective potential and the phases  of the theory in presence of a massive and a quartic de-  formation, showing how the same bound on h holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  As a consistency check, note that the determinant of  the Hessian H of the effective potential in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (4)  det H = −  �√  Σ  4π + 4v2  �  6hη − 2Σ ,  (9)  when restricted to the Higgsed and unHiggsed directions  of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (6) reads, respectively  det HH = −24hη2 ,  det HuH = (6h − 32π2)η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (10)  These quantities are both negative everywhere (except  P ∗) precisely when the bound in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (8) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' This  has to be the case since a stable quantum vacuum P ∗  should be at the same time a minimum of Veff in two di-  rections and a maximum in the third direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Indeed,  while in the path integrals for δφm and δρ the integration  contours run along the real axis, in the one for δσ it runs  along the imaginary axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Correspondingly, the Hessian  of Veff must have two positive and one negative eigenval-  ues in the neighborhood of P ∗ and its determinant should  be negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Our next task will be to verify whether the condition  shown in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (8) is satisfied at the RG fixed points of  tricritical QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' To this end we shall compute the β  3 This result is consistent with ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [45], where the κ → ∞ limit  of the stability bound can be recovered as a particular case of  their eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='10), namely in the limit λB → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We find perfect  agreement using the dictionary 4π2λ2  Bx6 = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  In addition, a  discussion on the stability of the multicritical point in the large  N bosonic vector model can also be found in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='2 of  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Our stability bound agrees with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='22) therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  3  function for h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' At the leading order in the 1/Nf expan-  sion and to all orders in κ and h, the result is [43, 46]  βh(h, κ) =  1  π2Nf  �  − 9  256h3 + 9  4h2  + 128π2(π2 − 128κ2)  (π2 + 64κ2)2  h −16384π4(π2 − 192κ2)  3(π2 + 64κ2)3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (11)  This β function is a cubic polynomial in h and gauge  interactions only affect the linear and constant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Furthermore, it is an even function of κ, so we can assume  κ > 0 while solving for its zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Let us first inspect the zeros of βh for κ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  In  this limit eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (11) reproduces the known results for the  β function of the sextic coupling in the free ungauged  O(N) model from ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [47], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='4  βh(h, ∞) =  1  π2Nf  �  − 9  256h3 + 9  4h2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (12)  Hence, we find a double zero at h∗ = 0 and a single zero  at h∗ = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The sextic operator (φ†φ)3 has dimension  exactly 3 in the free CFT at h∗ = 0, and is marginally  irrelevant for h > 0 and marginally relevant for h < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  On the other hand we have ∂hβh(h∗ = 64, ∞) < 0 making  the operator relevant there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Note that h∗ = 64 is outside  the window (8) of vacuum stability, while h∗ = 0 is at the  boundary of the window, and the corresponding theory  has a stable vacuum since it is a free CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  As a next step, we shall inspect the zeros of βh for  κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In this limit eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (11) becomes  βh(h, 0) =  1  π2Nf  �  − 9  256h3  +9  4h2 + 128 h − 16384  3  �  ,  (13)  and therefore we find one fixed point with irrelevant  (φ†φ)3 at h∗ = 32(  √  17 − 1)/3 and two fixed points  with relevant (φ†φ)3 at h∗  = 256/3 and at h∗  =  −32(  √  17 + 1)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Note that only the fixed point at  h∗ = 32(  √  17 − 1)/3 satisfies the vacuum stability bound  (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Having two singlet relevant deformations, φ†φ and  (φ†φ)2, this CFT can be identified with tricritical QED3  at large Nf and k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Besides these special values of κ, we can solve numeri-  cally for the zeros of βh and plot them as functions of κ,  as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We see that there are three fami-  lies of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' A family of zeros with relevant (φ†φ)3  (depicted in red in the figure) that exists for any value  of κ ≥ 0, and two families of zeros with (φ†φ)3 irrele-  vant/relevant (depicted in blue/green, respectively) that  exist for κ ≤ κ0 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='229, above which they annihilate  4 In particular, our result (12) matches eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (2a) of [47], using the  dictionary N = 2Nf and 8π2λ = 3hN2  f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='4  60  40  20  0  20  40  60  80  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Families of fixed points of βh as a function of κ for  tricritical QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  The fixed points corresponding to stable  vacua must lie inside the gray region given by 0 < h < 16π2  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  and move to the complex plane, until they reappear at  (κ, h) = (∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In the figure, the region highlighted in  gray corresponds to the values of h satisfying the vac-  uum stability bound in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  The blue curve is in-  side the stability region for κ between κ = 0 (where  h∗ = 32(  √  17 − 1)/3) and κ = π/8  √  3 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='227 (where  h∗ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The red curve is inside the stability region in  the interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='518 ≲ κ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Outside these values  of κ there is no CFT with a stable vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Note that  only the family of fixed points corresponding to the blue  curve properly deserves the name of “tricritical QED”  since the sextic operator is irrelevant there, so the sec-  ond order transition is reached by tuning the mass and  the quartic coupling, while also the sextic coupling re-  quires tuning to reach the family of fixed points on the  red curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  From the results of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [43, 46] we can also extract the  anomalous dimension of the lowest-lying singlet φ†mφm,  at the leading order in 1/Nf and to all orders in κ  γφ†φ = 128  3Nf  π2 − 128κ2  (π2 + 64κ2)2 ,  (14)  which does not depend on h at this order of the large Nf  expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' As a consistency check, note that for κ = 0  we recover the results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [21] for tricritical QED3,  while for κ → ∞ the anomalous dimension vanishes, as  we expect since the theory is simply the one of 2Nf free  real scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  4  Critical QED  The second theory we consider is critical QED3, where  the mass of the Nf complex scalars is still tuned to zero,  but the quartic coupling λ is at its non-trivial critical  point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' λ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The matter Lagrangian is  Lmatter = (Dµφm)†(Dµφm) +  σ  �  Nf  φ†mφm ,  (15)  where Dµ = ∂µ + iaµ is the covariant derivative and σ  is the Hubbard-Stratonovich field with scaling dimension  2 + O(1/Nf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We did not include a sextic term in the  Lagrangian, as from the equation of motion of σ we get  that φ†mφm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' As in the case of tricritical QED3, the  continuous global symmetry is SU(Nf) × U(1)m and the  theory further enjoys parity symmetry when κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In  the limit κ → ∞ the gauge field decouples and theory  becomes the critical O(2Nf) vector model restricted to  the U(1)-singlet sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' As it turns out, at the leading  order in the 1/Nf expansion, critical QED3 exists as a  stable CFT for any value of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' One can verify this by  computing the effective potential Veff of the theory at  large Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Following a similar strategy as in the case of  tricritical QED3, we write φm =  �  Nfvm +δφm and σ =  �  NfΣ + δσ, and we path-integrate over the fluctuations  δφm and δσ to get  Veff(v2, Σ) = Nf  �  Σv2 − 1  6π Σ  3  2  �  ,  (16)  with again the condition Σ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The gap equations for v  and Σ read, respectively  2Σv = 0 ,  v2 −  √  Σ  4π = 0 ,  (17)  which imply that the vacuum is (v, Σ) = P ∗ ≡ (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In  order for P ∗ to be a stable vacuum, we require that, in  the neighborhood of P ∗, one eigenvalue of the Hessian is  positive and one is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' This is equivalent to  det H = −  √  Σ  4π − 4v2 < 0 ,  (18)  which is satisfied everywhere (except at most P ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Since  the eigenvalues never change sign, this condition turns  out to be also sufficient and we conclude that the vacuum  of critical QED3 is always stable for any value of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  From the results of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [43, 46] we can also extract  the anomalous dimension of the lowest-lying singlet σ, at  the leading order in 1/Nf and to all orders in κ  γσ = −  16  3π2Nf  9π4 − 896π2κ2 + 4096κ4  (π2 + 64κ2)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (19)  As a consistency check, note that for κ = 0 we recover  the results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [22], while for κ → ∞ we recover the  results of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [48, 49] for the critical O(2Nf) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  FERMIONIC THEORIES  Gross-Neveu QED  The third theory we consider is Gross-Neveu QED3,  the theory of Nf massless Dirac fermions ψm (m =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' , Nf), coupled to U(1)k and with the quartic cou-  pling g at the UV fixed point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' g = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The matter  Lagrangian is  Lmatter = ¯ψm /Dψm +  σ  �  Nf  ¯ψmψm +  y  �  Nf  σ3 ,  (20)  where Dµ = ∂µ + iaµ is the covariant derivative and σ  is the Hubbard-Stratonovich field with scaling dimen-  sion 1 + O(1/Nf), whose equation of motion imposes  ¯ψmψm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  We have included in the Lagrangian the  classically marginal coupling y (held fixed in the large  Nf limit), which is generated for any finite κ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Simi-  larly to the bosonic case, the continuous global symmetry  is SU(Nf) × U(1)m, and theory also enjoys parity sym-  metry if κ = y = 0 and Nf is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In the limit κ → ∞  we recover the O(2Nf) Gross-Neveu model restricted to  the U(1)-singlet sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  We would like to study the vacuum stability of Gross-  Neveu QED3, and to this end we shall compute the effec-  tive potential Veff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Similarly to what we did for bosonic  QEDs, we will perform the computation at the leading  order in the 1/Nf expansion and to all orders in κ and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Therefore, we write σ =  �  NfΣ+δσ, being Σ a v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' that  scales as O(N 0  f ), and we path-integrate over ψm and δσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Importantly, in contrast to the bosonic case, for fermionic  vector models the Hubbard-Stratonovich field has to be  path-integrated along the real axis, in order to get a con-  vergent path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' All in all we get  Veff(Σ) = Nf  �  yΣ3 + 1  6π |Σ|3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (21)  In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (21) the first term is the classical contribution,  whereas the second term comes from quantum fluctua-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Due to the real path-integration contour of σ, the  stability of the vacuum at Σ = 0 now requires that  det H = 6  �  y sign(Σ) + 1  6π  �  |Σ| > 0 ,  (22)  and therefore we get the stability bound5  |y| < 1  6π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (23)  Again, this condition implies that the local minimum  is also a global one, since the second derivative never  5 This result is consistent with ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [45], where the κ → ∞ limit of  the stability bound can be recovered as a particular case of their  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='10), namely in the limit |λF | = 1 − |λB| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We find  perfect agreement using the dictionary x6/λF = −16πy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  5  changes its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' As is turns out, a cubic scalar potential  for Σ – which would be classically unbounded from below  since Σ can have both signs, corresponding to the sign  of the effective fermion mass – is allowed by quantum  corrections as long as y lies within the region of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  This is due to the fact that fermionic self-interactions are  attractive and tend to stabilize the vacuum, as opposite  to what happens in the bosonic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In the appendix, we  discuss the effective potential and the phases of the the-  ory in presence of a massive and a quartic deformation,  showing how the same bound on y holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Next, we shall check whether the condition shown in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (23) is satisfied at the RG fixed points of Gross-Neveu  QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' To this end we shall compute the β function for  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' At the leading order in the 1/Nf expansion and to all  orders in κ and y we find [43]  βy(y, κ) =  32  3π2Nf  �  −864y3 + 3(4096κ4 + 640π2κ2 − 3π4)  (π2 + 64κ2)2  y  + 4π3(320κ2 − 3π2)κ  (π2 + 64κ2)3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (24)  This β function is a cubic polynomial in y with no  quadratic term, and gauge interactions only affect the  linear and constant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Since the zeros of βy are odd  functions of κ, we can assume κ > 0 without loss of gen-  erality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Before discussing the zeros of βy for generic κ,  let us first study the special cases for κ = 0 and κ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  When κ = 0 from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (24) we get  βy(y, 0) =  96  π2Nf  �  −96 y3 − 3y  �  ,  (25)  and so we find only one fixed point corresponding to the  real zero at y∗ = 0, which lies in the middle of the sta-  bility region of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Since ∂yβy(y∗ = 0, 0) < 0, the  operator σ3 is relevant here, meaning that tuning of y is  needed to reach this fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  In the limit κ → ∞ the β function agrees with the  findings of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [45] for the ungauged Gross-Neveu model6  βy(y, ∞) =  32  π2Nf  �  −288 y3 + y  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (26)  This β function has a fixed point at y∗ = 0 with σ3 ir-  relevant, and two (parity-related) zeros at y∗ = ±  √  2/24  with σ3 relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Only the fixed point at y∗ = 0 lies  within the vacuum stability bound, and corresponds to  the usual Gross-Neveu CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  In general, we can solve numerically for the zeros of  βy as functions of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The results are presented in fig-  ure 2, which shows that there are three families of ze-  ros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' A first family of fixed points, depicted by the red  6 In particular, our result (26) matches eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='65) of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [45], using  the dictionary NF = Nf and λF  6 = −16πy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='10  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Families of fixed points of βy as a function of κ for  Gross-Neveu QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The fixed points corresponding to stable  vacua must lie inside the gray region given by |y| <  1  6π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  curve in the figure, has a relevant σ3 operator and exists  for any value of κ, interpolating from (κ, y) = (0, 0) to  (κ, y) = (∞, −  √  2/24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Two additional families of fixed  points are depicted by the blue and green curve in the fig-  ure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The operator σ3 is irrelevant/relevant there, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' They exist for every κ ≥ κ1 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='273, below which  they annihilate and become complex conjugate, and they  reach the values y = 0 and y = +  √  2/24, respectively, for  κ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The fixed points on the red curve instead are  stable only for κ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The fixed points on the blue  curve always lie within the region of vacuum stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Finally, the fixed points on the green curve have a sta-  ble vacuum only if κ1 ≲ κ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' For 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='162 ≲ κ ≲ κ1  there is a ‘blind spot’ where there is no CFT with a stable  vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  The anomalous dimension of the lowest-lying singlet σ  at the leading order in 1/Nf and to all orders in κ is [43]  γσ = −  16  3π2Nf  9π4 − 896π2κ2 + 4096κ4  (π2 + 64κ2)2  ,  (27)  which does not depend on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  As already observed in  ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [46] for the special case of κ = 0, we find that we find  that this result is identical to that for the anomalous  dimension of σ in critical bosonic QED3, see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (19), up  to O(1/N 2  f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' As a consistency check, in the limit κ →  ∞ we indeed recover the anomalous dimension for the  Hubbard-Stratonovich field of the O(2Nf) Gross-Neveu  model [50], while for κ = 0 we recover the results of  refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [21, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  6  Fermionic QED  The fourth theory we consider is fermionic QED3,  where the mass of the Nf Dirac fermions is still tuned  to zero, but there is no quartic fermionic self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  The matter Lagrangian is  Lmatter = ¯ψm /Dψm ,  (28)  where Dµ = ∂µ+iaµ is the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Similarly  to the Gross-Neveu case, the continuous global symme-  try is SU(Nf) × U(1)m, and theory also enjoys parity  symmetry if κ = 0 and Nf is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In the limit where  κ → ∞, it describes 2Nf free Majorana fermions re-  stricted to the U(1)-singlet sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Fermionic QED3 has  no marginal deformations and it leads to a stable CFT  for any κ, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Indeed, since the large-Nf effec-  tive potential does not depend on gauge interactions, the  stability analysis is the same as the one for a free fermion  theory, which is clearly always stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  The anomalous dimension of the lowest-lying singlet  ¯ψmψm, at the leading order in 1/Nf and to all orders in  κ reads [43]  γ ¯  ψψ = 128  3Nf  π2 − 128κ2  (π2 + 64κ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (29)  As a consistency check, in the limit κ → ∞ we get γ ¯  ψψ =  0, as it should be for a theory of free fermions, while for  κ = 0 we recover the result of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Again, we  have that at this order of the large-Nf expansion, the  fermionic result agrees with the bosonic one, γ ¯  ψψ = γφ†φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  DISCUSSION  In this work, we have studied the fixed points of either  bosonic or fermionic three-dimensional large-Nf QEDs  coupled to a Chern-Simons term at level k, with fixed  κ = k/Nf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' For each of these theories we computed the  effective potential, as well as the β functions of the clas-  sically marginal couplings, at the leading order in the  1/Nf expansion and to all orders in κ and in the cou-  plings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  For tricritical bosonic QED3 and Gross-Neveu  QED3 our analysis shows that a CFT with a stable vac-  uum can only exist when the parameter κ is restricted to  lie in certain intervals, while there is no such restriction  for critical bosonic QED3 and fermionic QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  In a forthcoming paper [43] we provide details about  the derivation of the results presented here, in particular  regarding the computation of the β functions, and we  couple the 3d large Nf models studied in this work to  a 4d real scalar field via a bulk/boundary interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  This allows us to give an explicit Lagrangian construction  of interacting conformal boundary conditions for the 4d  CFT of a free scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We are grateful to Ofer Aharony and Marco Serone for  discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' LD acknowledges support from the program  “Rita Levi Montalcini” for young researchers and from  the INFN “Iniziativa Specifica ST&FI”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' EL would like  to thank Nikolay Bobev and Balt van Rees for their sup-  port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' PN would like to thank Thomas Dumitrescu for  discussions, and Lorenzo Maffi for a useful conversation  on global stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' PN is grateful to ICTP and University  of Trieste for the kind hospitality during the preparation  of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The work of PN is supported by the Mani  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Bhaumik Institute for Theoretical Physics and by a  DOE Early Career Award under DE-SC0020421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Appendix: Deformations and phase diagrams  In this appendix, we discuss the phase diagrams of the  QED theories described above upon turning on relevant  deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Since the effective potential does not de-  pend on the gauge interactions at leading order in the  large Nf expansion, our discussion is the same as for the  case of ungauged vector models, which constitute a par-  ticular case of the analysis performed in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' [45, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Let us first consider the bosonic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Tricritical  QED3 admits two relevant deformations, the mass m2  and the quartic coupling λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The Lagrangian in the pres-  ence of such terms reads  Lmatter = (Dµφm)†(Dµφm) + σ  �  φ†mφm  �  Nf  − ρ  �  +  h  �  Nf  ρ3 + λ ρ2 +  �  Nfm2ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='1)  The effective potential is easily found to be  Veff = Nf  �  Σv2 − Ση − 1  6π Σ  3  2 + h η3 + λ η2 + m2 η  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='2)  The gap equations for v2 and Σ  2Σv = 0 ,  v2 − η −  √  Σ  4π = 0 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='3)  do not depend on the scalar potential and they can be  used to express the effective potential as a function of η  only:  Veff  Nf  =  �  h η3 + λ η2 + m2 η  if η > 0 ,  �  h − 16π2  3  �  η3 + λ η2 + m2 η  if η < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='4)  Note that the condition for this potential to be bounded  from below gives the same stability bound as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (8)  0 < h < 16π2  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='5)  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Phase diagram of tricritical QED3 as a function of  λ and m2, in the range 0 < h < 16π2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We have depicted  in blue the second-order line and the tricritical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Above  the dashed gray lines, there is a unique minimum (an unHig-  gsed vacuum in green, a Higgsed vacuum in yellow), whereas  below such lines the two distinct minima coexist and become  degenerate on the first-order line, depicted in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Its loca-  tion is specified by f(h), here we have chosen for definiteness  a value h < 8π2/3, so that the transition happens for m2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  As we have previously shown, tricritical QED3 admits  an IR fixed point h∗ satisfying the stability bound above  only if 0 ≤ κ < π/8  √  3 and if the flow starts from a UV  value of h in the basin of attraction of h∗, hgreen < h <  hred (see figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' From this new Veff, the vacuum is  determined by the gap equations for η, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  �  3h η2 + 2λ η + m2 = 0  if η > 0 ,  (3h − 16π2)η2 + 2λ η + m2 = 0  if η < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='6)  From these results we can determine the phase diagram  of the theory, which is depicted in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' First, let us  introduce two useful auxiliary functions of the couplings  η+ = −λ +  √  λ2 − 3hm2  3h  ,  η− = λ −  �  λ2 + (16π2 − 3h)m2  16π2 − 3h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='7)  If λ ≥ 0 and m2 > 0, there is a unique minimum  at η = η− < 0 (unHiggsed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  If λ ≥ 0 and m2 < 0 there is a unique minimum at  η = η+ > 0 (Higgsed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  If λ ≥ 0 and m2 = 0 there is a second-order phase  transition between the two vacua, which coalesce  to η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  If λ < 0 and m2 > λ2  3h, there is a unique minimum  at η = η− < 0 (unHiggsed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  If λ < 0 and m2 < −  λ2  16π2−3h, there is a unique  minimum at η = η+ > 0 (Higgsed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  If λ < 0 and −  λ2  16π2−3h ≤ m2 ≤ λ2  3h (dashed lines in  the figure), there are two competing local minima,  one at η = η+ > 0 and one at η = η− < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The  two distinct minima become exactly degenerate at  m2 = λ2f(h), where a first-order phase transition  happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  The function f(h) is found to be  f(h) =  8π2 − 3h + 8π2 sin  �  1  3 arctan  8π2−3h  √  3h(16π2−3h)  �  3h(16π2 − 3h)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='8)  It is straightforward to show that  f  �16π2  3  − h  �  = −f(h) ,  1  16π2 − 3h ≤ f(h) ≤ 1  3h ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='9)  so that f(h) is an odd function with respect to h = 8π2/3  (where it vanishes), and it is monotonically decreasing  from f(0) = +∞ to f(16π2/3) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Thus, the first-  order line (λ < 0, m2 = λ2f(h)) – depicted in red in  the figure – is at m2 > 0 (m2 < 0) if 0 < h < 8π2/3  (8π2/3 < h < 16π2/3), and at m2 = 0 if h = 8π2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' It  terminates at the tricritical point (λ, m2) = (0, 0), where  it merges with the second-order line (λ > 0, m2 = 0) –  depicted in blue in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Both transitions are between a phase (at η = η−, green  in the figure) described by a U(1)k Chern-Simons the-  ory with an unbroken SU(Nf) global symmetry, and a  phase (at η = η+, yellow in the figure) where U(1)k is  Higgsed and the SU(Nf) part of the global symmetry is  broken down to SU(Nf−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The tricritical point is where  the tricritical QED3 CFT sits, whereas any point on the  second-order line is described by the critical QED3 CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Indeed, at low energies, any λ > 0 flows to the critical  value λ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Note that the location of the phase tran-  sitions depends on the regularization scheme (we have  used dimensional regularization in this work), but the  existence of the various phases does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Let us now discuss the phase diagram for critical  QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The only relevant deformation is the mass term  Mσ (whereas the quartic term σ2/λ is obviously irrele-  vant), and so we consider  Lmatter = (Dµφm)†(Dµφm) +  σ  �  Nf  φ†mφm +  �  NfM σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='10)  The effective potential reads  Veff = Nf  �  Σv2 − 1  6π Σ  3  2 + M Σ  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='11)  8  which is always stable, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The gap equations  for v2 and Σ  2Σv = 0 ,  v2 −  √  Σ  4π + M = 0 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='12)  imply that there are two classes of vacua: if M > 0,  then v2 = 0 and Σ = 16π2M, so that the theory is in  the unHiggsed phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' if M < 0, then v2 = −M and  Σ = 0, so that the theory is in the Higgsed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' For  M = 0 there is a second-order phase transition where  the vacuum is (v2, Σ) = (0, 0), and this is described by  the critical QED3 CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We see that any horizontal line  at λ > 0 intersecting the phase diagram of figure 3 re-  produces this one-dimensional phase diagram of critical  QED3, as it should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' An alternative way to arrive to  the same conclusion is to allow ourselves to use the gap  equation for Σ to express Veff as a function of v2 only  Veff  Nf  = 16π2  3  (v2 + M)3 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='13)  with the condition v2 ≥ max(−M, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Note that this  potential is always bounded from below, confirming that  the vacuum is always stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Interestingly, the upper sta-  bility bound that we found for tricritical QED3 amounts  to require that the sextic coupling h does not exceed the  value it has, effectively, in critical QED3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Finally, let us discuss the phase diagram when we are  outside the stability bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  This scenario is depicted  in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  By investigating the gap equations, one  can show that if h is outside the stability interval, then  there is a region of the phase diagram where the gap  equations (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='6) do not have any solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' This region  stays in the half-plane of m2 < 0 (m2 > 0) for h < 0  (h > 16π2/3) and it is delimited by the curves m2 = λ2  3h  and m2 =  λ2  3h−16π2 (yellow and green lines in the figure,  respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Moreover, when the gap equations have a  solution, this always corresponds to a metastable mini-  mum, since the effective potential is unbounded from be-  low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Curiously, while the first-order transition for λ < 0  disappears, the second-order transition for λ > 0 is still  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' We interpret this as the fact that the sextic term  in critical QED3 is always (strongly) irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Hence,  starting from tricritical QED3 with any sextic coupling  h, as soon as we switch on the relevant quartic deforma-  tion with positive λ, at low energies the theory always  flows to the critical theory, irrespectively of the initial  value of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The metastable vacua of figure 4 become sta-  ble in the limit λ → +∞, when the runaway behavior is  suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Let us now consider the fermionic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Gross-Neveu  QED3 admits two relevant deformations, the mass ˆm2  and the quartic coupling ˆλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The Lagrangian reads  Lmatter = ¯ψm /Dψm +  σ  �  Nf  ¯ψmψm  +  y  �  Nf  σ3 + ˆλ σ2 +  �  Nf ˆm2 σ ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='14)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Phase diagram of tricritical QED3 as a function of λ  and m2, in the range h < 0 (top) and h > 16π2/3 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  We have depicted in blue the second-order line and the multi-  critical point, in green the unHiggsed vacuum, and in yellow  the Higgsed vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' In the dark gray regions, no solutions  to the gap equations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  and leads to the following effective potential  Veff  Nf  =  ��  y +  1  6π  �  Σ3 + ˆλ Σ2 + ˆm2 Σ  if Σ > 0 ,  �  y −  1  6π  �  Σ3 + ˆλ Σ2 + ˆm2 Σ  if Σ < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='15)  Note that the condition for this potential to be bounded  from below gives the same stability bound as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' (23)  |y| < 1  6π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='16)  As we have previously shown, Gross-Neveu QED3 admits  an IR fixed point y∗ satisfying the stability bound above  only if κ ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='273 and the flow starts from a UV value  9  of y in the basin of attraction of y∗, yred < y < ygreen  (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The discussion of the phase diagram of  Gross-Neveu QED3 is now exactly the same as the one  of tricritical QED3, using the dictionary  η ↔  Σ  24/3π ,  h ↔ 16π3  �  y + 1  6π  �  ,  λ ↔ 28/3π2 ˆλ ,  m2 ↔ 24/3π ˆm2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='17)  Obviously, now the phases have a different low-energy  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Since the sign of Σ gives the sign of the  effective fermion mass, phases with positive (negative)  Σ are characterized by a low-energy U(1) Chern-Simons  theory at level k ± Nf/2, respectively, both with an un-  broken SU(Nf) global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' The multicritical point  at the origin is described by the Gross-Neveu QED3 CFT,  whereas each point on the second-order line (ˆλ > 0,  ˆm2 = 0) is described at low energies by the fermionic  QED3 CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Indeed, any positive ˆλ flows to ˆλ = ∞ or,  equivalently, the quartic fermion coupling g ∼ ˆλ−1 flows  to g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content='  Finally, fermionic QED3 has only one relevant defor-  mation, given by the mass term m ¯ψmψm to be added  to the Lagrangian (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
+page_content=' Clearly, positive and negative  m give rise exactly to the same phases we got in Gross-  Neveu QED3 for positive and negative Σ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNE3T4oBgHgl3EQflwrt/content/2301.04611v1.pdf'}
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